LSE Gabriel M Ahlfeldt
email: g.ahlfeldt@lse.ac.uk@www.ahlfeldt.com

Wolfgang Maennig
email: w.maennig@arcor.de@www.uni-hamburg.de

External productivity and utility effects of city airports

Keywords: aircraft noise, air traffic, commercial properties, residential properties, Accessibility, aircraft noise, commercial properties, residential properties, real estate prices, Berlin Externe Produktivitäts-und Nutzeneffekte von City-Airports Dieser Beitrag verwendet Mikro-Daten aus der Kaufpreissammlung Berlins um am Beispiel der drei Berliner finden signifikant negative Effekte des Fluglärms auf die Immobilienpreise, welche bei unter ca. 55dB ihre Signifikanz verlieren. Die preismindernden Effekte des Fluglärms verringern sich signifikant, wenn andere Lärmquellen existieren. Eine Nichtberücksichtigung dieses Zusammenhanges kann zu verzerrten Schätzern führen. Vor dem Hintergrund nur begrenzter Evidenz von lokalen Erreichbarkeits-bzw. Anbindungseffekten durch Erreichbarkeit, Anbindung Fluglärm, Gewerbeimmobilien, Wohnimmobilien, Immobilienpreise

External productivity and utility effects of city airports

While large cities depend on major airports carrying out hub functions to provide various international non-stop connections, smaller downtown business airports are much appreciated by businessmen due to their accessibility. Neighbourhood activists usually oppose these airports, mainly because of extensive noise pollution and emissions. Opposition obviously becomes stronger the more central airports are located, since population density is typically found to be much higher in downtown areas. As a consequence, local authorities are confronted with two conflicting interests, emphasizing the role of downtown airports as a location factor to attract businesses on the one hand and the necessity of protecting local residents' living quality on the other. The case of Berlin stands exemplarily for such conflicts. Political opposition to the scheduled closure of city-airport Tempelhof became large enough to enforce a public referendum, which finally failed. To make appropriate decisions, politicians have to rely on valid information about the extent to which residents and businesses are effectively exposed to the external effects mentioned.

Quantifying (city) airport externalities, however, is difficult in practice as residents and firms have no incentive to reveal the true (dis)amenties they incur from a nearby airport. Instead, they exaggerate (perceived) benefits and costs in a political bargaining process in order to seek rents.(Social) cost benefit analyses rely on large sets of assumptions regarding the cost of travel time and noise, which are difficult to ascertain. Spatial property market analyses offer a way to circumvent these problems. Following bid-rent theory, all kind of positive and negative accessibility effects and environmental externalities can be assumed to capitalize into property prices as they affect the willingness to pay of the marginal buyer. An evaluation of urban property markets Property price effects of airports have, therefore, attracted scholars' attention. [START_REF] Bell | The Impact of Airport Noise on Residential Real Estate[END_REF] provides a survey on the impact on residents' physical condition and introduces effects on property prices. Most empirical studies available so far focus on North America [START_REF] Mieszkowski | An Estimate of the Effects of Airport Noise on Property Values[END_REF][START_REF] Nelson | Airport Noise, Location Rent, and the Market for Residential Amenities[END_REF][START_REF] Uyeno | Density of Residential Land Use and the Impact of Airport Noise[END_REF] or United Kingdom, where Manchester Airport has attracted much attention [START_REF] Collins | Aircraft Noise and Residential Property Values: An Artificial Neural Network Approach[END_REF][START_REF] Pennington | Aircraft Noise and Residential Property Values Adjacent to Manchester International Airport[END_REF][START_REF] Tomkins | Noise versus access: The impact of an airport in an urban property market[END_REF]. For Continental Europe evidence is available for airports located in Oslo, Basel, Paris and Amsterdam [START_REF] Navrud | The State-Of-The-Art on Economic Valuation of Noise[END_REF]. However, with the exception of [START_REF] Weigt | Auswirkungen von Flughäfen auf den Immobilienmarkt am Beispiel des Marktsegmentes "individuelles Wohnen[END_REF], there is still little evidence available for German Airports. Surveys on the empirical literature reveal that airports are clearly found to adversely affect property values [START_REF] Nelson | Airports and Property Values: A Survey of Recent Evidence[END_REF][START_REF] Van Praag | Using Happiness Surveys to Value Intangibles: The Case of Airport Noise[END_REF]. Meta-analyses on hedonic price aircraft noise studies are provided by [START_REF] Schipper | Thirty years of aircraft noise value studies: a metaanalysis[END_REF] and [START_REF] Nelson | Meta-Analysis of Airport Noise and Hedonic Property Values: Problems and Prospects[END_REF]. An alternative approach to the evaluation of airport externalities was used by [START_REF] Nitsch | Fly or Cry: Is Airport Noise Costly[END_REF], who analysed residents' preferences as stated in a public poll conducted by opponents to the closure of Tempelhof Airport in Berlin, one of the airports analysed in this study. This article adds to the existing literature in numerous important ways: First, it analyses the effects of three airports, Tegel (IATA Code: TXL), Tempelhof (IATA Code: THF) and Schönefeld (IATA Code: SXF) in one city. Two of the three airports (TLX and THF) are clearly city airports, whereas most airports in the studies mentioned above are outside cities. Second, it investigates the impact of airport externalities on three different submarkets. Two residential submarkets are considered to assess airport effects on household utility: typically owner-occupied detached and semidetached houses and typically renter-occupied multi-family houses. The third submarket comprises commercial properties. This facilitates the evaluation of external productivity effects, which have been the focus of less attention in the literature. Third, we assess not only negative effects related to aircraft noise, but also positive externalities arising from access to flight connections, which we approximate using(weighted average) road network distances to airports. Although we find strong evidence of adverse productivity and utility effects arising from noise exposure, the evidence is less compelling for positive accessibility effects. Nevertheless, estimated noise effects can be biased if accessibility effects are not accounted for in empirical models.

Fourth, the impact of aircraft noise is estimated not only conditional on alternative noise sources, but also in interaction with other sources. We show that estimated noise effects may be severely biased if these interactions are not accounted for appropriately. Finally, we show how a simple regression-based interpolation approach can help to overcome data limitations by generating a continuous noise surface where only limited (discrete) data are available.

We use GIS tools and projected GIS maps to analyse a highly disaggregated dataset covering a broad range of structural, location and neighbourhood characteristics. Our models use potentiality variables and include attributes such as neighbourhood historic quality, which has received little attention in the literature until recently. Where appropriate, we correct for a spatial autoregressive structure in the price-generating process or a spatial structure in the error term by applying SAR models. The remainder of this article is organized as follows. In the next section we discuss the airports within the study area in more detail. Section 3 discusses the empirical strategy and data issues. Results are presented in Section 4 and conclusions in the final section.

BERLIN AIRPORTS

The official inauguration of Tempelhof airport was in 1923. After complete redevelopment during the national socialist regime, Tempelhof was clearly Germany's most important air hub with a maximum capacity of 6 million passengers a year, exceeding the effective 1934 numbers by a factor of thirty. These dimensions, the facility design and architectural and historical particularities have frequently been discussed [START_REF] Carré | Berlin-Tempelhof[END_REF][START_REF] Demps | Flughafen Tempelhof[END_REF][START_REF] Meuser | Vom Fliegerfeld zum Wiesenmeer. Geschichte und Zukunft des Flughafens Tempelhof[END_REF][START_REF] Schmitz | Flughafen Tempelhof. Berlins Tor zur Welt[END_REF]. Tempelhof later became internationally prominent as Berlin's most important access point for the 1948-49 airlift established to supply West Berlin residents during the Berlin Blockade. To provide the necessary capacity, two more airports were conceptualised, one of which was Tegel Airport, jointly operated by the French since 1948.

By the mid 20 th century, Berlin possessed a decent infrastructure for air traffic and was preparing itself to benefit from the rapidly growing market. However, Berlin soon lost its status as Germany's pre-eminent hub, mainly due to the loss of market access following Germany's division (Redding, Sturm, & Wolf, in press). West Berlin became completely surrounded by the Soviet zone of occupation. While the most important West Germany counterpart of the airlift -Frankfurt -emerged as Germany's new pre-eminent hub, generating more and more traffic and continuously expanding facilities, improvements in air traffic infrastructure in West Berlin remained relatively modest.

As no reserve space for extension of facilities was available in Tempelhof due to its downtown location, Tegel Airport was opened for civilian air traffic in 1960 to meet the demands generated by increasing national and international air traffic, and the fact that a flight connection was the only way of travelling between West Berlin and West Germany avoiding border controls. In 1974,a new civilian terminal in the south of Tegel airfield replaced the existing facilities which subsequently have been used for military and governmental purposes only.

Following the inauguration of the new Tegel Airport, Tempelhof Airport was closed until 1984 when it was reopened mainly for smaller airplanes utilized by business travellers. Despite minor extensions during the following decades, Tegel Airport kept moderate size. Even experiencing a considerable capacity overload [START_REF] Steinke | Haupstadtairport Berlin-Tegel. Sechseck am Limit[END_REF], the number of served passengers at Berlin's Tegel Airport has hardly exceeded 13 million per year, a relatively small number compared to 52 million at Fankfurt or even over 67 million at London Heathrow in 2005. Figure 1 shows passenger traffic at Berlin airports since reunification in 1990.

[Figure 1 about here]

As noted above, the capacity of both airports is restricted by their central location and good accessibility. Compared to the city airports at Tegel and Tempelhof, the location of Schönefeld, which served East Berlin and its surroundings during the division period, is remote. In recent years, Schönefeld, which will be redeveloped as the new Berlin Brandenburg International (BBI) Airport, has become much appreciated by low-cost carriers due to low operating costs. However, Tegel continues being the most important airport for business flights and the only airport in Berlin to offer intercontinental connections.

While

EMPIRICAL STRATEGY

Following standard rent theory, spatial variation in the value of urban land, net of commuting cost, reflects productivity or utility differentials that drive bid rent functions of residents and firms. In principle, access to airports may be expected to increase the productivity of firms by providing fast access to other regions and national markets and reducing transaction, communication and information costs. If accessibility to other regions' markets significantly impacts on the economic performance of regions and cities, then city areas close to transportation nodes such as airports should particularly benefit from regional integration given that journeys to and from airports are time-consuming. If productivity effects are significant we would, all other factors being equal, expect increasing bids and equilibrium land prices in closer proximity to airports. Within air corridors, however, the exposure to aircraft noise should exhibit an adverse effect on worker productivity and therefore decrease bids and equilibrium prices, conditional on access to fight connections and other factors. Similarly, households living closer to airports may experience an increase in utility due to fast access to flight connections and travel opportunities and a decline in utility from the exposure to aircraft noise, which is a clear disamenity. The proximity effect of an airport is a net-effect of both, which following conventional praxis in the real estate economics literature can be identified in a hedonic regression analysis [START_REF] Rosen | Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition[END_REF]. Although we cannot observe the rate of owner occupancy directly, comparison of submarkets a) and b) still allow for an evaluation of the impact of aircraft noise on a) an owner-occupied and b) a rental market. The price of multi-family properties with rented units reflects the expected cash flow of discounted rent streams. Given the lower mobility cost for renters, we expect renters to be less risk averse with regards to the potential (dis)utility effect of aircraft noise, which they will usually be uncertain about before moving in. This could result in a lower marginal price effect of aircraft noise within the residential market (b) compared to the owner-occupied market a).

Besides distinguishing into three separate submarkets, we first separately focus on the impact areas of the three airports in order to increase homogeneity within the reduced study areas, which encompass very different submarkets and potentially exhibit heterogeneity in implicit prices of various housing and location characteristics. In the second step we pool the data across the entire city area to estimate an average treatment effect for the noise exposure as well as airport accessibility effects. Last, we investigate how individual sources of noise are perceived in interaction, i.e.

if the disamenity effect of aircraft noise increases or decreases in the presence of street noise.

Empirical specification

We employ a standard hedonic specification controlling for structural (S) as well as locational (L) and neighborhood (N) characteristics in order to estimate the conditional impact of access to airports and aircraft noise on the log of price per square meter of land (P). Given that the airport proximity effect potentially is a net-effect of accessibility and noise, failure of controlling for either of the two effects may lead to biased coefficients.

(1) , where is a full set of year fixed effects and is an error term. All other Greek letters are parameters. EAST is a dummy denoting all transactions within the eastern part of the formerly separated city. This specification is set up to detect localized externalities at an intra-city scale as this article contributes to the discussion of the optimal location of airports within a city or region.

All factors that impact on the study area as whole are captured by time effects .In order to capture the gradual reduction in the price differential between both city parts [START_REF] Ahlfeldt | The Train has Left the Station: Do Markets Value Inter-City Access to Intra-City Rail Lines? German Economic Review[END_REF] the EAST dummy is interacted with a set of year dummies. Hedonic controls include a conventional set of structural characteristics as well as broad and established sets of neighborhood attributes and environmental (dis)amenities (Ahlfeldt, 2010a) . Table A1 in the appendix provides a description of variables used in the hedonic models. AA is an indicator of airport accessibility to airport j.

As a default we use road distance measures to terminals (DA) of all airports.

(2)

At city-wide scale we also use an alternative airport accessibility indicator. Instead of estimating the marginal price effects for all three airports individually, we estimate the marginal price effect for the average distance to airport terminals (ADA), weighted by airports' proportions at overall air traffic (n j /N). While being slightly more restrictive, this functional specification still accounts for the size of the airports in the study areas and correspondingly heterogeneous effects while at the same time reducingthe sensitivity to correlations of airport distances with unobserved neighborhood or location characteristics.

(3)

Given the non-linear log scale of the decibel scale and the unknown subjective perception of noise, we estimate the unknown non-linear relation between aircraft noise level and the (log) of land price per square meter g(AN) rather than assuming a log-linear functional form ex-ante. We [START_REF] Lokshin | Difference-based semiparametric estimation of partial linear regression models[END_REF]. Informed by non-parametric estimates, the true functional relationship can then be approximated by an appropriate parametric specification.

Note that the vector ΣL l in specification (1) includes street noise as the most important alternative source of noise, so that specification (1) yields and estimate of aircraft noise effects, conditional on the effect of street noise. Specification (1), however, treats both sources of noise as independent, whichnot necessarily has to be true. Let´s assume that household utility (U) is a function of both air noise (A) and street noise (R) and other factors (Z) that are independent from both:

. Clearly, we expect that and . If A and R are not independent, it follows that and . In principle, second-order derivatives may be negative, if both sources of noise are perceived to amplify each other, or positive, if the marginal utility effect of one source of noise becomes smaller in the presence of another. This rationale applies analogically to marginal productivity effect. As a result, the marginal effect on property prices for each of the noise sources may depend on the presence of the alternative noise source.

The hypothesis of a dependency among the two sources of noise can be tested empirically by introducing an interactive term of air noise (AN) and street noise (SN) into specification (1).

(4)

Parameter then gives the change in marginal utility (productivity) effect as the level of alternative noise increases by 1 db. If one of the two noise sources is perceived as dominating, so that the presence of an alternative source of noise adds less to the perceived disutility (disproductivity) of residents (workers), will be positive. In line with the common strategy in applied urban economics research we control for various location attributes by a distance to the nearest feature measures (e.g. distance to the nearest green space, rail station, etc.). The implicit assumption underlying the inclusion of these variables is that the value of these features, discounted by distance, is traded against the land price. Similarly, building on the traditional framework of rent theory, hedonic studies typically control for the distance to the central business district (CBD). As Berlin exhibits a highly polycentric structure, with two dominating business areas, we include the minimum distance to the western (Breitscheidplatz) or the eastern CBD (metro station "Stadtmitte") in our specifications.1 Moreover, we calculate an employment potentiality as a detailed labour market accessibility indicator following Ahlfeldt (2010a). Transaction i receives the employment potentiality of the precincts v it falls within (EP iv ) , which is the aggregate of employment within all 1201 precincts in Berlin and 206

surrounding municipalities within a 50 km buffer zone from the city´s outboundaries, weighted by (car) travel time (tt vw ). 2

(5)

Analogically we calculate a green and water potentiality as the distance weighted sum of surrounding water or green areas to better capture the endowment with natural amenities. This is particularly important in this analysis since Tegel Airport lies within a major recreational area and our objective is to estimate the impact of the adverse environmental quality due to noise emissions net of the utility derived from these amenities. Green (GP) and water (WP) potentialities are calculated as the distance-weighted sum of the surface area for green and water spaces, respectively, at the level of 15,937 statistical blocks, which are connected by a straight-line distance matrix (in km). To reflect car (employment) and walking (green and water) speed we employ spatial discount parameters of 0.1 (car) and 2 (green and water) following Ahlfeldt (2010a) and [START_REF] Ahlfeldt | Built Environment as Determinant for Attractiveness of Location[END_REF] and [START_REF] Ahlfeldt | Substitutability and Complementarity of Urban Amenities: External Effects of Built Heritage in Berlin[END_REF] respectively. 3 All potentialities enter the empirical specifications in logarithms so that coefficients can be interpreted as elasticities. Besides the potentiality variables another variable is worth mentioning, which is less common in the applied urban and real estate economics literature: the number of designated landmarks within 600 m, a threshold based on [START_REF] Ahlfeldt | Substitutability and Complementarity of Urban Amenities: External Effects of Built Heritage in Berlin[END_REF]. This variable accounts for the historic quality of the neighbourhood, which is receiving increasing attention in the literature. 4Note that with relatively few exceptions, we find a systematic spatial structure in the error term, which is typical for micro level spatial analyses. We use spatial autoregressive (SAR) models to obtain unbiased and efficient estimates in the presence of spatial dependency. LM tests in most of these cases reject a spatial-lag model in favor of an error-corrections model.

(

) 6
where W is a binary row standardized weights matrix indicating transactions that are neighbours, is a parameter and is a random error term. 5 In few cases, however, a weak autoregressive structure seems to be resent in the price generating process so that a spatial-lag model is employed as a robustness check.

(

) 7
where y is our endogenous variable, Z is a vector of variables included in specification (1), a is the respective coefficient vector and the lag parameter. Note that in spatial lag-models, coefficients need to be adjusted before the usual interpretation applies. It can be shown that using a row standardized spatial weights matrix, the appropriate "spatial" multiplier for the estimated coefficients is 1/(1-) (see e.g. Won [START_REF] Kim | Measuring the benefits of air quality improvement: a spatial hedonic approach[END_REF]. Spatial error-correction models as well as spatial lag-models are estimated using maximum likelihood techniques.

Data

In the present analysis we make use of an exhaustive record of 32,763 transactions of developed Officially, local authorities are required to determine noise protection zones where land use activities are restricted for all airports. For Tempelhof, however, the noise protection zone defined on the basis of an equivalent long-term sound pressure level of more than 67 dB(A) zone hardly exceeds the territory of the airport and is therefore of little use in the present analysis. The best available data that could be obtained were from the Berlin airports operating company (Flughafengesellschaft) in form of an electronic map for which sound pressure levels ranging from 50-55, 55-60, 60-65, 65-67 and more than 67 dB(A) are defined. Based on these discrete information, we employ a simple regression based interpolation approach in order to generated a detailed continuous noise surface that is compatible with the official information for Tegel airport.

Therefore, in the first step we define an auxiliary 100 m×100m grid and a new coordinate system with an origin in the airfield centroid and the x-axis running parallel to the air corridor. Moreover, we define a 350m buffer around the outmost zone where we assume a noise pressure of 45-50dB.

In the next step, a naive average of noise pressure (e.g. 52.5 for the 50-55 zone) as well x-and ycoordinates within the auxiliary coordinate system are assigned to the newly generated grid and the western (W) quadrants of the coordinate system yields predicted values of noise exposure for about 10,000 gird points (g) in the area.

(8)

, where lower case letters form the set of parameters and is an error term. Based on the estimated parameter vectors the level of noise exposure can be predicted for the about 10,000 grid points (g) in the area.

Naturally, this approach is better suited for producing reliable interpolations rather than extrapolations, so that we only keep grid points with a predicted value larger than 45dB. Overall, the procedure yields a reasonable fit as suggested by a R 2 of 81.5 and a close fit of obtained and imputed noise level along the zone boundaries (see Figure2). One limitation of this approach is that physical obstacles within a pre-defined noise-zone are not taken into account by the interpolated values. We note, however, that there are no evident obstacles, e.g. high-rise buildings, elevated roads or railways, evident for the noise impact area.

[Figure 2 about here]

For Schoenefeld airport, noise information is even more restricted. As the airport lies outside the boundaries of Berlin, with only a relatively small part of the air corridor crossing Berlin territory, no detailed noise maps were included into the noise report. The only available information therefore is a map of the area of restricted development, which, however, already takes into account that noise levels will increase considerably when the new international airport BBI will be inaugurated. Based on this zone of restricted development we further define a 3 km buffer zone. Fig- ure3shows the study area and the areas exposed to considerable aircraft noise. Evidently, the noise emissions follow the extensions of runways, which run parallel in east-west direction in each case. Note that for Tegel airport the available noise data covers a much larger area, but we restrict the visualization to the area where noise exposure exceeds a threshold of 45 dB so that the scale is compatible with Tempelhof. We make use of the GIS-environment to assign transactions to noise levels and zones displayed in Figure 3.

[Figure 3 about here]

EMPIRICAL RESULTS

Airport impact areas: Residential

As discussed, Tegel Airport during the study period was the most important airport within the region and by far the most important of the two city airports. We start our empirical analyses for the residential submarkets a) and b) within the TXL impact area where aircraft noise exceeds a 40 dB level. Below this threshold, aircraft noise should hardly play an important role within an urban environment where the usual alternative noise sources are present. Table 1 presents a series of estimations following equation (1) that permit inference on the disutility effects of aircraft noise.

Column (1) shows results for a set of mutually exclusive 5-dB grid cell dummies for submarket a), starting at a 45-50 dB noise level. Coefficients on the grid cell dummies give the average price differential within the respective zones relative to the base zone with a noise level of 40-45 dB.

Results indicate non-significant price effects up to a level of 50-55 dB and negative and significant price discounts at higher noise levels. While properties within the 55-60 dB zone sell at moderate discounts of about 7% compared to otherwise comparable properties, properties that are ex- posed to an equivalent sound pressure of more than 70 dB sell at discounts of more than 40%.8

For an average property in our sample this implies an absolute reduction in sales price of close to €88,000.9 These results are in line with a large negative impact on household utility and indicate a non-linear impact with a discontinuity around 55 dB. Price differentials remain virtually unchanged if estimated conditional on airport accessibility, measured as the road distance to the TXL terminal (column 2).

These results stand in sharp contrast to the corresponding findings for submarket b shown in column(3). While, as discussed, a smaller discount might be expected for submarket b) and renter occupied multi-family houses, the entire absence of significant effects is certainly surprising. For none of the noise zones, however, are there significant price differentials observable. Not even are there negative coefficients that systematically increase in magnitude at higher noise levels. Again, results hardly change if noise effects are estimated conditional on access to the airport (column 4).

Note that correcting for a spatial structure in the error terms detected in models (3) and (4) hardly affects the results for either market (columns 5 and 6). Figure 4 shows the results of a semiparametric regression of the noise treatment (conditional on structure, location, neighbourhood and airport access). We plot the conditional mean in transaction prices at different noise level relative to the area average of the 40-45-dB base zone (in log differences). The results pretty much confirms Table2 presents the results for a similar set of estimates for the impact area of Tempelhof airport.

Since our generated noise data does not cover noise levels below 45 dB, we define a 500 m buffer distance to the 45 dB area as base zone. Due to the much smaller size of the airport, the noise level, even for properties within the air corridor and very close to the airport, hardly exceeds a level of 60 dB. Overall, the pattern of results resembles the findings for the Tegel Airport impact area, although even for the dummy variables denoting the areas with the highest noise level, there are no coefficients that are negative and significant. The large and negative, albeit not significant, price differential for the 60+ dB zone, however, is nonetheless remarkable. It implies a negative price differential of about 27% compared to the control zone and even more compared to areas with lower noise levels. In terms of magnitude this price differential even exceeds the one for similar noise levels within the Tegel Airport neighbourhood. For an average property within the 60+ zone the relative discount implies an absolute discount of about €84000, which, despite the lower noise level, is close to the maximum noise effect in proximity to Tegel Airport. In contrast, similar to the case of TXL, there is hardly evidence for a negative noise effect on submarket b). Again, conditioning on airport access (columns 2 and 4) as well as accounting for spatial dependency (columns 5 and 6) hardy affects the pattern of results, which also becomes evident in the semi-parametric estimates in Figure 5. Prices for multi-family houses (b) even tend to increase when moving into areas with higher noise level. While properties in submarket a) similarly exhibit conditional mean prices that increase in noise at lower levels, there is a sharp discontinuity at approximately55dB, after which the relationship is reversed.

As discussed, for Schönefeld Airport, detailed noise records are not available. The best information we have is the zone of restricted development. In order to assess whether properties within this zone sell at a significant discount due to noise emissions, we run a set of There is a significant price discount for submarket a) properties of about 27%, pointing to considerable disutility effect. For an average property within the zone this implies a considerable discount of about €70,000. Although generally within the same range, this is a slightly lower magnitude compared to the maximum noise effects found for Tegel and Tempelhof airport. At the same time no significant discounts are found for submarket b). These findings are robust to controlling for airport accessibility and spatial dependency, which following the LM-test scores is addressed by spatial-lag models (5). Note that no spatial dependency is evident in the column (4) estimates and that the column (6) results are provided as a robustness check only.

Airport impact areas: Commercial

As discussed in Section 2, we expect aircraft noise not only to have an adverse effect on household utility, but also on productivity of workers and employees and, consequently, the value of commercial land. Our record of property transactions covers commercial properties within the noise impact area of Tegel and Tempelhof airport. Analogically to Tables 1-3 for the residential submarkets, Table 4 presents estimated average price differentials within different zones of noise exposure for the commercial property market for the Tegel (columns 1-3) and Tempelhof (columns 4-6) impact areas. Again, taking the 40-45 dB noise zone as a basis, coefficients in column

(1), similarly to the results for the owner-occupied residential submarket a), indicate negative price differentials for the 55-60dB zone. At higher noise levels, however, no significant effects are found. Once airport accessibility is accounted for (column 2), however, all coefficients become negative and considerably increase in magnitude. At the same time, the negative and significant coefficient on road distance to the airport in column (2) points to significant proximity benefits. Moreover, noise effects for the65-70 dB zone is large and negative (price discounts up to 85%), conditional on airport accessibility. Apparently, the negative productivity effects related to Ahlfeldt (2010a) for main roads and urban rail stations. Although negative and large, the coefficient for the 60-65-dB is not significant and of smaller magnitude than for the 55-60-dB zone, which seems somewhat anomalous. Closer inspection of the data reveals that this effect is most likely attributable to relatively high prices for commercial properties within a medium-size retail center at Residenzstrasse. Correction for spatial dependency (3) leaves the results largely unchanged. At first glance, the coefficient seems to be much lower for the 65-70dB zone. The decrease, however, is partially attributable to application of a lag-model, for which coefficients need to be corrected, as described in Section 2 (coefficient Rho takes a value of 0.36). In any case, the coefficient for the 60-65-dB noise zone is not significant in the lag-model.

Estimated noise effects on commercial properties are similar within the Tempelhof Airport impact area. There is a large, negative and highly statistically significant discount of about 75% within the 55-60-dB zone in addition to a smaller effect for the 50-55-dB zone (columns 1-3). Estimated noise effects are hardly affected by controlling for airport accessibility (2) and spatial dependency (3). The notable difference is that negative noise effects seem to be compensated by positive accessibility effects in the case of Tegel, but not Tempelhof, which is plausible in light of the relatively small number of flight connections offered by Tempelhof airport.

Overall, these results strongly indicate the presence of localized positive and, in particular, negative productivity effects of (city) airports. Although positive effects seem to be limited to airports offering a large array of flight connections), these effects can be large enough to partially com- pensate for the negative effect of noise. Our results further suggest a discontinuity in the productivity effect of aircraft noise around 50-55dB, which is even more apparent than in the disutility effects for households. A (conditional) discount of about 75% within the area exposed to heavy aircraft noise indicates a considerable reduction in worker efficiency, making respective properties much less desirable for commercial purposes.

City-wide effects

The results presented so far consistently point to adverse productivity and utility effects related to exposure to aircraft noise within all airport impact areas, as well as a potential discontinuity in the noise perception at a threshold of about 55 dB. Only for submarket b, comprising renter-occupied multi-family houses, could a negative effect not been found. In the remainder of the article, we pool our data separately for each submarket across the whole city area in order to estimate the average treatment effects for aircraft noise and airport accessibility. While calibrating the hedonic models based to the full data-base allows us to exploit all available price variation and to achieve potentially higher parameter stability, the pooled models may be slightly less efficient in predicting hedonic prices within the airport impact areas as marginal prices for selected attributes may slightly vary across space.

We start with submarket a), the 1-and two-family houses, and repeat column (1) estimates form Tables 1 and2 for the whole city area. Results presented in Table 5, column (1) are in line with the finding for the TXL and THF impact areas. There is a negative and statistically significant discount beyond 55 dB. The maximum percentage discount of about 47% is within the same range as in Table 1, even slightly larger. In column (2), we extend the specification by individual road distance to airport measures for all airports (see Equation 2) to account for airport accessibility. While there is a negative and significant impact for distance to TXL and THL, which is in line with a positive utility effect from access to flight connections, the opposite is true for SXF. Esti- 3) where neighbourhood effects corresponding to the airport impact areas used in the previous section are included to control for unobserved neighbourhood particularities. In column (4), finally, we replace individual accessibility variables by the (weighted) average distance to airport (AVA) measure defined in equation ( 3). This is our preferred accessibility treatment due to presumably lower correlations with unobserved characteristics of the airport neighbourhoods. We find a positive effect for proximity to flight connections, with property prices decreasing by about 2.2% per 1 km increase in average distance to airports.

Table 6 repeats column 5 estimates for submarket b), the renter occupied multi-family houses. In line with the previous findings for the individual airport impact areas, there are no significant noise discounts (columns 1-4). Individual airport accessibility effects are inconsistent (2-3) and the average distance to airport treatment effect insignificant (4). Similarly, no compelling accessibility effects are revealed for commercial properties in Table 7. Results for the productivity effects of aircraft noise are more ambiguous. For the zone of highest noise exposure (65-70 dB)

there is a large and significant discount of about 50% in our preferred column (4) model, which is in line with previous findings. Contrary to Table 4 results for the TXL and THF impact areas, there is no adverse effect for the 55-60 dB zone in all models. Moreover, the problem with the medium size retail center at "Residenzstrasse" within the 60-65 dB noise zone of TXL airport, is considerably aggravated. The large and positive coefficient indicates that the pooled model is less capable to explain the relatively high prices for commercial properties in the center. The SAR model, for which results are presented in column (5), to some degree "cures" these inconsistencies. After correcting for the spatial structure in the error term, we find a large and significant discount of about 47% within the 55-60 dB zone, which, however, is still considerably less than suggested by and relatively large, but not significant coefficients for the higher noise zones (60-65 and 65-70 dB).

Note that we don´t estimate SAR models for submarkets a) and b) at city-wide scale due to the large sample sizes. Spatial LM test scores presented in Table 5 and 6 notes strongly indicate the appropriateness of spatial error correction models. In contrast to lag-models, error-correction models leave OLS coefficients unbiased if the underlying models are appropriately specified.

Given the consistency of OLS and SAR coefficient estimates for both submarkets in Tables 1 and2, there is reason to believe that potential problems of spatial dependency are limited to inefficient standard errors at the city-wide level, too. Since for both submarkets noise effects are generally estimated at very high levels of statistical significance, we believe that qualitative and quantitative interpretations of OLS coefficients are justified.

Marginal price effects and treatment heterogeneity

In the last step of our empirical analyses we turn our attention to the marginal price effect of aircraft noise. Average treatment estimates at the city-level basically confirm previous findings from the narrower samples indicating negative and significant effects for submarkets a) and c). Noise effects become crucial beyond a threshold level of about 55 dB. This non-linearity needs to be taken into account when defining a parametric specification with the objective of revealing marginal noise effects. As a somewhat pathological result, we consistently find no effects for submarket b). Although at a city wide level airport accessibility does not seem to be a very critical determinant for household utility or firm productivity, the neighbourhood analysis of Tegel Airport shows that estimated aircraft noise effects may be considerably biased if airport accessibility is not accounted for. In addition, there is another important source of bias that has not been addressed in the previous steps and has often been overlooked in the literature: The interaction with alternative noise sources, in our case, street noise. As discussed in section 3, the marginal (dis)utility and (dis)productivity effects of aircraft noise may be expected to be larger if no alternative noise is present. Under this assumption, estimated aircraft noise effects will be biased if the spatial distribution of aircraft noise is correlated with street noise. In order to address this potential interaction effect we estimate specification (4), which includes an interactive term of aircraft noise and street noise. 10 Results are presented in Table 8, starting with submarket a) and omitting the interaction effect in column (1). Throughout Table 8 only the variables of interest are displayed to save space. Full estimation results including hedonic characteristics are presented in Table A2 in the appendix for selected models (3, 6, 9) that stand exemplarily for the three submarkets. If the interaction of street noise and air noise is not accounted for, we find a negative and significant (log-)linear impact of both noise sources where, notably, street noise seems to have much greater (dis)utility effects than aircraft noise. While for an average 10 dB increase in street noise there is a price discount of about 5%, a respective increase in air noise yields only a relatively moderate 1% effect (column2). This relationship changes considerably once the interaction between the two noise sources is accounted for (column 2). There is a positive and significant coefficient on the noise interactive term, revealing that the marginal price effect for one type of noise diminishes in the presence of another. Equivalently interesting, estimated individual noise effects increase considerably, in particular for air craft noise, whose impact is now within the same range as street noise.

An average 10 dB increase in air or street noise now yield price discounts of 5% and 7%. As discussed, our previous findings indicate a discontinuity in the utility and productivity effect around 55 dB. We therefore extend the model by a dummy variable indicating areas with 55 or more dB noise level in order to test for a significant level shift, conditional on the log-linear average effect 10 Our measure of street noise does not include sources of noise, especially not aircraft noise. in column (3). Indeed, a significant discount of approximately 10% is evident for properties within that zone, while the marginal price effect of an average 10 dB increase is considerably reduced to 2% and no longer statistically significant. The coefficient on the interactive term is slightly reduced and sharply fails the 10% significance criterion (p-value: 0.14).

In column (4-6) we apply the same models as in (1-3) to subsample b). If the interaction between noise sources is not accounted for we, similar to the previous results, find the "pathological" positive noise effects for air noise and also for street noise (column 4). Once the interaction is considered, however, these effects are reversed (5). Individual noise effects are now negative, significant and within a similar range to the other residential submarket a). An average 10 dB increase in noise level yields a 6% (3%) reduction in property prices in the case of air (street) noise. As in the case of submarket a) the coefficient on the interactive term is positive and statistically significant, again pointing to considerable treatment heterogeneity. Interestingly, there results even remain unchanged if a level-shift at the 55 dB level is allowed for, indicating that there is no discontinuity at this threshold for this submarket. These findings are most notable as they highlight the potential of severe bias in estimated noise effects if interaction effects are not accounted for appropriately.

The pattern of results for the commercial property market (submarket c) in columns (7-10) exhibits some similarities. Without interactive term, estimated noise effects are small and insignificant for street noise and positive and significant for air noise (column 7). With the interactive term (column 8), both coefficients on individual noise sources are negative and of roughly the same magnitude as for the other submarkets, although not statistically significant at conventional levels.

Similarly, the interactive term exhibits a positive, but not significant coefficient. Previous results had shown the strongest discontinuity for the commercial property prices, which is confirmed when we extend the present specification by the dummy for 55 or more dB (column 9). While the three coefficients of interest considerably increase in magnitude and the coefficients on street noise and the interactive term even become statistically significant, there is still a negative (conditional) price shift of about 40% once the 55 dB threshold is crossed. Submarket c) seems to be the only submarket where spatial misspecifications give some cause for concern. We therefore repeat column (9) estimates employing an SAR (error) model. Results do not change qualitatively. There is a negative and highly statistically significant discount for property exposed to 55 dB or more of now about 44%, while individual noise effects are not estimated precisely taking as a basis conventional criteria. These findings, nevertheless, confirm the presence of very strong adverse effects on the productivity of office workers. We note that the slight instability of noise estimates for the commercial property market might be partially caused by a relatively low number of observations of traded commercial properties within areas exposed to high air noise levels, which, however, is in line with firms' aversion to aircraft noise.

CONCLUSION

This paper contributes to the assessment of external effects of (city) airports by providing an indepth investigation of three airports in Berlin, Germany. While we find strong evidence of negative productivity and utility effects reflected in significant property price discounts within areas that are exposed to high levels of aircraft noise, evidence of positive accessibility effects is less compelling. For residential properties, an average treatment effect of approximately 5-6% is evident for every 10-dB increase in aircraft noise, which is within the range for results available in the literature [START_REF] Nelson | Meta-Analysis of Airport Noise and Hedonic Property Values: Problems and Prospects[END_REF]. Moreover, for the submarket of one-and two-family houses, there is evidence of a significant discontinuity in the noise perception when a threshold of 55dB is crossed. Within the zones of highest noise exposure, properties sell at discounts of more than 40%, corresponding to €85,000 for an average property. For commercial properties, the discontinuity is even more pronounced. Conditional mean property prices decrease relatively abruptly by approximately 40% once the threshold is crossed, indicating a strong adverse effect on office worker productivity. Positive accessibility effects could only be found at the city level for one-and two family houses, where a 1-km increase in the weighted average distance to flight connections reduces prices by 2.2%, and for commercial properties within the narrower impact area of Tegel airport.

Our results support the notion that airport externalities are composite effects of positive and negative effects, so that failure to control for either of the effects can result in biased coefficients for the other. Even more crucial, a significant interaction effect with alternative sources of noise is evident, which can lead to severe bias if not appropriately accounted for. Although there are nonsignificant or even positive noise effects for the submarket of multi-family houses, significant negative effects within the usual range are evident once the interaction with street noise is accounted for. Consistently for all submarkets, the positive interaction effect indicates that the marginal price effect of either street or aircraft noise decreases in the presence of an alternative noise source.

Based on our findings, it is possible to inform planners and authorities about productivity and utility effects of city airports, which are quite controversial in general and especially in the case of Berlin. Overall, our results provide little justification for location of airports within densely developed downtown areas. Although at the city level there is hardly any evidence of positive accessibility effects, such effects within the narrower impact area seem, if present at all, to be more than compensated by adverse noise effects. As a result, the net effect is clearly dominated by adverse productivity and utility effects, making a more remote airport location desirable from a welfare economics point of view. More generally, our results confirm recent findings on limited productivity effects of intra-city access to inter-city transport hubs [START_REF] Ahlfeldt | The Train has Left the Station: Do Markets Value Inter-City Access to Intra-City Rail Lines? German Economic Review[END_REF], which is some- what surprising in light of the strong emphasis in economic geography on the benefits arising from good access to regional and international markets. [START_REF] Lokshin | Difference-based semiparametric estimation of partial linear regression models[END_REF] are conditional on the control variables used in Table 1 and4.

  a natural starting point of a welfare analysis of airport externalities based on the revealed preferences of market participants.

  Tegel Airport will continue in operation until 2011, when the new BBI Airport is scheduled to be inaugurated, Tempelhof airport was closed on October 31, 2008. The external effects of Tempelhof Airport were judged differently by citizens: Some neighbourhood activist movements favoured the final shutdown, while the Interest Group of City Airport Tempelhof (ICAT) successfully held a referendum in favour of Tempelhof Airport remaining in operation. The poll was held on April 27, 2008 and won the approval of the majority of voters, but failed to achieve the minimum quorum of 25% 'Yes' votes of the total electorate. Legal claims of airlines opposing Tegel's closure have not been successful.

  Our empirical analyses are conducted for three distinct submarkets, a) one/two family houses, town houses and villas, b) multi-family houses and c) commercial properties. While submarkets a) and b) serve as a basis for an evaluation of household utility, submarket c) analogically yields insights into productivity effects. Distinguishing the residential market into submarket a) and b) potentially yields further interesting insights, given the very distinct tenure structure. Sub-market a) is closest to the vast majority of the existing literature on the impact of air-craft noise, which has focused on owner occupied detached housing. A particularly interesting characteristic of the housing market in Berlin (submarket b) is the very low rate of owner occupancy.

  dummy variables denoting 5 db grid cells (e.g. 50-55) and employ difference based semi-parametric estimation techniques

  properties that took place between January 1, 2000 and December 31, 2007 within the boundaries of the Federal State ofBerlin, Germany. 6 This study period stops almost 17 month before April 27, 2008, when the referendum confirmed that Tempelhof would be closed. Our data set is a complete record, covering transactions for commercial (1,474) and residential properties (31,289), which following rent theory facilitate the evaluation of the impact of accessibility and environmental quality on the productivity of land (commercial land) as well as on household utility and location desirability (residential land). Throughout our empirical analyses, we distinguish residential transactions into transactions of properties of a) on/two-family houses, townhouses and villas (15,199 observations) and b) multi-family houses (14,998 observations).6 Only relatively few observations had to be excluded from the full record due to missing values in crucial characteristicss. No signs for a sample selection bias were found.

  The transaction data provided by the Committee of Valuation Experts inBerlin (2008) includes the usual parameters such as age, floor space, plot area, storeys as well as information on land use, physical condition and building type. Employing a GIS-environment, property transactions were geo-referenced based on geographic coordinates and merged with the framework of the Urban and Environmental Information System of the Senate Department of Berlin(Senatsverwaltung für Stadtentwicklung Berlin, 2006). Within this GIS-environment, additional environmental control variables capturing the impact of natural and environmental amenities, transport and public infrastructure and built heritage, as well as noise emissions and airport accessibility variables could be generated. All distances are precise at least at a 6 digit level and accurate to the level of addresses when referring to transactions. When referring to precincts or blocks, distances strictly refer to their geographic centroids. Within the GIS-environment, neighborhood data are merged that were available for 15,937 statistical blocks (population by age and origin, all referring to 2005, and employment at workplace, referring to 2003), 338 traffic cells (rate of unemployment, referring to 2005) or 191 zip codes (purchasing power, referring to 2008). 7 With the exception of purchasing power, which was bought from the market research organization GfK, these data were provided by the State Statistical Institute Berlin-Brandenburg. The primary variables of interest used to assess the external effects are indicators of access to the flight connections offered by the three (city) airports as well as the exposure to aircraft noise within the affected neighborhoods. Access to the airports is measured by the effective road distance from every individual transaction to the terminal buildings of the three airports. A distance matrix is created on the basis of the full Berlin road map built-inin MS Mappoint 2009. From an official report (Laermkartierung nach Umgebungsrichtline, 09.07.2007),data on exposure to street 7 Data on employment at workplace include all employees contributing to social insurances.

  noise for Tegel Airport were available at a very detailed level of 10x10m grid cells. The noise map for Tegel Airport covers approximately the northern half of the city, including the air corridors. Within this area, noise levels are recorded for all developed properties and expressed in an equivalent long-term sound pressure index (Lden) in the standard log decibelscale (dB). These official records refer to the effective sound pressure at facades and take into account all physical obstacles that potentially affect noise patterns.

  points(g). A regression of average noise pressure (NL) on third order polynomial vectors of x-(X = x+x²+x³) and y-(Y=+y+y²+y³) coordinates (suppressing negative signs), interactions of both (X x Y=xy+(xy)²+(yx)³) and a full set of interactive terms with dummies denoting the northern (N)

  by productivity gains from quick access to the wide array of flight connections offered by the city's most important business airport. These results indicate that estimated aircraft noise effects can be biased if accessibility effects are not controlled for. More generally, they highlight the importance of disentangling positive and negative externalities emanated by transport infrastructure as shown by

  remain virtually unaffected as they are in column (

Fig. 1 -Fig. 2 -

 12 Fig. 1 -Passenger traffic at Berlin airports

Fig. 3 -Fig. 4 -

 34 Fig. 3 -Aircraft noise in Berlin

Fig. 5 -

 5 Fig. 5 -Semi-parametric noise effects: THF

  Table1findings. While for submarket b) price differentials hardly deviate from zero for all noise levels, for submarket a) prices continuously decline with in noise levels beyond 50 dB and become negative beyond 55dB. At the same time, the marginal impact increases with noise level, supporting the notion of a non-linear effect of aircraft noise on household utility.

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  buffer surrounding this zone. Results in Table3, again, reveal a relatively clear pattern of results.

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Table 1 and 2 type regressions using a dummy for the zone of restricted development and a study area within a 3 km

  Table 4 results. In line with the SAR model in Table 4, column (3), we find negative

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  Table 2 and 4. Notes: Endogenous variable is log of price per square meter land in all models. Baseline specification is equation (1). Controls are defined in Table A1 in the appendix. Submarket A covers one/two family houses, townhouses and villas, submarket B covers multi-family houses. Robust standard errors are in parenthesis. **/*/+ denote significance at the 1/5/10% level. Spatial LM statistics for model 2 [4] are: LM error : 112.86 robust LM error : 83.23, LM lag : 46.89, robust LM lag : 17.05 [LM error : 230.04 robust LM error : 136.22, LM lag : 95.31, robust LM lag : 1.46] Notes: Endogenous variable is log of price per square meter land in all models. Baseline specification is equation (1). Controls are defined in Table A1 in the appendix. Submarket A covers one/two family houses, townhouses and villas, submarket B covers multi-family houses. Robust standard errors are in parenthesis. **/*/+ denote significance at the 1/5/10% level. Spatial LM statistics for model 2 [4] are: LM error : 49.64, robust LM error : 16.38, LM lag : 42.82, robust LM lag : 8.56 [LM error : 220.08 , robust LM error :183.78, LM lag : 43.89, robust LM lag : 7.60].

	Tab 1 -Residential submarkets -TXL impact area Tab. 2 -Residential submarkets -THF impact area Tab 3 -Residential submarkets -SXF impact area
		(1) (1) (1)	(2) (2) (2)	(3) (3) (3)		(4) (4) (4)	(5) (5) (5)	(6) (6) (6)
		OLS OLS OLS	OLS OLS OLS	OLS OLS OLS		OLS OLS OLS	SAR (error) SAR (error) SAR (error) SAR (error) SAR (error) SAR (error)
	dB 45-50 dB 45-50 SXF Zone of rest.	-0.009 0.031 -0.231**	-0.011 0.036 -0.305**	0.033 0.028 0.086		0.004 0.025 0.091	-0.007 0.04 -0.276**	0.027 0.017 0.034
	(Dummy) (Dummy)) Develop. (dummy)	(0.019) (0.029) (0.048)	(0.019) (0.03) (0.053)	(0.035) (0.022) (0.149)		(0.028) (0.022) (0.185)	(0.021) (0.039) (0.065)	(0.04) (0.029) (0.213)
	dB 50-55 dB 50-55 Distance to	-0.012 0.077+	-0.013 0.088* -0.033**	0.046 -0.003		0.008 -0.007 0.002	-0.018 0.106* -0.024	0.061 -0.034 -0.021
	F o r (0.021) -0.068** (0.023) -0.073* (0.033) F o r (0.04) 0.031 (0.067) -0.326 (0.234) F o r SXF Airport (km) (Dummy) dB 55-60 (Dummy) dB 60-65 (Dummy) (Dummy) dB 55-60 (Dummy) dB>60 (Dummy) Submarket A Year Effects Yes Year x East Effect Yes Structural Controls Yes F o r	(0.021) -0.069** (0.023) -0.078* (0.033) (0.041) 0.045 (0.066) -0.309 (0.234) (0.013) A Yes Yes Yes	(0.031) 0.041 (0.032) 0.056 (0.037) (0.031) 0.028 (0.032) -0.023 (0.061) B Yes Yes Yes		(0.03) -0.021 (0.037) -0.012 (0.038) (0.031) 0.025 (0.032) -0.03 (0.061) (0.036) B Yes Yes Yes	(0.024) -0.060* (0.027) -0.106** (0.039) (0.051) 0.014 (0.079) -0.319 (0.244) (0.017) A Yes Yes Yes	(0.048) 0.051 (0.05) 0.05 (0.056) (0.041) -0.019 (0.044) -0.041 (0.071) (0.045) B Yes Yes Yes
	dB 65-70 (Dummy) Distance to THF Airport (km) Location Controls Neighb. Controls	P -0.229** (0.061) P P Yes Yes P	-0.232** (0.061) 0.015 (0.012) Yes Yes	0.115+ (0.065) Yes Yes		0.053 (0.064) -0.011 (0.01) Yes Yes	-0.252** (0.062) 0.019 (0.018) Yes Yes	0.033 (0.078) -0.011 (0.015) Yes Yes
	dB>70 (Dummy) Distanceto TXL Airport (km) Submarket Year Effects Year x East Effect Structural Controls Observations R-squared	e e r -0.519** -0.528** (0.085) (0.085) -0.002 (0.004) e e r A A Yes Yes Yes Yes Yes Yes e e r 1176 1176 0.83 0.83 e e r	B Yes Yes Yes 158 0.91	-0.038* (0.017) B Yes Yes Yes 158 0.91	-0.528** (0.091) -0.007 (0.021) A Yes Yes Yes 1176	0.027 (0.04) B Yes Yes Yes 158
	Submarket Year Effects Year x East Effect Structural Controls Location Controls Neighb. Controls Observations R-squared	A Yes Yes Yes Yes Yes 1591 0.74	R e A Yes Yes Yes R e Yes Yes 1591 4695 B Yes Yes Yes Yes Yes R R 0.74 0.61 e e		B Yes Yes Yes Yes Yes 4695 0.61	A Yes Yes Yes Yes Yes 1591	B Yes Yes Yes Yes Yes 4695
	Location Controls Neighb. Controls Observations R-squared	Yes Yes 2998 0.81	Yes Yes 2998 0.81	v i e w Yes Yes 3502 0.61 v i e v i e v i e w w w	Yes Yes 3502 0.61	Yes Yes 2998	Yes Yes 3502
					O n l O n l O n l O n l
						y y y y

Notes: Endogenous variable is log of price per square meter land in all models. Baseline specification is equation (1). Controls are defined in Table A1 in the appendix. Submarket A covers one/two family houses, townhouses and villas, submarket B covers multi-family houses. Robust standard errors are in parenthesis. **/*/+ denote significance at the 1/5/10% level. Spatial LM statistics for model 2 [4] are: LM error : 14.49, robust LM error : 10.36, LM lag : 5.35.13, robust LM lag : 1.23 [LM error : 1.02 , robust LM error : 1.36, LM lag : 0, robust LM lag : 0.34]

Table 4 -Commercial properties -TXL and THF impact area

 4 Notes: Endogenous variable is log of price per square meter land in all models. Baseline specification is equation (1). Controls are defined in Table A1 in the appendix. Submarket C covers commercial properties. Robust standard errors are in parenthesis. **/*/+ denote significance at the 1/5/10% level. Spatial LM statistics for model 2 [5] are: LM error : 2.13 , robust LM error : 2.91, LM lag : 10.92, robust LM lag : 11.07 [LM error : 5.20 , robust LM error :6.41, LM lag : 0.21, robust LM lag : 1.41] The spatial lag parameter Rho takes a value of 0.36 in model (3). Notes: Endogenous variable is log of price per square meter land in all models. Baseline specification is equation (1). Controls are defined in Table A1 in the appendix. Submarket B covers multi-family houses. Robust standard errors are in parenthesis. **/*/+ denote significance at the 1/5/10% level. Spatial LM statistics for model 4 are: LM error : 2220.73, robust LM error : 4351.75, LM lag : 2131.02, robust LM lag : 3.25. Notes: Endogenous variable is log of price per square meter land in all models. Baseline specification is equation (1). Controls are defined in Table A1 in the appendix. Submarket C covers commercial properties. Robust standard errors are in parenthesis. **/*/+ denote significance at the 1/5/10% level. Spatial LM statistics for model 4 are: LM error : 421.76 , robust LM error :247.05, LM lag : 184.76, robust LM lag : 10.05.

		(1)	(2)	(3)	(4)	(5)	(6)
		OLS	OLS	SAR (lag)	OLS	OLS	SAR (error)
	dB 45-50	-0.13	-0.306	-0.453+	0.082	0.072	0.106
	(Dummy)	(0.284)	(0.308)	(0.26)	(0.122)	(0.12)	(0.097)
	dB 50-55	0.035	-0.226	-0.306	-0.568**	-0.577**	-0.646**
	(Dummy)	(0.332)	(0.351)	(0.289)	(0.2)	(0.207)	(0.155)
	dB 55-60	-0.958+	-1.399*	-1.490**	-1.471**	-1.544**	-1.653**
	(Dummy)	(0.488)	(0.557)	(0.469)	(0.277)	(0.296)	-0.239
	dB 60-65	0.155	-0.883	-0.85
	(Dummy)	(0.415)	(0.693)	(0.554)
	dB 65-70	-0.431	-1.609+	-1.12
	(Dummy)	(0.6)	(0.835)	(0.701)
	Distanceto		-0.372+	-0.323*		0.118	0.227
	Airport (km)		(0.203)	(0.157)		(0.179)	(0.14)
	Airport	TXL	TXL	TXL	THF	THF	THF
	Submarket	C	C	C	C	C	C
	Year Effects	Yes	Yes	Yes	Yes	Yes	Yes
	Year x East Effect	Yes	Yes	Yes	Yes	Yes	Yes
	Structural Controls	Yes	Yes	Yes	Yes	Yes	Yes
	Location Controls	Yes	Yes	Yes	Yes	Yes	Yes
	Neighb. Controls	Yes	Yes	Yes	Yes	Yes	Yes
	Observations	159	159	159	105	105	105
	R-squared	0.8	0.81		0.94	0.94

F o r P e e r R e v i

e w O n l y Tab. 8 -Marginal price effects and treatment heterogeneity

  Notes: Endogenous variable is log of price per square meter land in all models. Baseline specification is equation (1). Controls are defined in Table A1 in the appendix. Submarket A covers one/two family houses, townhouses and villas, submarket B covers multi-family houses, submarket C covers commercial properties. Full estimation results for models (3), (6) and (6) are presented in Table A2 in the appendix. Robust standard errors are in parenthesis. **/*/+ denote significance at the 1/5/10% level. Spatial LM statistics for model 9 are: LM error : 484.30 , robust LM error : 280.33, LM lag : 215.603, robust LM lag : 11.64. Notes: Endogenous variable is log of price per square meter land in all models. Baseline specification is equation (1). Variables are defined in Table A1. Submarket A covers one/two family houses, townhouses and villas, submarket B covers multi-family houses, submarket C covers commercial properties. Robust standard errors are in parenthesis. **/*/+ denote significance at the 1/5/10% level.

	(1) Tab. A2 -Hedonic estimates		(2)		(3)			(4)	(5)		(6)	(7)	(8)	(9)	(10)
	(OLS)	(OLS) (1)	(OLS)	(2)	(OLS)	(OLS)	(3)	(OLS)	(OLS)	(OLS)	(OLS)	(SAR)
	AverageDistance to Airport (ADA) (km) Street Noise (dB) Floor Space Index (FSI) -0.020** (0.004) Coeff. -0.005** 1.445** -0.020** (0.004) S.E. -0.007** 0.043 6.0 (0) (0.001) Plot Area (m²) -0.0001** 0 Air Noise (dB) -0.001* -0.005* Storey 0.017+ 0.009 7.0 (0.001) (0.002) Street Noise 0.00007+ x Air Noise (0.00004) (0.00004) -0.021** (0.004) Coeff. -0.007** 0.450** (0.001) -0.000* -0.002 -0.002 (0.002) 0.00006 dB>55 -0.103** (Dummy)) (0.018) Age (Years) -0.009** 0.001 -0.006** Age (Years) squared 0.000** 0 0.000** F Condition: Good 0.209** 0.011 0.437** o Condition: Bad -0.314** 0.012 -0.411** r Dist. to Centre (km) -0.055** 0.003 -0.061** Submarket A A A Year Effects Yes Yes Yes Year x East Effect Yes Yes Yes Structural Controls Yes Yes Yes Location Controls Yes Yes Yes Neighb. Controls Yes Yes Yes Emp. Potentiality (log) 0.040* 0.018 0.007 Dist. to Station (km) -0.075** 0.008 -0.074** P Dist. to Main St. (km) -0.024 0.0160 0.114** e Dist. To School (km) 0.052** 0.003 0.027** Landmarks within 600m 0.003** 0 0.001** e Dist. toWater (km) 0.009 0.010 0.022 r WaterPotentiality (log) 0.011* 0.005 0.009	-0.002 (0.007) S.E. 0.002** 0.007 (0) 0 0.003** 0.005 (0.001) 0.001 0 0.014 0.014 0.005 B Yes Yes Yes Yes Yes 0.035 0.010 0.040 0.002 0 0.016 0.008	-0.004 (0.007) Coeff. -0.003** 0.348** (0.001) 0.000** -0.006** -0.018 (0.001) 0.00015** 0.00015** -0.004 (0.007) S.E. -0.003** 0.032 (0.001) 0 -0.005** 0.011 (0.002) (0.00002) (0.00002) -0.004 (0.015) -0.003+ 0.002 0 0 0.743** 0.047 -0.465** 0.064 -0.131** 0.024 B B Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes -0.287 0.185 -0.343** 0.105 -0.619* 0.247 0.000** 0 0.090** 0.028 0.054 0.097 0.094+ 0.051	0.036 (0.036) -0.001 (0.002) 0.007* (0.003) C Yes Yes Yes Yes Yes	0.033 (0.036) -0.007 (0.005) -0.008 (0.013) 0.00023 (0.00018) (0.00019) (0.0002) 0.012 -0.031 (0.036) (0.054) -0.010+ -0.006 (0.006) (0.006) -0.013 -0.018 (0.013) (0.017) 0.00033+ 0.0002 -0.500** -0.576** (0.158) (0.187) C C C Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
	SXF Effects Observations R-squared Dist. to Green (km) Green Potentiality (log) 0.022** Yes 15199 0.73 -0.009 Dist. toIndustry (km) 0.041**	Yes 15199 0.73 0.011 R 0.006 e Yes 15199 0.73 -0.106** -0.046** 0.003 0.032**		Yes 14998 0.65 0.020 0.011 0.012	Yes 14998 0.65 -0.217* -0.098+ 0.218**	Yes 14998 0.65 0.109 0.058 0.50	Yes 1474 0.76	Yes 1474 0.76	Yes 1474 0.76	Yes 1474
	Proportion (%) Foreign Proportion (%) Young Proportion (%) Old Proportion (%) Unemp. -0.010** 0.002* 0.001 0.001** P. Power (1000€/cap) 0.015** Av. Dist. to Air. (AVA) -0.021**		0.001 v -0.004** 0 0.007** i 0 0.002** e 0.001 -0.014** 0.002 0.034** w 0.004 -0.004	0.001 0.001 0.001 0.002 0.004 0.007	0.002* -0.003 0 0.006 0.085** 0.012	0.001 0.002 0.002 0.008 0.026 0.036
	SXF Zone Street Noise Air Noise Street Noise x Air Noise dB> 55 Submarket	-0.151** -0.007** -0.002 0 -0.103**	A	0.027 0.001 0.002 0 0.018	0.186 -0.003** O 0.12 0.001 -0.005** 0.002 0.000** 0 n l 8.0 -0.010+ -0.013 0.000+ -0.004 0.015 -0.500** y B	C	9.0 0.006 0.013 0 0.158
	Year Effects		Yes			Yes			Yes
	Year x East Eff.		Yes			Yes			Yes
	Observations		15199			14998			1474
	R-squared		0.73			0.65			0.76

http://mc.manuscriptcentral.com/cres Email: regional.studies@fm.ru.nlRegional Studies

This specification implicitly treats both centres as perfect substitutes, which is in line with the definition of the Senate Department (Senatsverwaltung für Wirtschaft Arbeit und Frauen,

2004). This specification also avoids collinearity problems compared to the alternative of introducing distances to both CBD individually.2 The internal distance for precincti is calculated on the basis of its surface area (area i ) (see[START_REF] Keeble | Regional Accessibility and Economic Potential in the European Community[END_REF].

The car discount parameter (0.1) is based on a gravity type urban labour market accessibility model for the metropolitain area for Berlin. The walking (2) discount parameter was set to yield and exponential cost function that converges towards zero at a maximum walking distance of 2 km.

4 See[START_REF] Coulson | Gracing the Land of Elvis and Beale Street: Historic Designation and Property Values in Memphis[END_REF] and[START_REF] Ahlfeldt | Substitutability and Complementarity of Urban Amenities: External Effects of Built Heritage in Berlin[END_REF] for recent examples for the U.S. and Europe

.5 We define transactions as neighbors if they occur within a 500 m radius. In very few cases where not transaction occured within the threshold, we define the nearest transaction as neighbor. The specification generally producessimilar resuls to an alternative weights matrix with inverse distance weights. We prefer the binary weights matrix since inverse distance weights in some of our models with a limited geographic scope and few observations tend to produce a strong spatial smooth.

The percentage impact (PI) is approximated from the coefficient b according to the standard interpretation for dummy variables in semi-log models : PI = (exp(b)-1)×100[START_REF] Halvorsen | The Interpretation of Dummy Variables in Semilogarithmic Equations[END_REF].

From the percentage impact (PI) the average absolute impact (AI) is derived according to following formula : AI = PI/(1-PI)×P×S, where P and S are the mean sales price and lot size of properties.

R31 -Housing Supply and Markets < R3 -Production Analysis and Firm Location < R -Urban, Rural, and Regional Economics, R41 -Transportation: Demand|Supply|Congestion|Safety and Accidents < R4 -Transportation Systems < R -Urban, Rural, and Regional Economics, R53 -Public Facility Location Analysis; Public Investment; Capital Stock < R5 -Regional Government Analysis < R -Urban, Rural, and Regional Economics

http://mc.manuscriptcentral.com/cres Email: regional.studies@fm.ru.nl


Maria J Esteban

Simona Rota Nodari

GROUND STATES FOR A STATIONARY MEAN-FIELD MODEL FOR A NUCLEON

In this paper we consider a variational problem related to a model for a nucleon interacting with the ω and σ mesons in the atomic nucleus. The model is relativistic, and we study it in a nuclear physics nonrelativistic limit, which is of a very different nature than the nonrelativistic limit in the atomic physics. Ground states are shown to exist for a large class of values for the parameters of the problem, which are determined by the values of some physical constants.

Introduction

This article is concerned with the existence of minimizers for the energy functional

E(ϕ) = R 3 |σ • ∇ϕ| 2 (1 -|ϕ| 2 ) + dx - a 2 R 3 |ϕ| 4 dx (1.1)
under the L 2 -normalization constraint

R 3 |ϕ| 2 dx = 1. (1.2)
More precisely, for a large class of values for the parameter a, we show the existence of solutions of the following minimization problem

I = inf E(ϕ) ; ϕ ∈ X, R 3 |ϕ| 2 dx = 1 , (1.3)
where

X = ϕ ∈ L 2 (R 3 , C 2 ) ; R 3 |σ • ∇ϕ| 2 (1 -|ϕ| 2 ) + dx < +∞ . (1.4)
We remind that σ denotes the vector of Pauli matrices (σ 1 , σ 2 , σ 3 ),

σ 1 = 0 1 1 0 , σ 2 = 0 -i i 0 , σ 3 = 1 0 0 -1 .
The Euler-Lagrange equation of the energy functional E under the L 2 -normalization constraint is given by the second order equation

-σ • ∇ σ • ∇ϕ (1 -|ϕ| 2 ) + + |σ • ∇ϕ| 2 (1 -|ϕ| 2 ) 2 + ϕ -a|ϕ| 2 ϕ + bϕ = 0 , (1.5)
where b is the Lagrange multiplier associated with the L 2 -constraint (1.2). Hence a solution of the minimization problem (1.3) is a solution of the equation (1.5).

Date: August 10, 2012.

Moreover, Lemma 2.1 below proves that any ϕ ∈ X satisfies |ϕ| 2 ≤ 1 a.e. in R 3 . So, a minimizer for (1.3) is actually a solution of

-σ • ∇ σ • ∇ϕ 1 -|ϕ| 2 + |σ • ∇ϕ| 2 (1 -|ϕ| 2 ) 2 ϕ -a|ϕ| 2 ϕ + bϕ = 0. (1.6)
Solutions of (1.6) which are minimizers for I are called ground states. The equation (1.6) is a equivalent to the system iσ • ∇χ + |χ| 2 ϕ -a|ϕ| 2 ϕ + bϕ = 0,

-iσ • ∇ϕ + 1 -|ϕ| 2 χ = 0 . (1.7)
As we formally derived in a previous paper ( [START_REF] Esteban | Symmetric ground states for a stationary relativistic meanfield model for nucleons in the non relativistic limit[END_REF]), this system is the nuclear physics nonrelativistic limit of the σ-ω relativistic mean-field model ( [START_REF] Walecka | A theory of highly condensed matter[END_REF][START_REF] Walecka | Theoretical Nuclear and Subnuclear Physics[END_REF]) in the case of a single nucleon.

In [START_REF] Esteban | Symmetric ground states for a stationary relativistic meanfield model for nucleons in the non relativistic limit[END_REF], we proved the existence of square integrable solutions of (1.7) in the particular form

ϕ(x) χ(x) =       g(r) 1 0 if (r) cos ϑ sin ϑe iφ       , (1.8)
where f and g are real valued radial functions. This ansatz corresponds to particles with minimal angular momentum, that is, j = 1/2 (for instance, see [START_REF] Thaller | The Dirac equation[END_REF]). In this model, the equations for f and g read as follows:

   f ′ + 2 r f = g(f 2 -ag 2 + b) , g ′ = f (1 -g 2 ) , (1.9)
where we assumed f (0) = 0 in order to avoid solutions with singularities at the origin, and we showed that given a, b > 0 such that a -2b > 0, there exists at least one nontrivial solution of (1.9) such that (f (r), g(r)) -→ (0, 0) as r -→ +∞ .

(1.10)

In this paper, we prove the existence of solutions of the above nuclear physics nonrelativistic limit of the σ-ω relativistic mean-field model without considering any particular ansatz for the nucleon's wave function.

Note that (1.6) is the Euler-Lagrange equation of the energy functional

F (ϕ) = R 3 |σ • ∇ϕ| 2 1 -|ϕ| 2 dx - a 2 R 3 |ϕ| 4 dx (1.11)
under the L 2 normalization constraint. In the Appendix, we prove that the energy functional F is not bounded from below. So, trying to find solutions of (1.6) which minimize the energy F is hopeless and the definition of ground states for (1.6) based on this functional is not clear.

In our previous work ( [START_REF] Esteban | Symmetric ground states for a stationary relativistic meanfield model for nucleons in the non relativistic limit[END_REF]), we showed that for all the solutions of (1.9) which are square integrable, g 2 (r) < 1 in [0, +∞). Hence, according to this result, we conjecture that a solution of (1.6) has to satisfy |ϕ| 2 ≤ 1 a.e. in R 3 . As we prove in the Appendix, this assumption is also justified when we consider the intermediate model

ϕ = u 0 (1.12)
with u : R 3 → R and a > b. Moreover, in the physical literature finite nuclei are described via functions ϕ such that, in the right units, |ϕ| 2 ≤ 1 and |ϕ| is rather flat near the center of the nucleus, and is equal to 0 outside it, see [START_REF] Ring | Relativistic Mean Field Theory in Finite Nuclei[END_REF][START_REF] Greiner | Nuclear Models[END_REF]. Note that if |ϕ| 2 ≤ 1 a.e. in R 3 , then F (ϕ) = E(ϕ), and the ground states of (1.6) can be defined without further specification as the minimizers of E.

The main result of our paper is the following Theorem 1.1. If I < 0 there exists a minimizer of (1.3). Moreover, I < 0 if and only if a > a 0 where a 0 is a strictly positive constant. In particular, 10.96 ≈ 2 S 2 < a 0 < 48.06, where S the best constant in the Sobolev embedding of H 1 (R 3 ) into L 6 (R 3 ).

Remark 1. The upper estimate for a 0 is obtained by using a particular test function and is probably not optimal .

The proof of the above theorem is an application of the concentration-compactness principle ( [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case[END_REF][START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case[END_REF]) with some new ingredients. The main new difficulty is due to the presence of the term R 3 |σ•∇ϕ| 2

(1-|ϕ| 2 )+ dx in the energy functional. As we will see below, to rule out the dichotomy case in the concentration-compactness lemma we have to choose ad-hoc cut-off functions allowing us to deal with possible singularities of the integrand. This is also necessary in order to show the localization properties of

R 3 |σ•∇ϕ| 2
(1-|ϕ| 2 )+ dx. In the next section, we will establish a concentration-compactness lemma in X and then apply it to prove our main result. The Appendix contains some auxiliary results about various properties of the model problem that we consider here.

Proof of Theorem 1.1

To prove this theorem, we are going to apply a concentration-compactness lemma that we state below. The reader may refer to [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case[END_REF] and [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case[END_REF] for more details on this kind of approach. The particular shape of the energy functional, where the kinetic energy term is multiplied by a function which could present singularities as |ϕ| gets close to 1 creates some complications in the use of concentration-compactness, that we deal with by using very particular cut-off functions.

Let us introduce

I ν = inf E(ϕ) ; ϕ ∈ X, R 3 |ϕ| 2 dx = ν (2.1)
where ν > 0 and I 1 = I, and we make a few preliminary observations.

Lemma 2.1 ([6]). Let ϕ ∈ X. Then, ϕ ∈ H 1 (R 3 , C 2 ) and |ϕ| 2 ≤ 1 a.e. in R 3 .
Proof. First, by a straightforward calculation, we obtain

R 3 |∇ϕ| 2 dx = R 3 |σ • ∇ϕ| 2 dx ≤ R 3 |σ • ∇ϕ| 2 (1 -|ϕ| 2 ) + dx < +∞. Hence, ϕ ∈ H 1 (R 3 , C 2 ). Next, let n ∈ C 2 such that |n| = 1. Note that for ϕ ∈ X, ½ Re{n•ϕ}≥1 (σ • ∇ϕ) = 0, a.e. in R 3 . Define the functions f = (Re{n • ϕ} -1) +
and ψ = f n. (Note that for 2 complex vectors A, B ∈ C 2 , A • B denotes the scalar product Σ 2 i=1 A i B, where z stands for the complex conjugate of any complex number z).

We have

f ∈ H 1 (R 3 , R) and ψ ∈ H 1 (R 3 , C 2 ). Moreover, for k = 1, 2, 3, ∂ k ψ = ∂ k f n and ∂ k f = Re {n • ∂ k ϕ} ½ Re{n•ϕ}≥1 = n • ∂ k ψ .
Hence, we obtain

R 3 |∇f | 2 dx = R 3 |∇ψ| 2 dx = R 3 3 k=1 Re {Re {n • ∂ k ϕ} n • ∂ k ψ} dx = R 3 3 k=1 Re {n • ∂ k ϕ} Re {n • ∂ k ψ} dx = R 3 3 k=1 Re {∂ k f n • ∂ k ϕ} dx = R 3 Re {∇ψ • ∇ϕ} dx = R 3 Re {(σ • ∇ψ) • (σ • ∇ϕ)} dx = R 3 Re (σ • ∇ψ) • ½ Re{n•ϕ}≥1 (σ • ∇ϕ) dx = 0
As a consequence, f = 0 a.e. in R 3 that means Re{n • ϕ} ≤ 1 a.e. for all n ∈ C 2 such that |n| = 1. This clearly implies that |ϕ| ≤ 1 a.e. in R 3 .

In what follows, we say that a sequence {ϕ n } n is X-bounded if there exists a positive constant C independent of n such that

ϕ n 2 L 2 + R 3 |σ • ∇ϕ n | 2 (1 -|ϕ n | 2 ) + dx ≤ C . (2.2) Lemma 2.2. Let {ϕ n } n be a minimizing sequence of (2.1), then {ϕ n } n is X- bounded, bounded in H 1 (R 3 ) and I ν > -∞.
Proof. Indeed, since {ϕ n } n is a minimizing sequence, there exists a constant C such that

C ≥ E(ϕ n ) ≥ R 3 |σ • ∇ϕ n | 2 (1 -|ϕ n | 2 ) + dx - a 2 ν ≥ R 3 |σ • ∇ϕ n | 2 dx - a 2 ν = R 3 |∇ϕ n | 2 dx - a 2 ν ≥ - a 2 ν.
As a conclusion, ϕ n H 1 is bounded independently of n and I ν is bounded from below.

Lemma 2.3. For all ν ∈ (0, 1), I ν ≤ 0. Moreover, the strict inequality I < 0 is equivalent to the strict concentration-compactness inequalities

I < I ν + I 1-ν , ∀ν ∈ (0, 1) . (2.3) Proof. Indeed, let ϕ ∈ D(R 3 ) such that R 3 |ϕ| 2 = ν and R 3 |σ•∇ϕ| 2 (1-|ϕ| 2 )+ dx < +∞, and let ϕ γ (x) = γ -3/2 ϕ(γ -1 x) for γ > 1. Then I ν ≤ E(ϕ γ ) = 1 γ 2 R 3 |σ • ∇ϕ| 2 1 -1 γ 3 |ϕ| 2 + dx - 1 γ 3 a 2 R 3 |ϕ| 4 dx ,
and letting γ → +∞, we prove I ν ≤ 0.

By a scaling argument, we obtain

I ϑν ≤ inf ϑ 1/3 R 3 |σ • ∇ϕ| 2 (1 -|ϕ| 2 ) + dx - ϑ a 2 R 3 |ϕ| 4 dx|ϕ ∈ X, R 3 |ϕ| 2 dx = ν ,
and, if I ν < 0, we may restrict the infimum I ν to elements ϕ satisfying

K(ϕ) = R 3 |σ • ∇ϕ| 2 (1 -|ϕ| 2 ) + dx ≥ δ > 0 ,
for some δ > 0. Indeed, if there is a minimizing sequence

{ϕ n } n of I ν such that K(ϕ n ) -→ n 0, then, by Sobolev embeddings, ϕ n -→ n 0 in L p (R 3
) for 2 < p ≤ 6 and I ν ≥ 0. As a conclusion, if I ν < 0, then, for all ϑ > 1 and for all ν > 0,

I ϑν < ϑ inf E(ϕ)|ϕ ∈ X, K(ϕ) > 0, R 3 |ϕ| 2 dx = ν = ϑI ν .
(2.4)

Hence, a straightforward argument (see lemma II.1 of [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case[END_REF]) proves that (2.3) is equivalent to I < 0.

In order to prove Theorem 1.1 we need to analyse the possible behaviour of minimizing sequences for I. This is done in the following lemma.

Lemma 2.4. Let {ϕ n } n be a X-bounded sequence such that R 3 |ϕ n | 2 dx = 1 for all n ≥ 0.
Then there exists a subsequence that we still denote by {ϕ n } n such that one of the following properties holds:

(1) Compactness up to a translation: there exists a sequence {y n } n ⊂ R 3 such that, for every ε > 0, there exists 0 < R < ∞ with

B(yn,R) |ϕ n | 2 dx ≥ 1 -ε;
(2) Vanishing: for all 0 < R < ∞ sup

y∈R 3 B(y,R) |ϕ n | 2 dx -→ n 0;
(3) Dichotomy: there exist α ∈ (0, 1) and n 0 ≥ 0 such that there exist two Xbounded sequences, {ϕ n 1 } n≥n0 and {ϕ n 2 } n≥n0 , satisfying the following properties:

ϕ n -(ϕ n 1 + ϕ n 2 ) L p -→ n 0, for 2 ≤ p < 6, (2.5)
and

R 3 |ϕ n 1 | 2 dx -→ n α and R 3 |ϕ n 2 | 2 dx -→ n 1 -α, (2.6) dist(supp ϕ n 1 , supp ϕ n 2 ) -→ n +∞. (2.7)
Moreover, in this case we have that

lim inf n→+∞ E(ϕ n ) -E(ϕ n 1 ) -E(ϕ n 2 ) ≥ 0 , (2.8
)

which implies I ≥ I α + I 1-α .
Proof of Lemma 2.4. Let {ϕ n } n be a X-bounded sequence such that R 3 |ϕ n | 2 dx = ν for all n ≥ 0. We remind that X-bounded means that there exists C > 0 such that

ϕ n 2 L 2 + R 3 |σ • ∇ϕ n | 2 (1 -|ϕ n | 2 ) + dx ≤ C.
Moreover, thanks to Lemma 2.1, if {ϕ n } n is a X-bounded sequence then {ϕ n } n is bounded in L ∞ (by the constant 1) and in H 1 (R 3 ). Then, along the lines of [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case[END_REF], we introduce the so-called Lévy concentration functions

Q n (R) = sup y∈R 3 |x-y|<R |ϕ n | 2 dx, (2.9)
K n (R) = sup y∈R 3 |x-y|<R |σ • ∇ϕ n | 2 (1 -|ϕ n | 2 ) + dx (2.10)
for R > 0. Note that Q n and K n are continuous non-decreasing functions on [0, +∞), such that for all n ≥ 0 and for all R > 0

Q n (R) + K n (R) ≤ C since {ϕ n } n is X-bounded.
Then, up to a subsequence, we have for all R > 0

Q n (R) -→ n Q(R), (2.11)
K n (R) -→ n K(R), (2.12)
where Q and K are nonnegative, non-decreasing functions. Clearly, we have that

α = lim R→+∞ Q(R) ∈ [0, 1],
and we denote l = lim R→+∞ K(R).

If α = 0, then the situation (2) of the lemma arises as a direct consequence of Definition (2.9). If α = 1, then (1) follows, see [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case[END_REF] for details. Assume that α ∈ (0, 1), we have to show that (3) holds.

First of all, consider ε > 0, small, and

R ε > 0 such that Q(R ε ) = α -ε and K(R ε ) ≤ l -ε. Then, for n large enough, Q n (R ε ) -Q(R ε ) < 1/n, K n (R ε ) -K(R ε ) < 1/n ,
and by definition of the Lévy functions Q n , extracting subsequences if necessary, there exists

y n ∈ R 3 such that |x-yn|<Rε |ϕ n | 2 dx -Q n (R ε ) ≤ 1 n , |x-yn|<Rε |σ • ∇ϕ n | 2 (1 -|ϕ n | 2 ) + dx -K n (R ε ) ≤ 1 n . Next define R n > R ε such that Rε<|x-yn|<Rn |ϕ n | 2 dx = 3 n + ε .
Necessarily, R n → +∞ as n → +∞. Indeed, if R n ≤ M for some M > 0, then Q(M ) > α, which is impossible. We then deduce that for n large enough,

Rn 8 ≤|x-yn|≤Rn |ϕ n | 2 dx ≤ 3 n + ε Let ξ, ζ be cut-off functions: ξ, ζ ∈ D(R 3 ) such that ξ(x) =              1 |x| ≤ 1 1 -exp   1- 1 1 -exp 1 -1 2-|x|   1 < |x| < 2 0 |x| ≥ 2 ζ(x) =              0 |x| ≤ 1 exp   1 - 1 1 -exp 1 -1 2-|x|   1 < |x| < 2 1 |x| ≥ 2
,

and let ξ µ , ζ µ denote ξ • µ , ζ • µ . We define ϕ n 1 (•) = ξ Rn 8 (• -y n )ϕ n (•) = ξ Rn 8 ,yn (•)ϕ n (•) (2.13) ϕ n 2 (•) = ζ Rn 2 (• -y n )ϕ n (•) = ζ Rn 2 ,yn (•)ϕ n (•) (2.14)
with R n → +∞. (2.7) follows easily from these definitions. Furthermore, (2.5) and (2.6) are obtained in the following way:

lim n→+∞ R 3 |ϕ n -(ϕ n 1 + ϕ n 2 )| 2 dx = lim n→+∞ Rn 8 ≤|x-yn|≤Rn |(1 -ξ Rn 8 -ζ Rn 2 )ϕ n | 2 dx ≤ lim n→+∞ Rn 8 ≤|x-yn|≤Rn |ϕ n | 2 dx ≤ ε,
Now by taking a sequence of ε tending to 0, and by taking a diagonal sequence of the functions ϕ n , and calling it by the same name, we find

Rn 8 ≤|x-yn|≤Rn |ϕ n | 2 dx -→ n 0,
and, since {ϕ n 1 } n and {ϕ n 2 } n are bounded in H 1 (R 3 ), we also obtain

lim n→+∞ ϕ n -(ϕ n 1 + ϕ n 2 ) L p -→ n 0,
for 2 ≤ p < 6. Next, we have to prove that {ϕ n 1 } n≥n0 and {ϕ n 2 } n≥n0 are X-bounded. To this purpose, we show that

lim n→+∞ R 3 |σ • ∇ϕ n 1 | 2 (1 -|ϕ n 1 | 2 ) + dx - R 3 ξ 2 Rn 8 ,yn |σ • ∇ϕ n | 2 (1 -|ϕ n 1 | 2 ) + dx = 0 (2.15) and lim n→+∞ R 3 |σ • ∇ϕ n 2 | 2 (1 -|ϕ n 2 | 2 ) + dx - R 3 ζ 2 Rn 2 ,yn |σ • ∇ϕ n | 2 (1 -|ϕ n 2 | 2 ) + dx = 0 (2.16)
Indeed, if (2.15) and (2.16) hold, we obtain that for all ε > 0, there exists n 0 ≥ 0 such that for all n ≥ n 0 , we have

R 3 |σ • ∇ϕ n 1 | 2 (1 -|ϕ n 1 | 2 ) + dx ≤ R 3 ξ 2 Rn 8 ,yn |σ • ∇ϕ n | 2 (1 -|ϕ n 1 | 2 ) + dx + o(1) n→+∞ ≤ R 3 |σ • ∇ϕ n | 2 (1 -|ϕ n | 2 ) + dx + o(1) n→+∞ ≤ C + o(1) n→+∞ ,
and

R 3 |σ • ∇ϕ n 2 | 2 (1 -|ϕ n 2 | 2 ) + dx ≤ R 3 ζ 2 Rn 2 ,yn |σ • ∇ϕ n | 2 (1 -|ϕ n 2 | 2 ) + dx + o(1) n→+∞ ≤ R 3 |σ • ∇ϕ n | 2 (1 -|ϕ n | 2 ) + dx + o(1) n→+∞ ≤ C + o(1) n→+∞ .
To prove (2.15) we proceed as follows. We remark that

R 3 |σ • ∇ϕ n 1 | 2 (1 -|ϕ n 1 | 2 ) + dx - R 3 ξ 2 Rn 8 ,yn |σ • ∇ϕ n | 2 (1 -|ϕ n 1 | 2 ) + dx = A n + B n ,
where

A n := R 3 |σ • (∇ξ Rn 8 ,yn )ϕ n | 2 (1 -|ϕ n 1 | 2 ) + dx = Rn 8 ≤|x-yn|≤ Rn 4 |σ • (∇ξ Rn 8 ,yn )ϕ n | 2 (1 -|ϕ n 1 | 2 ) + dx ≤ Rn 8 ≤|x-yn|≤ Rn 4 |σ • (∇ξ Rn 8 ,yn )ϕ n | 2 1 -ξ 2 Rn 8 ,yn dx := C n , and
|B n | ≤ 2 (C n ) 1 2 R 3 |σ • ∇ϕ n | 2 (1 -|ϕ n | 2 ) + dx 1 2
.

Let us now prove that C n tends to 0 as n goes to +∞. Using spherical coordinates, we obtain

C n ≤ Rn 4 Rn 8 π 0 2π 0 |(σ • e r ) ϕ n (s, θ, φ)| 2 ξ ′ Rn 8 (s) 2 1 -ξ 2 Rn 8 (s) s 2 sin θ ds dθ dφ ≤ 64 R 2 n Rn 4 Rn 8 π 0 2π 0 |ϕ n (s, θ, φ)| 2 ξ ′ 8 Rn s 2 1 -ξ 2 8
Rn s s 2 sin θ ds dθ dφ

≤ 64 R 2 n max 1≤r≤2 (ξ ′ (r)) 2 1 -ξ 2 (r) +∞ 0 π 0 2π 0 |ϕ n (s, θ, φ)| 2 s 2 sin θ ds dθ dφ = O 1 R 2 n since max 1≤r≤2 (ξ ′ (r)) 2 1-ξ 2 (r) ≤ C. Indeed, since ξ 2 (r) = 1 if and only if r = 1, (ξ ′ (r)) 2 1-ξ 2 (r)
is a continuous function on [START_REF] Esteban | Symmetric ground states for a stationary relativistic meanfield model for nucleons in the non relativistic limit[END_REF][START_REF] Greiner | Nuclear Models[END_REF]. Moreover, by a straightforward calculation, we obtain lim

r→1 + (ξ ′ (r)) 2 1-ξ 2 (r) = 0 = lim r→2 - (ξ ′ (r)) 2 1-ξ 2 (r)
. Hence, we can conclude, that

(ξ ′ (r)) 2 1-ξ 2 (r) is bounded in [1, 2]. As a conclusion, since R n → +∞, we obtain lim n→+∞ R 3 |σ • ∇ϕ n 1 | 2 (1 -|ϕ n 1 | 2 ) + dx - R 3 ξ 2 Rn 8 ,yn |σ • ∇ϕ n | 2 (1 -|ϕ n 1 | 2 ) + dx = 0.
With the same argument, we prove (2.16). Finally, it remains to show that

lim inf n→+∞ E(ϕ n ) -E(ϕ n 1 ) -E(ϕ n 2 ) ≥ 0.
First of all, using the definitions (2.13) and (2.14), we obtain

lim n→+∞ R 3 |σ • ∇ϕ n | 2 (1 -|ϕ n | 2 ) + dx ≥ lim n→+∞ R 3 |σ • ∇ϕ n | 2 (1 -|ϕ n 1 | 2 -|ϕ n 2 | 2 ) + dx.
Next, we remark that

R 3 |σ • ∇ϕ n | 2 (1 -|ϕ n 1 | 2 -|ϕ n 2 | 2 ) + dx - R 3 |σ • ∇ϕ n 1 | 2 (1 -|ϕ n 1 | 2 ) + dx - R 3 |σ • ∇ϕ n 2 | 2 (1 -|ϕ n 2 | 2 ) + dx = R 3 |σ • ∇ϕ n | 2 (1 -|ϕ n 1 | 2 -|ϕ n 2 | 2 ) + dx - R 3 ξ 2 Rn 8 ,yn |σ • ∇ϕ n | 2 (1 -|ϕ n 1 | 2 ) + dx - R 3 ζ 2 Rn 2 ,yn |σ • ∇ϕ n | 2 (1 -|ϕ n 2 | 2 ) + dx + o(1) n→∞ = R 3 1 -ξ 2 Rn 8 ,yn -ζ 2 Rn 2 ,yn |σ • ∇ϕ n | 2 (1 -|ϕ n 1 | 2 -|ϕ n 2 | 2 ) + dx + o(1) n→∞ ≥ o(1) n→∞ .
As a conclusion,

lim n→+∞ R 3 |σ • ∇ϕ n | 2 (1 -|ϕ n | 2 ) + dx ≥ lim n→+∞ R 3 |σ • ∇ϕ n 1 | 2 (1 -|ϕ n 1 | 2 ) + dx + lim n→+∞ R 3 |σ • ∇ϕ n 2 | 2 (1 -|ϕ n 2 | 2 ) + dx,
and, using (2.5) and the localization properties of ϕ n 1 and ϕ n 2 , we have

I = lim n→+∞ E(ϕ n ) ≥ lim inf n→+∞ E(ϕ n 1 ) + lim inf n→+∞ E(ϕ n 2 ) ≥ I α + I 1-α .
Proof of Theorem 1.1. Assume that I < 0. By Lemma 2.2, any minimizing sequence {ϕ n } n is X-bounded, and then we can use Lemma 2.4 to it. It is easy to rule out vanishing and dichotomy whenever I < 0. Vanishing cannot occur. Indeed, If vanishing occurs, then, up to a subsequence, ∀R < +∞ we have lim

n→+∞ sup y∈R 3 B(y,R) |ϕ n | 2 = 0.
(2.17)

This implies that ϕ n converges strongly in L p (R 3 ) for 2 < p < 6 and, as a consequence, I ≥ 0. Clearly, this contradicts I < 0. Moreover, if dichotomy occurs, we have

I = lim n→+∞ E(ϕ n ) ≥ lim inf n→+∞ E(ϕ n 1 ) + lim inf n→+∞ E(ϕ n 2 ) ≥ I α + I 1-α
which contradicts Lemma 2.3, since I < 0. Hence, for n large enough, there exists

{y n } n ∈ R 3 such that ∀ε > 0, ∃R < +∞, B(yn,R) |ϕ n | 2 ≥ 1 -ε.
We denote by φn (•) = ϕ n (•+y n ). Since { φn } n is bounded in H 1 , { φn } n converges weakly in H 1 , almost everywhere on R 3 and in L p loc for 2 ≤ p < 6 to some φ. In particular, as a consequence of weak convergence in H 1 , σ • ∇ φn converges weakly to σ • ∇ φ in L 2 . Moreover, thanks to the concentration-compactness argument, { φn } n converges strongly in L 2 and in L p for 2 ≤ p < 6.

Lemma 2.5. Let {f n } n and {g n } n be two sequences of functions such that f n : R 3 → R + , g n : R 3 → C 2 , f n converges to f a.e., g n converges weakly to g in L 2 and there exists a constant C, that does not depend on n, such that

R 3 f n |g n | 2 dx ≤ C. Then R 3 f |g| 2 dx ≤ lim inf n→+∞ R 3 f n |g n | 2 dx.
Proof. Given a function h : R 3 → R + , let T k be the function defined by

T k (h)(x) = h(x) if h(x) ≤ k k if h(x) > k for all k ∈ [0, ∞).
Hence, the following properties are satisfied for all k ∈ [0, ∞):

T k (f n ) -→ n T k (f ) a.e. in R 3 , (2.18)
T k (f n )|g| 2 -→ n T k (f )|g| 2 in L 1 , (2.19)
T k (f n )g ⇀ n T k (f )g in L 2 , (2.20)
T k (f n )g L 2 -→ n T k (f )g L 2 , (2.21)
where to obtain (2.19) and (2.21), we use Lebesgue's dominated convergence theorem. Moreover, as a consequence of (2.20) and (2.21), we have

T k (f n )g -→ n T k (f )g in L 2 . (2.22)
Next, we have

0 ≤ lim inf n→+∞ R 3 T k (f n )|g n -g| 2 dx = lim inf n→+∞ R 3 T k (f n )|g n | 2 dx + lim inf n→+∞ R 3 T k (f n )|g| 2 dx -lim inf n→+∞ R 3 T k (f n ) g n • g dx + R 3 T k (f n ) g n • g dx = lim inf n→+∞ R 3 T k (f n )|g n | 2 dx + R 3 T k (f )|g| 2 dx -2 R 3 T k (f )|g| 2 dx
thanks to (2.19), (2.22) and the fact that g n converges weakly to g in L 2 . As a consequence,

R 3 T k (f )|g| 2 dx ≤ lim inf n→+∞ R 3 T k (f n )|g n | 2 dx (2.23) Since lim inf n→+∞ R 3 T k (f n )|g n | 2 dx ≤ lim inf n→+∞ R 3 f n |g n | 2 dx ≤ C,
we can pass to the limit for k that goes to +∞ in (2.23) and we obtain

R 3 f |g| 2 dx ≤ lim inf n→+∞ R 3 f n |g n | 2 dx.
By applying Lemma (2.5) to

f n = 1 (1-| φn| 2 ) + and g n = |σ • ∇ φn |, we obtain R 3 |σ • ∇ φ| 2 (1 -| φ| 2 ) + dx ≤ lim inf n→+∞ R 3 |σ • ∇ φn | 2 (1 -| φn | 2 ) + dx.
Hence, φ ∈ X, R 3 | φ| 2 dx = 1, and

E( φ) ≤ lim inf n→+∞ E( φn ) ≤ E( φ).
As a conclusion, the minimum of I is achieved by φ. Finally, it remains to prove that there exists a 0 > 0 such that for all a > a 0 we have I < 0.

It is clear that I < 0 for a large enough. Since I is non-increasing with respect to a, we may denote by a 0 the least positive constant such that I < 0 for a > a 0 . We have to prove that a 0 > 0 or in other words I = 0 for a small enough. Using Sobolev and Hölder inequalities, we find, for ϕ ∈ X such that R 3 |ϕ| 2 dx = 1,

E(ϕ) ≥ 1 S 2 R 3 |ϕ| 6 dx 1/3 - a 2 R 3 |ϕ| 6 dx 1/3 .
Hence, if a ≤ 2 S 2 , I = 0. This implies a 0 > 2 S 2 . According to [START_REF] Talenti | Best constant in Sobolev inequality[END_REF] the best constant for the Sobolev inequality

u L q (R m ) ≤ C ∇u L p (R m )
with 1 < p < m and q = mp (m-p) is given by

C = π -1/2 m -1/p p -1 m -p 1-1/p Γ(1 + m/2)Γ(m) Γ(m/p)Γ(1 + m -m/p) 1/m . In particular, S = 1 √ 3π 4 √ π 1/3
, and 2 S 2 = 3π 4/3 2 1/3 ≈ 10.96. To obtain an upper estimate for a 0 , we consider the following test function φ

(x) = fR (|x|) 0 where fR (|x|) = f |x| R , f (|x|) =    cos(|x|) |x| ≤ π 2 0 |x| > π 2 and R ∈ (0, 1) is such that | fR | 2 dx = 1. This implies R = 2 π 2/3 3 π 2 -6 1/3 .
Next, we denote by ā the positive constant such that E(ϕ) = 0. By definition,

ā = 2 R 3 |σ•∇ φ| 2 (1-| φ| 2 )+ dx | φ| 4 = 2 R 3 |∇ fR| 2 1-| fR| 2 dx | fR | 4 ,
and, by a straightforward calculation, we obtain

R 3 |∇ fR | 2 1 -| fR | 2 dx = π 4 6 R = π 10/3 3 2/3 (2(π 2 -6)) 1/3 , | fR | 4 = π 2 (2π 2 -15) 32 R 3 = 3(2π 2 -15) 8(π 2 -6) .
As a consequence, ā = 8π 10/3 2 3 (π 2 -6)

2/3 3(2π 2 -15) ≈ 48.06
Since the energy functional E is decreasing in a, if a > ā then I ≤ E( φ) < 0. As a conclusion, a 0 ≤ ā + ε for all ε > 0.

Appendix A

A.1. We begin this section by proving that if (ϕ, χ) a solution of (1.7) with ϕ ∈ H 1 (R 3 ) of the form (1.12), then |ϕ| 2 ≤ 1 a.e. in R 3 . As we saw before, ϕ is a solution of

-σ • ∇ σ • ∇ϕ 1 -|ϕ| 2 + |σ • ∇ϕ| 2 (1 -|ϕ| 2 ) 2 ϕ -a|ϕ| 2 ϕ + bϕ = 0, (A.1) or equivalently, ∆ϕ |ϕ| 2 -1 - |σ • ∇ϕ| 2 (|ϕ| 2 -1) 2 ϕ -a|ϕ| 2 ϕ + bϕ = 0 , or still, ∆ϕ - |∇ϕ| 2 |ϕ| 2 -1 ϕ -(a|ϕ| 2 ϕ -bϕ)(|ϕ| 2 -1) = 0 ,
because for functions ϕ of the form (1.12), |σ • ∇ϕ| 2 = |∇ϕ| 2 and σ • (∇ϕ ∧ ∇ϕ) = 0 a. e.

For any K > 1, we define the truncation function T K (s) by T K (s) = s if 1 < s < K, and T K (s) = 0 otherwise. Multiplying the above equation by ϕ T

K (|ϕ| 2 ) ∈ L 2 (R 3 ), we obtain - R 3 |∇ϕ| 2 T K (|ϕ| 2 ) - R 3 (∇ϕ • ϕ) ∇T K (|ϕ| 2 ) - R 3 |∇ϕ| 2 |ϕ| 2 -1 |ϕ| 2 T K (|ϕ| 2 ) - R 3 (a|ϕ| 2 -b)(|ϕ| 2 -1)|ϕ| 2 T K (|ϕ| 2 ) = 0. (A.2)
Moreover, for all K > 1,

∇T K (|ϕ| 2 ) = 2ϕ • ∇ϕ 1 < |ϕ| 2 < K 0 |ϕ| 2 ≤ 1 or |ϕ| 2 ≥ K .
Therefore, if a -b > 0 the l.h.s of (A.2) is negative and this implies that either |ϕ| 2 ≤ 1 or |ϕ| 2 ≥ K a.e. As a conclusion, taking the limit K → +∞, if a -b > 0 then any solution ϕ of (A.1) of the form (1.12) satisfies |ϕ| (f ′ (r)) 2 1-f 2 (r) ≤ C, for some constant C > 0. Next, for all integers n > 0, define the rescaled functions ξ n (x) := n 3/2 ξ(nx). This change of variables leaves invariant the L 2 (R 3 ) norm. Then for n large, consider the function g n (x) := max R 3 {ξ n , f } . Note that the measure of the set {x ∈ R 3 ; g n = ξ n } tends to 0 as n goes to +∞. This function satisfies

2 ≤A. 2 .ln 2 ,

 222 1 a.e. in R 3 , and in the equation (A.1) we can replace the term (1 -|ϕ| 2 ) by (1 -|ϕ| 2 ) + without changing its solution set. The same happens for solutions of the form (1.8). Let us next prove that the functional F , defined by (1.11), is not bounded from below. Consider the function ξ introduced in the proof of Lemma 2.4. Let us denote A := R 3 |ξ(x)| 2 dx.Then, let us define the radially symmetric functionf (r) = ewhere ξ(|x|) = ξ(x) for all x, and take a := R 3 f (|x|) 2 dx. Note that supp(f ) ⊂

2 R 3 |ξ| 4 dx + R n R 3 (f ′ (r)) 2 1 3 n 2 R 3 f 2 >

 234323232 R 3 |g n (x)| 2 dx = A + a + o(1), as n goes to +∞. In order to normalize it in the L 2 norm, let us finally define the rescaled functiong R n (x) := g n x R , R > 0 and choose R n such that R 3 |g Rn n (x)| 2 dx = 1. As n goes to +∞, R n → R := (A + a) -1/3 > 0. We compute now the energy F of the vector function ϕ Rn n -f 2 (r) dx -a R (x) 4 dx + o(n 3 ) ,because whenever ξ Rn n ≥ f Rn , ξ Rn n 1 and because the sequence {R n } n is bounded. This clearly shows that F is unbounded from below.

Ceremade, Université Paris-Dauphine, Place de Lattre de Tassigny, F-75775 Paris Cédex 16, France E-mail address: esteban@ceremade.dauphine.fr

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France E-mail address: rotanodari@ann.jussieu.fr

Acknowledgment. This work was partially supported by the Grant ANR-10-BLAN 0101 (NONAP) of the French Ministry of Research. The authors would like to thank Éric Séré for useful comments and for the proof of Lemma 2.1. They also thank the Isaac Newton Institute, where this paper was finalized.