Alberto Benegiamo
email: alberto.benegiamo@eurecom.fr

Patrick Loiseau
email: patrick.loiseau@eurecom.fr

Giovanni Neglia
email: giovanni.neglia@inria.fr.

Dissecting Demand Response Mechanisms: the Role of Consumption Forecasts and Personalized Offers *

Demand-Response (DR) programs, whereby users of an electricity network are encouraged by economic incentives to re-arrange their consumption in order to reduce production costs, are envisioned to be a key feature of the smart grid paradigm. Several recent works proposed DR mechanisms and used analytical models to derive optimal incentives. Most of these works, however, rely on a macroscopic description of the population that does not model individual choices of users.

In this paper, we conduct a detailed analysis of those models and we argue that the macroscopic descriptions hide important assumptions that can jeopardize the mechanisms' implementation (such as the ability to make personalized offers and to perfectly estimate the demand that is moved from a timeslot to another). Then, we start from a microscopic description that explicitly models each user's decision. We introduce four DR mechanisms with various assumptions on the provider's capabilities. Contrarily to previous studies, we find that the optimization problems that result from our mechanisms are complex and can be solved numerically only through a heuristic. We present numerical simulations that compare the different mechanisms and their sensitivity to forecast errors. At a high level, our results show that the performance of DR mechanisms under reasonable assumptions on the provider's capabilities are significantly lower than

Introduction

Demand Response (DR hereinafter) programs are envisioned to be a key feature of the Smart Grid paradigm [START_REF] Albadi | A summary of demand response in electricity markets[END_REF]. By means of economic incentives (discounts or penalties), DR schemes encourage users to rearrange their consumption in response to the network state, thus mitigating the grid overload and driving wholesale prices down.

Several analytical models are available in the literature, which describe and quantify the effects of DR mechanisms. Whatever their specifics are, these schemes need to model how users react to the incentives. Ideally the models should capture the most realistic features of a practical DR mechanism while maintaining tractability.

Among these contributions, the authors of [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] study how an energy provider should select time-dependent discounts to minimize its production costs. They assume that the percentage of users who shift their consumption from slot i to slot j is a decreasing function of the temporal distance between slots i and j and a concave and increasing function of the discount offered in slot j (R j ), independent from discounts in other slots. The same user's model as in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] is adopted also in [START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF], where the optimization problem is extended in order to account for battery storages and distributed renewable sources available into a specific microgrid. Authors of [START_REF] Subramanian | Dynamic Price Optimization Models for Managing Time-of-Day Electricity Usage[END_REF] propose a day ahead pricing scheme which maximizes the provider's profitability and capacity utilization. Users are assumed to reschedule their consumption by comparing the utility v i they get by scheduling a task in each timeslot i; therefore they allocate their consumption proportionally to these utilities, i.e., they consume a fraction v i T j=1 v j of their total energy demand in timeslot i. The resulting optimization problem is non convex but some relaxation techniques are introduced, which allow one to calculate a solution within a reasonable amount of time. In [START_REF] Yang | Quantifying the benefits to consumers for demand response with a statistical elasticity model[END_REF], a more realistic model is proposed where each user first calculates the welfare (defined as utility minus timedependent cost) she gets from consuming electricity in each of the possible timeslots, and then allocates all the consumption to the slot returning the largest welfare. As we show below (see Sec. 4.4) this model can lead to a much more complex optimization problem than the one presented in [START_REF] Yang | Quantifying the benefits to consumers for demand response with a statistical elasticity model[END_REF]. Finally, the authors of [START_REF] Song | Demand Side Management in Smart Grids Using a Repeated Game Framework[END_REF] propose a full-fledged game theoretical model, but their results hold only if users experience a large number of interactions without any change in the system.

We claim that these studies rely on too strong assumptions, which jeopardize their usability for practical purposes. Interestingly, we observe that the assumptions are sometimes hidden in the macroscopic models the pa-pers start from. In particular in this paper we focus on [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] and show that its model requires personalized offers and a very precise forecast of the baseline consumption of each user. The implementation of these features may require potentially significant costs in terms of communication, measurement and computation infrastructure. Besides highlighting these implicit requirements in the analytical framework in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] (and then also in [START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF]), we explore their potentials considering four DR mechanisms with different levels of complexity:

1. the base mechanism corresponds to an optimization problem similar to the one considered in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF], it requires personalized offers and individual consumption forecasts; the energy production cost is optimized over the discount values, each of which is offered to a given fraction of the population, 2. the optimized mechanism takes full advantage of personalized offers and consumption forecasts by minimizing the cost over both the discount values and the population fractions to which the discounts are offered, 3. the robust mechanism relies on personalized offers, but does not need individual consumption forecasts, 4. finally the broadcast mechanism (analogous to that in [START_REF] Yang | Quantifying the benefits to consumers for demand response with a statistical elasticity model[END_REF]) needs neither of the two features.

Interestingly, contrarily to prior studies, we find that the cost-minimization problems resulting from our DR mechanisms are not convex (even for the base mechanism). Nevertheless, simple heuristics can identify (potential) minima in a reasonable amount of time in realistic scenarios. Then, our numerical results show that the simpler robust and broadcast mechanisms achieve significantly lower cost reductions than the optimized mechanism, which is difficult to implement, but that the gap reduces when the population's flexibility increases. The paper is organized as follows. In Sec. 2 we discuss how the macroscopic models considered in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF][START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF][START_REF] Subramanian | Dynamic Price Optimization Models for Managing Time-of-Day Electricity Usage[END_REF] hide some implicit assumptions about the user rationality or about the interactions between the provider and the user. We define our microscopic model in Sec. 3 and then describe different DR mechanisms and their corresponding optimization problems in Sec. 4. Finally, we evaluate their performance numerically in a realistic scenario in Sec. 5.

Due to space constraints proofs, examples and some additional numerical results are in the companion technical report [START_REF] Benegiamo | Dissecting Demand Response Mechanisms: the Role of Consumption Forecasts and Personalized Offers[END_REF].

In this section, we describe in more detail the macroscopic models proposed in the literature for day-ahead price optimization. Consider a finite time horizon discretized in a set T of N timeslots and a large population S of users. The baseline aggregate energy consumption in slot j is denoted by E 0 j .

The energy provider charges a flat rate B, but it can offer discount rates to incentivize the users to move some of their consumption so as to reduce the energy production cost. Due to consumption shifts, the actual aggregate consumption in time slot j is E 1 j . Observe that a usual assumption in the literature (including the papers mentioned above) is that the introduction of a DR scheme neither reduces nor increases users' demand; it merely rearranges users' consumption in a more cost effective way, so that

N j=1 E 0 j = N j=1 E 1 j . (1)
We denote the amount of consumption shifted from slot j to slot i = j as E j→i , and the amount of consumption the users refuse to shift away from j as E j→j . Then we have

E 1 i = E 0 i + N z=1 E z→i - N k=1 E i→k .
We now start to further detail the model considering some specific assumptions made in previous works. In [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] and [START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF], the electricity provider offers an energy price discount R i ≥ 0 in each slot i. The users are assumed to react to these incentives by shifting a fraction of their baseline consumption from slot j to slot i (|j -i| slots away) according to the following formula:

E j→i = E 0 j S j (R i , |j -i|). (2)
S j (R i , |j -i|) is called the aggregate sensitivity function and is increasing in the discount R i and decreasing in the temporal shift |j -i|, in order to take into account the user discomfort. The provider selects the vector of discounts R in order to minimize its total cost, equal to the sum of the electricity generation costs and the loss of revenues due to the discounts. In particular the optimization problem considered in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] is the following:

min R i j =i R i E j→i + i c i E 1 i (3) s.t. 0 ≤ R i ≤ B ∀i = 1, . . . N, (4)
where c i (•) is the cost of electricity production in slot i. Eq. ( 4) guarantees that discounts R are non negative and smaller than the flat rate B, so that the money stream goes toward the provider.

As it often happens, the devil is hidden in the details, and in this case in Eqs. ( 2) and (3). Our first remark is that the cost of lost revenues i j =i R i E j→i in Eq. ( 3) implicitly assumes the possibility to reward only the consumption actually shifted from j to i, i.e., E j→i , but this quantity cannot be directly measured. The actual consumption E 1 i can be measured, and then E j→i can be quantified provided that we have good estimates of the sensitivity function S j (R i , |j -i|) and of the baseline consumption E 0 i . Let us assume for a moment that S j (R i , |j -i|) is known from historical data and that the aggregate baseline consumption may be predicted with a reasonably high level of accuracy on a large set of users. Then it seems possible to solve the macroscopic problem in Eqs. ( 3) and ( 4), but we need to consider also what should happen at the microscopic scale. While the estimates for the aggregate baseline consumption can be adequately precise, finally the billing is done at the user's granularity and each user expects to receive the price discount corresponding to the energy consumption she actually moved. If the energy bill's reduction does not correspond to her forecast, the user is likely to opt out of the program (in particular if she has experienced underpayments) or to reduce her efforts and milk occasional discounts. It appears then that Eq. ( 3) implicitly requires very precise predictions of individual consumptions.

We now observe that the form of the sensitivity function S j (R i , |j -i|) in Eq. ( 2) indicates that the amount of energy shifted from j to i depends on the discount R i but not on the other discounts. We can then ask ourselves which individual decisions may lead to this aggregate behavior, an issue ignored both in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] and [START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF]. As long as a rational individual is offered two different discounts R i and R k , it seems natural that her decision to move some consumption from j to i or from j to k or to keep it in j will take into account both the discounts. To stress the point, consider a case when both S j (R i , |j -i|) and S j (R k , |j -k|) are positive, but moving the consumption from i to k is both less inconvenient (i.e., |j -k| < |j -i|) and more rewarding (i.e., R k > R i ). There is then no reason why the user would move consumption to i. The conclusion is the same for all the users and then we should have E j→i = 0 at the aggregate level, in contradiction with Eq. ( 2). We can then conclude that the expression of the sensitivity function in Eq. ( 2) is not suited to model the situation when a user is offered two or more rewards, but it can capture the case when the user decides between moving from j to i in exchange of a discount R i or staying in j. If every user is offered a single discount to move to a given slot, but different users can receive different offers, then Eq. ( 2) can reasonably describe the macroscopic effect of such personalized offers. The details are described in Sec. 4.1, here we only highlight that Eq. ( 2) requires then that the electricity provider i) calculates an offer for each user, ii) communicates individually to the user, iii) considers the individual offer when billing the user. This is clearly more demanding than simply advertising to the whole population the same set of discounts.

We observe that the equivalent sensitivity function considered in [START_REF] Subramanian | Dynamic Price Optimization Models for Managing Time-of-Day Electricity Usage[END_REF] poses similar problems. Using our notation, we have

E j→i = v i k∈T v k E 0 j
, where v i is the net utility a user gets by consuming electricity in slot i and can be a function of the timeslot itself and of the discount R i . This formula tries to capture the effect of the whole set of discounts, but it is not clear again what is the underlying user's model: if slot i has a larger utility than slot k (v i > v k ), why should the user consume in k?

Starting from a Microscopic Model

In the previous section we made the point that, while aggregate population models may be convenient, it is necessary to explicitly consider the microscopic level: how the user takes the decisions and how the provider and the user are supposed to interact. In this paper we follow the opposite path in comparison to the existing works mentioned: we move from the microscopic level to the macroscopic one. In particular, in this section, we start from a clear model of rationality for the single user and then move to describe how aggregate quantities can be derived.

Each user u has a baseline energy consumption {e 0u j } j=1,...N that leads to the aggregate baseline consumption E 0 j = u∈S e 0u j for j = 1, . . . N . We assume in what follows that users are homogeneous, i.e., ∀ u, e 0u j = e 0 j j = 1, . . . N.

In [START_REF] Benegiamo | Dissecting Demand Response Mechanisms: the Role of Consumption Forecasts and Personalized Offers[END_REF] we show how the DR mechanisms perform when this assumption does not hold. User u is characterized by a private type D u = {d u j→i } j,i=1,...N where d u j→i indicates the discomfort due to shifting one unit of consumption from timeslot j to timeslot i. We assume that discomforts are expressed in monetary units; and that, ∀u ∈ S,

d u j→j = 0 and d u j→i > 0, ∀j, i = j, (6)
i.e., there is a strictly positive discomfort if and only if consumption is shifted from its original timeslot. The provider does not know the private type D u of each user u: from its point of view, each discomfort d u j→i is drawn from a known, continuous distribution F j→i on [0, α j,i ] (where possibly α j,i = +∞). Discomforts of distinct users are independent but note that we do not assume that, for a given user, the discomforts {d u j→i } j,i=1,...N are mutually independent.

Rational Users

We assume that a user simply chooses the option that maximizes her utility. In particular let T u j be a set of timeslots the user could move the baseline consumption e 0u j to in exchange for different discounts R u j = {R u j→k ≥ 0, k ∈ T u j }. The set pair (T u j , R u j ) defines the offer user u receives for timeslot j. The set of options includes the possibility to keep the consumption in j, i.e., j ∈ T u j . A rational user maximizes her utility by scheduling her consumption e 0u j to a timeslot i * ∈ arg max

k∈T u j R u j→k -d u j→k . (7)
We assume that if two or more timeslots are equally palatable, the whole consumption is shifted to only one of them, picked at random with equal probability.

Aggregation

We observe that the quantities d u j→k in Eq. ( 7) are random variables, then two different users could take different decisions while confronted with the same offers. The aggregate consumption E 1 i , for i ∈ T , would then be a random variable. Here we assume (as it is implicit in the other works) that we always work with large sets of the population so that the variability can be neglected by approximating actual random quantities with their expected values. In particular, if a subset Q containing a fraction q of the population receives an offer (T j , R j ), the corresponding consumption shifted from j to a time slot i, denoted as E Q j→i will be

E Q j→i = qE 0 j Prob i ∈ arg max k∈T j R u j→k -d u j→k , (8)
if the probability that a user has two or more equally palatable timeslots is zero. When discomforts are continuous random variables (as we consider in this paper), this is always the case if each user receives only one offer (the first three mechanisms introduced below) or if the discomforts {d u j→i } j,i=1,...N are mutually independent. In Sec. 4.4, we discuss how Eq. ( 8) should be modified if this probability is not zero. We denote Prob i ∈ arg max k∈T j R j→k -d u j→k simply as P j→i (R u j , T j ).

DR mechanisms

Under different assumptions on the provider's capabilities, we introduce different demand response mechanisms based on the microscopic model above, which are therefore practically implementable. We introduce and study the corresponding optimization problems.

We start by the base mechanism that leads to the same aggregate optimization problem considered in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF][START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF].

Base mechanism

This mechanism requires that the energy provider can manage personalized offers to its customers and moreover that it has perfect knowledge (or very precise estimates) of the baseline consumption of each user.

The population is segmented into N 2 disjoint subsets Q j→i , for j, i ∈ T , respectively including a fixed fraction q j→i of the population. Each user in Q j→i is simply offered to move her baseline consumption in slot j (e 0 j ) to slot i in exchange for a price discount R i .

The total consumption that is shifted from j to i is then

E j→i = q j→i E 0 j Prob R i -d u j→i > 0
as it can be obtained from Eq. ( 8), taking into account that in this case T j = {j, i} and R j -d u j→j = 0. We observe that the probability appearing on the right-hand side only depends on the reward R i and on the random variable d u j→i . If the discomfort is only a function of the temporal distance |j -i|, then the sensitivity function (the ratio of people who move from j to i) has the same properties than in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF], in particular:

S j (R i , |i -j|) = q j→i P j→i (R i ),
where for P j→i (•) we have made explicit the only variable it depends from.

As we discussed in Sec. 2, because the provider knows exactly the consumption shifted from each user, it can formulate the optimization problem [START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF][START_REF] Subramanian | Dynamic Price Optimization Models for Managing Time-of-Day Electricity Usage[END_REF]. In [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] it is stated that the problem is convex if i) the productions costs c j (•) are continuous piecewise linear and increasing and ii) the discomfort distributions F j→i (•) are continuous and concave. We show in [START_REF] Benegiamo | Dissecting Demand Response Mechanisms: the Role of Consumption Forecasts and Personalized Offers[END_REF] that this is not the case by providing a counterexample. Stronger hypotheses are required for the problem to be concave, as for example the linearity of the discomfort functions.

In particular in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] the numerical evaluation considers

S j (R i , |i -j|) = 1 N k=1 1 (|k-j|+1) R i B • (|i -j| + 1)
, that leads us to consider

q j→i = 1 |i-j|+1 N k=0 1 |k-j|+1 , P j→i (R i ) = R i B . (9)
This particular expression for P j→i can be obtained if d u j→i is a uniform random variable with support in [0, B]. The numerical results for the base mechanism in Sec. 5 are obtained considering the same expression for the fractions q j→i . Due to the non-convexity of the optimization problem (3-4) we cannot use one of the classic algorithms for convex optimization. For the results shown in section 5 we have adopted instead a multi-start approach: we have generated random starting points uniformly distributed in the problem domain and we have run per each point a descendent algorithm which converged on a local minimum; the optimal offers are therefore those returning the smallest cost among these minimizers. This approach does not guarantee convergence to the global optimum but its reliability can be improved by increasing the number of starting points.

Optimized Mechanism

We have now understood which DR mechanism can lead to the optimization problem ( 3), but now that we look at its implementation at the microscopic level and the need for personalized offers, some specifics of the base mechanism look arbitrary and unjustified. For example, given that discounts are not broadcast but each user receives an individual offer, why should the discounts offered to the two disjoint sets of users Q j→i and Q k→i be equal to the same value R i ? It is clear that the energy provider can further reduce the cost if it can independently choose R j→i and R k→i . Moreover, there is no reason to think that the size of the sets {Q j→i } should be fixed, the fractions {q j→i } can also be optimization variables.

We allow the provider to take advantage of these additional degrees of freedom that-we repeat-do not impose any additional requirement to the system. We call this new DR mechanism optimized. The load E j→i (R j→i , q j→i ) rescheduled from j to i is now E j→i (R j→i , q j→i ) = q j→i P j→i (R j→i )E 0 j and the cost minimization problem becomes: min

R,q cost opt. (R, q) = i z =i R z→i E z→i + i c i E 1 i (10) s.t. 0 ≤ R z→i ≤ B ∀z, i = 1, . . . N (11) 0 ≤ q z→i ≤ 1, z, i = 1, . . . N (12)
i q z→i ≤ 1, ∀z = 1 . . . N. (13)
Eq. ( 11) guarantees that discounts R are non negative and smaller than the flat rate B, Eq. ( 13) is a consequence of the fact that each user receives at most one offer for its baseline consumption in a given slot.

The optimization problem (10-13) can be solved with the same heuristic proposed for problem (3-4).

Robust Mechanism

The optimization problems (3-4) and (10-13) assume that the provider has perfect knowledge of each user's baseline consumption, so that it can correctly identify the consumption shifted and reduce accordingly the energy bill. This assumption is probably unrealistic. If the provider does not have such capability, then it can offer the user a discount for all the consumption in a given timeslot i and not just for the consumption moved to i. The population is then divided into N subsets Q i , each containing a fraction q i of the users. All users in Q i receive one and only one offer: they are encouraged to shift their consumption from any timeslot in the time horizon to timeslot i and they get the discount R i for all the electricity consumed in i, including the one originally in i.

We call this scheme robust, because it does not rely on estimates of individual consumption. It is clearly simpler than the previous two, because the provider needs only to measure the amount of consumption in i for the users who got the offer and to bill them accordingly.

The load E j→i (R i , q i ) shifted from j to i is E j→i (R i , q i ) = q i P j→i (R i )E 0 j . Note that users in Q i have no interest to move their baseline consumption away from i, then E i→i = q i E 0 i . The robust mechanisms leads to the following optimization problem: min

R,q cost rob. (R, q) = i,z R i E z→i + i c i E 1 i (14) s.t. 0 ≤ R i ≤ B i = 1, . . . , N (15)
0 ≤ q i ≤ 1, i = 1, . . . , N (16)
N i=1 q i ≤ 1. (17)
Note that in Eq. ( 14) the first sum includes also E i→i because all the final consumption in i from the users in Q i is paid at a discounted price. The term does not appear in Eq. ( 3) and Eq. (10).

The optimization problem (14-17) can be solved with the same heuristic proposed for problem [START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF][START_REF] Subramanian | Dynamic Price Optimization Models for Managing Time-of-Day Electricity Usage[END_REF].

Broadcast Mechanism

In the three mechanisms introduced above, the provider makes personalized offers to users in selected fractions of the population. This may not always be possible (due to the complexity it introduces for instance in billing) or desirable (for perceived fairness issues). Our last mechanism, which is the simplest (in its definition), does not assume personalized offers. The provider selects a single vector R of discounts for every time slot and broadcasts these discounts to all users (hence the name broadcast mechanism). Users then re-arrange their demand and pay the discounted price for their demand in each slot (hence this mechanism also does not rely on the need to estimate shifted demand).

As explained in Sec. 3.1, each individual user moves her demand from slot j to a slot (potentially j itself) that maximizes her net utility (discount minus discomfort). Recall that if several slots give equal net utility, the user chooses one of them randomly.

Until now, we have not made any assumption on the possible correlation of a given user's discomforts. This is because, in the previous three mechanisms, each user was receiving only one offer.In the broadcast mechanism, each user has several offers to compare to decide on his new demand schedule, we therefore need to describe the discomfort correlations.

Let us consider now the particular case when two slots, say h and k may appear equally attractive to a user, i.e. R h -

d j→h = R k -d j→k .
If we assumed that, for each user, the discomforts {d j→i } {i,j=1,••• ,N } were mutually independent, this event would have probability zero according to our assumption on F i→j , and therefore it would not appear at the aggregate level. As a result, the aggregate demand moved from j to i would be E j→i (R) = P j→i (R)E 0 j , where

P j→i (R) = Pr R i -d j→i ≥ max k =i {R k -d j→k } . (18)
However, rather than making the above independence assumption, we prefer to assume that the discomforts have the form d j→i = β j |i -j| t j , where t j is a constant independent of the user and β j is a random variable with concave Cumulative Distribution Function (CDF) F j (•). This model describes a symmetric delay sensitivity of users (users are indifferent between moving two hours earlier or two hours later) while keeping the flexibility of users having a different flexibility of demand of different times (since β and t are indexed by the origin timeslot j); but it also introduces correlations between the discomforts of a user. As a result, the fraction of demand shifting from j to i is

P j→i (R) = Pr (R i -d j→i ≥ max k =i {R k -d j→k }) 1 + 1 R 2j-i =R i , (19)
rather than (18). The denominator in (19) accounts for cases when a slot other than i (which has to be 2j -i) gives equal net utility for all users.

The broadcast mechanism then leads to the following optimization problem: min

R cost brd. (R) = z i R i E z→i + i c i E 1 i (20) s.t. 0 ≤ R i ≤ B ∀i = 1, . . . N. (21)
Unfortunately, due to indicator function in Eq. ( 19), the cost function (20) of the broadcast mechanism is not continuous, even in very simple scenarios with continuous production costs [START_REF] Benegiamo | Dissecting Demand Response Mechanisms: the Role of Consumption Forecasts and Personalized Offers[END_REF]. Discontinuity arises also in the macroscopic model in [START_REF] Yang | Quantifying the benefits to consumers for demand response with a statistical elasticity model[END_REF], but it seems to have been ignored.

In practice, we solve problem (20-21) using the same heuristic proposed for the previous problems, but we work on a continuous and smooth approximation of the cost function.

Ranking DR mechanisms

In [START_REF] Benegiamo | Dissecting Demand Response Mechanisms: the Role of Consumption Forecasts and Personalized Offers[END_REF] we prove that the optimized mechanism always performs better than the base and robust mechanisms, but the ranking cannot be extended further.

Numerical Results

In this section we evaluate the performance of the different DR mechanisms in the realistic scenario considered in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] and based on energy data about the Ontario province in Canada. In particular, the baseline consumption E 0 and the cost functions are estimated from the IESO energy portfolio [START_REF] Ieso Ontario | Market data[END_REF], consisting of nuclear plans, hydro gas powered stations and renewable and from typical costs associated to these energy sources. We considered the flat rate B = 110$/MWh and the timeslot-independent piecewise-linear cost function c(•) with derivative: c (E) = $10 for E ≤ C 1 , c (E) = $72.46 for E ∈ (C 1 , C 2 ) and c (E) = $91 for E ≥ C 2 , where C 1 = 16.3 GWh and C 2 = 17.9 GWh represent respectively the base to intermediate load capacity and intermediate to peak load capacity. We assume that discomforts take the form d j→i = β j |i -j|, where β j is an exponential random variable with cdf

F j (R; β) = 1 -e β µ
. µ is a parameter representing the population's flexibility. The larger it is, the smaller (in a stochastic order sense) the discomfort of the users to shift their consumption.

In Fig. 1, we report the cost savings of the DR schemes, normalized to the initial cost, for four different values of the µ parameter: 1 10 , 1 6 , 1 3 , 1. The dashed line represents the saving which could be achieved if users' demand could be rearranged at the provider's will without providing any discount (we indicate it as the dictatorial solution). Consistently with the results in Sec. 4.5, the optimized mechanism returns larger savings than the robust and the base ones. Interestingly, the robust mechanism performs consistently better than the base one despite the fact that it does not require the ability to estimate the demand shifted and it therefore "wastes" some discount by giving it to demand that was already scheduled in a given timeslot in the baseline demand. Moreover, as the population flexibility increases, the savings gap between the optimized scheme and the robust mechanism reduces, the latter being effectively close to exploiting all the population's flexibility. In Fig. 2, we focus on the case µ = 1 3 and analyze the components of the cost for each DR mechanism. Fig. 2 confirms that the optimized scheme provides the largest savings as it can minimize the production cost while paying the smallest amount of discounts. We indicate with wasted discounts the amount of discounts paid to consumption that would in any case have been scheduled in that timeslot. The base and optimized mechanisms do not waste any discount, while the robust mechanism and the broadcast scheme do, as they provide the discount R i to all the electricity consumed in i, including the part of E 0 i that remains in i.

Conclusions

In this paper, we have shown that macroscopic descriptions of DR mechanisms can hide important assumptions that can jeopardize the mechanisms' implementation. For this reason, our proposal moved from a microscopic description that explicitly models each user's decision. We have then introduced four DR mechanisms with various assumptions on the provider's capabilities. Interestingly, contrarily to previous studies, we find that the optimization problems that result from our mechanisms are complex and can be solved numerically only through a heuristic. Moreover, our results show that the performance of DR mechanisms under reasonable assumptions on the provider's capabilities are significantly lower than those suggested by previous studies, but that the gap reduces when the population's flexibility increases.

Figure 1 :

 1 Figure 1: Cost savings normalized to the initial cost, for various flexibility parameters µ.

Figure 2 :

 2 Figure 2: Analysis of the components of the cost savings. All the quantities are normalized to the initial cost.

Acknowledgements

This work was partly funded by the French Government (National Research Agency, ANR) through the "Investments for the Future" Program reference #ANR-11-LABX-0031-01.


Graziano Crasta

Virginia De Cicco
email: virginia.decicco@sbai.uniroma1.it

AND Guido De Philippis
email: guido.de-philippis@ens-lyon.fr

Francesco Ghiraldin
email: francesco.ghiraldin@math.uzh.ch

STRUCTURE OF SOLUTIONS OF MULTIDIMENSIONAL CONSERVATION LAWS WITH DISCONTINUOUS FLUX AND APPLICATIONS TO UNIQUENESS

Keywords: Mathematics Subject Classification, 35L65, Key words and phrases, conservation laws with discontinuous flux, uniqueness

We investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux. We show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this to prove the validity of a generalized Kato inequality for any pair of solutions. Applications to uniqueness of solutions are then given.

Introduction

Aim of this paper is to study the structure of solutions of conservation laws with discontinuous flux of the form [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF] div z A(z, u) = 0, in order to establish a general framework for studying uniqueness of solutions of the Cauchy problem associated to the evolutionary equation 1

(2) u t + div x F (t, x, u) = 0, in (0, +∞) × R N .

Here A (respectively F ) is discontinuous in its first variable z (respectively (t, x)). More precisely we will assume that A(z,

•) ∈ C 1 (R, R n ), A(•, v) ∈ SBV (R n , R n )
where SBV is the space of special function of bounded variation, see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]Chapter 4], and that A satisfies some mild structural assumptions listed in Section 2.

In recent years, the study of conservation laws with discontinuous flux has attracted the attention of many authors since they naturally arise in many models, see [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Andreianov | Entropy conditions for scalar conservation laws with discontinuous flux revisited[END_REF][START_REF] Audusse | Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies[END_REF][START_REF] Coclite | Conservation laws with time dependent discontinuous coefficients[END_REF][START_REF] Diehl | A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients[END_REF][START_REF] Garavello | Conservation laws with discontinuous flux[END_REF][START_REF] Karlsen | L 1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients[END_REF][START_REF] Mitrovic | New entropy conditions for scalar conservation laws with discontinuous flux[END_REF][START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF] and the references therein.

Even in the case the flux F is smooth it is well known that the Cauchy problem associated to [START_REF] Ambrosio | Lecture notes on optimal transport problems[END_REF] it is not well posed and some additional entropy conditions have to be imposed in order to recover uniqueness of the solution, see [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]. In the case of a discontinuous flux, these conditions are still not sufficient to select a unique solution to [START_REF] Ambrosio | Lecture notes on optimal transport problems[END_REF] and further dissipation conditions, involving the traces of the solutions on the set of discontinuities of the flux, must be imposed in order to ensure uniqueness.

The problem of existence and uniqueness for solutions of (2) has been mainly studied in the case of one space variable and of fluxes with just one point discontinuity (but the analysis can be easily extended to the case of finitely many discontinuity points). Assuming that the discontinuity is located at x = 0 and imposing the validity of Kruzhkov entropy inequalities separately on (-∞, 0) and (0, +∞), one can show that every pair u, v of bounded solutions satisfies [START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF] |u(T, x)v(T, x)| dx ≤ |u(0, x)v(0, x)| dx

+ T 0 W (u ± (t, 0), v ± (t, 0)) dt ,
where W is a quantity that depends only on the traces u ± , v ± of u and v at x = 0. The L 1 -contractivity of the semigroup associated with ( 2) is then obtained if W (u ± , v ± ) ≤ 0 for every pair of solutions. Several conditions have been proposed in literature in order to have that W ≤ 0, and different conditions lead to different physically relevant semigroups of solutions, see [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Garavello | Conservation laws with discontinuous flux[END_REF].

In [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] Andreianov, Karlsen and Risebro have proposed a general framework in order to study uniqueness for [START_REF] Ambrosio | Lecture notes on optimal transport problems[END_REF] in the model case of one space variable and for fluxes with finitely many discontinuity points. The validity of the inequality W ≤ 0 is axiomatized in the notion of L 1 -dissipative germ and given a germ G they show uniqueness of G-entropy solutions, see Definition 3.8 in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] and Definition 2.8 below.

Loosely speaking, at a point of discontinuity of the flux F , a germ G is a set of pairs (u -, u + ) satisfying the Rankine-Hugoniot condition, such that

W (u ± , v ± ) ≤ 0 ∀(u -, u + ), (v -, v + ) ∈ G
and, in the model case of flux with one single discontinuity at x = 0, a G-entropy solution is a solution of (2) satisfying Kruzkov's conditions outside the origin and whose traces at 0 belong to G. A similar analysis has been performed, always in the model case of one dimensional fluxes with one discontinuity point, independently by Garavello, Natalini, Piccoli and Terracina in [START_REF] Garavello | Conservation laws with discontinuous flux[END_REF] in terms of the notion of dissipative Riemannian solvers. Let us also mention that this analysis can be extended to the multidimensional case by assuming that the set of discontinuity of the flux is a regular submanifold, see [START_REF] Andreianov | Entropy conditions for scalar conservation laws with discontinuous flux revisited[END_REF], or by assuming a priori BV regularity of the solution, see [START_REF] Crasta | Kinetic formulation and uniqueness for scalar conservation laws with discontinuous flux[END_REF].

The main purpose of this paper is to provide a general framework to extend this analysis to solutions of (2) under quite general assumptions on the flux. In order to do this we introduce a rather weak notion of entropy solution, see Definition 2.3 below, and under a suitable genuine nonlinearity assumption on the flux we show that these solutions admits traces on the discontinuity set of the coefficients, see Theorem 1.1 below. Once the existence of traces has been established we prove that any pair of weak entropy solutions of (1) satisfies a generalized Kato inequality with a reminder term concentrated on the discontinuity set of the flux, see Theorem 1.2 below. It is then classical to show that this Kato type inequality leads to a quasi contractivity inequality for solutions of (2) of the form [START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF]. Once this inequality has been established, the analysis in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] in terms of germs and of G-entropy solutions can be straightforwardly extended to [START_REF] Ambrosio | Lecture notes on optimal transport problems[END_REF], see Theorem 2.9 below. As a byproduct of our results we can also obtain existence and uniqueness of solutions of (2) assuming Sobolev dependence of the flux F with respect to (t, x), see Theorem 2.11 below.

Let us now describe in a more detailed way our main results. First of all, the structural assumptions on A and the results in [START_REF] Ambrosio | A nonautonomous chain rule in W 1,p and BV[END_REF] guarantee the existence of a H n-1 -rectifiable set N (defined in [START_REF] Crasta | Kinetic formulation and uniqueness for scalar conservation laws with discontinuous flux[END_REF] below) that represents a universal jump set of A(•, v), independent of v.

We say that a distributional solution u ∈ L ∞ (R n ) of (1) a weak entropy solution (WES) of (1), if there exists a non-negative Radon measure µ such that µ(R n \ N ) = 0 and, for every

k ∈ R, (4) div z sign(u -k)[A(z, u) -A(z, k)] + sign(u -k) div a z A(z, k) ≤ µ,
see Definition 2.3 below. Here div a z A(•, k) denotes, for every k ∈ R, the absolutely continuous part of the measure div z A(•, k).

As we shall see in a moment, the notion of weak entropy solution is strong enough to guarantee that such solutions possess a reasonable structure. On the other hand, it is weak enough to include essentially all solutions of (1) obtained by approximation schemes. In particular, under our assumptions on the flux, the solutions constructed by Panov in [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF] are weak entropy solutions.

Assuming the genuine nonlinearity of the flux, and adapting to our setting the techniques developed by De Lellis, Otto and Westdickenberg in [START_REF] De Lellis | Structure of entropy solutions for multi-dimensional scalar conservation laws[END_REF] (see also [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF][START_REF] Vasseur | Strong traces for solutions of multidimensional scalar conservation laws[END_REF]), our first result ensures the existence of traces on N for weak entropy solutions. Loosely speaking, we have the following result, see Theorem 2.5 below for the precise statement.

Theorem 1.1 (Existence of traces). If u is a bounded weak entropy solution of (1), then u admits traces u ± on N (in a generalized sense, see Definition 2.4).

The existence of generalized traces of weak entropy solutions allows us to prove the validity of the following Generalized Kato Inequality, see Theorem 2.6 below for the precise statement.

Theorem 1.2 (Generalized Kato Inequality). Let u and v be weak entropy solutions. Then

(5) div z sign(u -v)[A(z, u) -A(z, v)] ≤ W (u ± , v ± ) H n-1 N ,
where

W (u ± , v ± ) = sign(u + -v + )[A + (z, u + ) -A + (z, v + )] -(sign(u --v -)[A -(z, u -) -A -(z, v -)] • ν N , (6)
where ν N is the measure-theoretic normal to the H n-1 rectifiable set N , and A ± (z, v) are the traces at z of the SBV function z → A(z, v).

In order to prove the above theorem, we combine Kruzkov's doubling of variables technique (see [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]) with Ambrosio's lemma on incremental quotients of BV functions (see [START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF]), to show that the left-hand side of ( 5) is a measure whose positive part is concentrated on N . Once this result has been established, the representation formula ( 6) is an easy consequence of the existence of traces.

Our main application concerns the study of uniqueness conditions for the Cauchy problem associated to the multidimensional evolutionary equation [START_REF] Ambrosio | Lecture notes on optimal transport problems[END_REF]. In this case, a bounded distributional solution u ∈ L ∞ ((0, +∞) × R N ) of ( 2) is a weak entropy solution to (2) if, for every k ∈ R,

(7) ∂ t |u -k| + div x sign(u -k)[F (t, x, u) -F (t, x, k)] + sign(u -k) div a x F (t, x, k) ≤ µ,
where µ is, as before, a non-negative measure concentrated on N . The Generalized Kato Inequality (5) implies the quasi-contractivity of the L 1 norm of the difference of solutions in the following sense:

if u, v ∈ C 0 ([0, +∞); L 1 (R N )) ∩ L ∞ ((0, +∞) × R N )
are weak entropy solutions of (2), then for every T > 0 and every R > 0

BR |u(T, x) -v(T, x)| dx ≤ BR+V T |u(0, x) -v(0, x)| dx + N ∩([0,T ]×BR+V T ) W (u ± , v ± ) dH N , (8)
where B r := {x ∈ R N : |x| < r} and V := A ∞ . As a consequence, if one prescribes an entropy condition stronger than (4) and implying the inequality W ≤ 0, then the Generalized Kato Inequality would give the standard contractivity inequality

(9) BR |u(T, x) -v(T, x)| dx ≤ BR+V T |u(0, x) -v(0, x)| dx
and hence the uniqueness of solutions to the Cauchy problems associated to [START_REF] Ambrosio | Lecture notes on optimal transport problems[END_REF], see Definition 2.7 and Theorem 2.9 below.

Let us also stress that existence of solutions satisfying these additional entropy conditions is not trivial and currently not known in the general setting here considered. Existence results are available assuming additional conditions on the structure of the flux field, see Remark 2.10 for a more detailed discussion.

In case F (•, u) ∈ W 1,1 and satisfies the assumptions listed in Section 2, it is straightforward to check that N = ∅, so that (5) implies contractivity of the semigroup associated to [START_REF] Ambrosio | Lecture notes on optimal transport problems[END_REF]. In particular, also using the results of Panov [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF], we can generalize to this situation the classical Kruzkov results concerning existence and uniqueness of solutions of (2), see Theorem 2.11 and Remark 2.10 below.

Let us conclude this Introduction by presenting the structure of the paper. In Section 2 below we state our main assumption on the flux A, we recall some of its consequence and we provide the precise statements of our main results. In Section 3 we prove Theorem 2.5, in Section 4 we prove Theorem 2.6 and eventually in Section 5 we provide the proofs of Theorems 2.9 and 2.11.

Acknowledgements. F.G. has been supported by ERC 306247 Regularity of area-minimizing currents and by SNF 146349 Calculus of variations and fluid dynamics. G.D.P. is supported by the MIUR SIR grant Geometric Variational Problems (RBSI14RVEZ).

Conflict of interest:

The authors declare that they have no conflict of interest.

Assumptions on the vector field and main results

In this section we state our main structural hypotheses on the vector field (assumptions (H1)-(H5) below) and prove some consequences of these assumptions.

Structural assumptions on the vector field

. Let A ∈ L ∞ (R n × R; R n ) be such that: (H1) There exists a set C A with L n (C A ) = 0 such that A(z, •) ∈ C 1 (R, R n ) for every z ∈ R n \C A and A(•, v) ∈ SBV (R n , R n ) for every v ∈ R n . (H2) There exists a constant M such that |∂ v A(z, v)| ≤ M ∀ z ∈ R n \ C A , v ∈ R.
(H3) There exists a modulus of continuity ω such that

|∂ v A(z, u) -∂ v A(z, w)| ≤ ω(|u -w|) ∀ z ∈ R n \ C A , u , w ∈ R.
(H4) There exists a function g ∈ L 1 (R n ) such that

|∇ z A(z, u) -∇ z A(z, w)| ≤ g(z)|u -w| ∀ z ∈ R n \ C A , u , w ∈ R,
where ∇ z A(z, v) denotes the approximate gradient of the map z → A(z, v).

(H5) The measure [START_REF] Coclite | Conservation laws with time dependent discontinuous coefficients[END_REF]

σ := u∈R |D z A(•, u)| satisfies σ(R n ) < ∞. Here D z A(•, u) is the distributional gradient of the map z → A(z, u) (which is a measure since A(•, u) ∈ BV
) and denotes the least upper bound in the space of non-negative Borel measures, see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]Definition 1.68].

Assumptions (H1)-(H5) imply that A satisfies the hypotheses of [START_REF] Ambrosio | A nonautonomous chain rule in W 1,p and BV[END_REF]. Let us summarize some consequence of this fact. First of all from the definition of σ we deduce that

v∈R |∇A(•, v)|L n ≤ σ a L n v∈R |D s A(•, v)| ≤ σ s ,
where ∇A(•, v) and D s A(•, v) are the approximate differential of A(•, v) and the singular part of the measure DA(•, v) respectively, and σ a L n and σ s are the the absolutely continuous and singular parts of σ. Moreover if we define ( 11)

N := z ∈ R n : lim inf r→0 σ(B r (z)) r n-1 > 0 ,
then N is a H n-1 rectifiable set2 , see Section 3 in [START_REF] Ambrosio | A nonautonomous chain rule in W 1,p and BV[END_REF]. Furthermore for H n-1 -a.e. point in R n \ N and every v ∈ R there exists the limit

Ã(z, v) := lim r→0 - Br (z) A(y, v)dy ,
and for H n-1 almost every z ∈ N and every v ∈ R there exists the traces of A on N defined as:

(12) A ± (z, v) := lim r→0 - B ± r (z) A(y, v)dy,
where we denoted

B ± r (z) = {w ∈ B r (z) : ± w -z, ν(z) ≥ 0}. In addition the functions v → Ã(z, v), A ± (z, v) are C 1 with derivatives given by ∂ v A(z, v) = ∂ v A(z, v) and ∂ v A ± (z, v) = (∂ v A(z, v)) ± respectively, see [4, Proposition 3.2].
Hence, if we denote by a the vector field [START_REF] De Lellis | Structure of entropy solutions for multi-dimensional scalar conservation laws[END_REF] a(z, v)

:= ∂ v A(z, v),
then a admits a precise representative for H n-1 -almost every z ∈ R n \ N as well as one sided traces on N that agree with ∂ v A (respectively with ∂ v A ± ).

In the sequel we shall assume the following genuine nonlinearity hypothesis:

(GNL) L 1 ({v : a ± (z, v) • ξ = 0}) = 0
for every ξ ∈ S n-1 and for H n-1 a.e. z ∈ N .

Remark 2.1. Let us point out that our hypotheses include (and actually are modeled on) the case

A(z, v) = A(w(z), v) where w ∈ SBV (R n ; R d ) ∩ L ∞ (R n ; R d ), A ∈ C 1 (R d × R, R n ) ∩ Lip(R d × R, R n ), and
L 1 ({v : ∂ v A(w, v) • ξ = 0}) = 0 for every ξ ∈ S n-1 and for every w ∈ R d .
Remark 2.2. Since we are dealing with bounded solutions, all our assumptions can be localized in the v variable. Moreover, it is not difficult to modify the proofs in order to localize also in the z variable, see Remark 3.5 in [START_REF] Ambrosio | A nonautonomous chain rule in W 1,p and BV[END_REF].

2.2. Main results. We consider the following scalar conservation law

(14) div z A(z, u(z)) = 0 ,
where A : R n × R → R n satisfies the structural assumption (H1)-(H5) and (GNL).

Definition 2.3 (Weak entropy solutions). A function u ∈ L ∞ (R n
) is a weak entropy solution (WES shortened) of ( 14) if u is a distributional solution of ( 14) and for every k ∈ R it holds

(15) div z sign(u -k)[A(z, u) -A(z, k)] + sign(u -k) div a z A(z, k) ≤ µ, where µ is a non-negative Radon measure independent of k and such that µ(R n \ N ) = 0. Here div a z A(z, k) = tr∇A(z, k) is the absolutely continuous part of div z A(•, k). Definition 2.4 (Traces). Let u ∈ L ∞ (R n
) and let J ⊂ R n be an H n-1 -rectifiable set oriented by a normal vector field ν. We let the set of traces of u at z 0 ∈ J be

Γ u,J (z 0 ) := (c -, c + ) : ∃r k ↓ 0 : u z0,r k → c -1 H -+ c + 1 H + in L 1 loc , where u z0,r k (z) := u(z 0 + r k (z -z 0 )), H ± := {z ∈ R n : ± z -z 0 , ν ≥ 0} and 1 A denotes the characteristic function of a set A.
The very same definition can be given component-wise for a vector field

B ∈ L ∞ (R n , R n ). Moreover it is immediate to see from the definition that if u ∈ L ∞ (R n ) and f ∈ C 0 (R) then Γ f (u),J (z 0 ) = (f (c -), f (c + )) : (c -, c + ) ∈ Γ u,J (z 0 ) . Theorem 2.5 (Existence of generalized traces). If u is a WES, then for H n-1 almost every z 0 ∈ N Γ u,N (z 0 ) = ∅. Moreover if (c -, c + ) ∈ Γ u,N (z 0 ) satisfies c -= c + then the traces are unique: Γ u,N (z 0 ) = {(c -, c + )}. Otherwise there exist a, b ∈ R such that Γ u,N (z 0 ) = {(v, v) : v ∈ [a, b]}.
Finally, the Rankine-Hugoniot condition holds:

A -(z 0 , c -) • ν(z 0 ) = A + (z 0 , c + ) • ν(z 0 ) ∀ (c -, c + ) ∈ Γ u,N (z 0 ).
Theorem 2.6 (Generalized Kato Inequality). Let u and v be WES. Then there exists a Borel function w : N → R such that the following Kato inequality holds true:

(16) div z sign(u -v)[A(z, u) -A(z, v)] ≤ w H n-1 N .
Furthermore, for H n-1 almost every z ∈ N 1 := {z ∈ N : w(z) = 0}, the functions u and v admit unique traces at z and the following representation formula holds:

w = W (u ± , v ± ) = sign(u + -v + )[A + (z, u + ) -A + (z, v + )] -(sign(u --v -)[A -(z, u -) -A -(z, v -)] • ν. (17)
The Generalized Kato Inequality yields a uniqueness result for the Cauchy problem for the evolutionary equation ( 18)

u t + div x A(t, x, u) = 0, in (0, +∞) × R N , u(0, x) = u 0 (x), x ∈ R N .
More precisely, if we prescribe an entropy condition stronger than [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF] and implying the inequality w ≤ 0, then the Generalized Kato Inequality gives the uniqueness of solutions to [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]. To this end let us recall the definition of dissipative germ introduced in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF], see Definition 3.1 there.

Definition 2.7 (Germ). Given two functions f ± ∈ C 0 (R), a set G ⊂ R 2 is said to be a dissipative germ associated to f ± if the following two conditions hold true:

(i) Every (u -, u + ) ∈ G satisfies the Rankine-Hugoniot condition f + (u + ) = f -(u -).
(ii) For every two pairs (u -, u + ) , (v -, v + ) ∈ G we have

W f ± (u ± , v ± ) := sign(u + -v + )[f + (u + ) -f + (v + )] -(sign(u --v -)[f -(u -) -f -(v -)] ≤ 0.
Following [START_REF] Andreianov | On vanishing viscosity approximation of conservation laws with discontinuous flux[END_REF] we now define G-entropy solutions associated to germs, compare with Definition 3 there and Definiton 3.8 in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]. H5) and (GNL) above. Let N ⊂ R × R N be the rectifiable set defined in [START_REF] Crasta | Kinetic formulation and uniqueness for scalar conservation laws with discontinuous flux[END_REF]. Assume that for every z = (t,

Definition 2.8 (G-entropy solutions). Let F : R × R N × R → R N be such that A := (u, F ) satisfies (H1)-(
x) ∈ N such that A ± (z, u) exist it is given a dissipative germ G z associated to f ± (u) := A ± (z, u) • ν(z) and let us set G = {G z } z∈N . We say that a bounded function u ∈ C 0 ([0, +∞); L 1 (R N )) is a G-entropy solution of (18) if (i) u is a weak entropy solution of (18) according to Definition 2.3. (ii) For H N -almost every x ∈ N any (u -, u + ) ∈ Γ u,N (z) belongs to the germ G z .
A straightforward consequence of Theorem 2.6 is then the following: H5) and (GNL) above. Then for any choice of G there exists at most one G-entropy solution of (18).

Theorem 2.9 (Uniqueness of G-entropy solutions). Let F : R × R N × R → R N be such that A := (u, F ) satisfies (H1)-(
Remark 2.10. Under mild requirements on the flux, the existence of weak entropy solutions can be obtained by the results of Panov, see [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF]. On the other hand, the existence of G-entropy solutions, i.e. additionally satisfying condition (ii) in Definition 2.8, is far from trivial and known only in some special cases. Positive results in this direction are available either in one space dimension for a flux with a finite number of discontinuity points, see for instance [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Garavello | Conservation laws with discontinuous flux[END_REF] and the references therein, or in many space dimensions and for the particular case of the vanishing viscosity germ, assuming that the jump set of the F is a C 2 submanifold [START_REF] Andreianov | On vanishing viscosity approximation of conservation laws with discontinuous flux[END_REF], see also [START_REF] Andreianov | Entropy conditions for scalar conservation laws with discontinuous flux revisited[END_REF] where a more general situation is considered.

If F (•, u) is a Sobolev function one can easily obtain from the above analysis uniqueness of (weak) entropy solutions.

Theorem 2.11. Let F : R × R N × R → R N be such that A := (u, F ) satisfies (H1)-(H5) and (GNL) above and assume that

F (•, u) ∈ W 1,1 (R × R N , R N ) for every u ∈ R. Then any two (weak) entropy solutions u, v ∈ C 0 ([0, +∞); L 1 (R N )) ∩ L ∞ ((0, +∞) × R N ) of (2) satisfy R N |u(T, x) -v(T, x)| dx ≤ R N |u(0, x) -v(0, x)| dx .

Proof of Theorem 2.5

In this section we prove Theorem 2.5. We start with the following well known Lemma.

Lemma 3.1. Let B ∈ L ∞ (R n , R n ) and assume that µ = div B is a Radon measure. Then |µ| ≪ H n-1 . Furthermore if J is a rectifiable set and Γ B,J (z) = ∅ for H n-1 almost every z ∈ J then it holds div(B) J = (B + -B -) • ν H n-1 J
where (B -(z), B + (z)) ∈ Γ B,J (z). In particular for every two pairs in Γ B,J (z), their projections along ν(z) have the same difference.

Proof. The fact that |µ| ≪ H n-1 is proved for instance in [START_REF] Lellis | Notes on hyperbolic systems of conservation laws and transport equations[END_REF]Lemma 2.4]. To show the second part we decompose µ as

µ = µ J + µ (R n \ J ) =: µ 1 + µ 2 with µ 1 ⊥ µ 2 .
Since µ 1 is a Radon measure and H n-1 J is σ-finite we can apply the Radon-Nikodym Theorem to get that

µ 1 = div(B) J = hH n-1 J for some h ∈ L 1 (H n-1 J ). Let now z 0 be a point such that Γ B,J (z 0 ) = ∅, h(z 0 + rz)H n-1 J -z 0 r * ⇀ h(z 0 )H n-1 {z • ν(z 0 ) = 0} and lim r→0 |µ 2 |(B r (z 0 )) r n-1 = 0.
Note that H n-1 almost every point z satisfies the above properties. Indeed, the first one follows by our assumptions, while the second and the third ones follow, respectively, from [5, Theorem 2.83] and [5, Equation 2.41].

Let us choose r k ↓ 0 with

B r k → B -(z 0 )1 H -+ B + (z 0 )1 H + ,
where

H ± = {± z, ν(z 0 ) ≥ 0}. Let ϕ ∈ C 1 c (R n ) and define ϕ r k (z) = r 1-n k ϕ((z -z 0 )/r k ).
Integrating by parts we get [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF] µ, ϕ

r k = 1 r n k B(z) • ∇ϕ z -z 0 r k dz = B r k (z) • ∇ϕ(z) dz. Moreover µ, ϕ r k = 1 r n-1 k µ 1 , ϕ • -z 0 r k + 1 r n-1 k µ 2 , ϕ • -z 0 r k = J -z 0 r k h(r k z + z 0 )ϕ(z)dH n-1 (z) + O |µ 2 |(B r k (z 0 )) r k n-1 .
Hence, passing to the limit as k goes to infinity in [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF], we get

h(z 0 ) {z•ν(z0)=0} ϕ(z) dH n-1 (z) = B -(z 0 ) • H - ∇ϕ(z) dz + B + (z 0 ) • H + ∇ϕ(z) dz .
Integrating by parts we obtain that h(z

0 ) = (B + (z 0 ) -B -(z 0 )) • ν(z 0 )
, and this concludes the proof.

Proof of Theorem 2.5. We divide the proof in several steps.

Step 1 (Definition of the measure for the kinetic equation). Let u be a WES, according to [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF] for every k ∈ R the distribution (20)

η k := div z sign(u -k)[A(z, u) -A(z, k)] + sign(u -k) div a z A(z, k)
is a Radon measure. We now claim that for every K ⋐ R and for every R > 0

(21) sup k∈K |η k |(B R ) ≤ C(K, R).
To see this note that µ -

η k ≥ 0 for every k ∈ K. Therefore if φ ∈ C 1 c (B R ) and χ ∈ C 1 c (B R+1 ) satisfies χ ≥ 0, χ ≡ 1 in B R , we have µ -η k , ( φ ∞ ± φ)χ ≥ 0, hence, since χφ = φ, ± η k , φ ≤ -η k , χ φ ∞ + 2 µ, χ φ ∞ .
The above inequality implies the validity of ( 21), since, by the very definition of η k , one has

sup k∈K | η k , χ | ≤ C(K, R).
In particular the map

C ∞ c (R n × R) ∋ Φ → η, Φ := R n ×R Φ(z, k) dη k (z) dk defines a Radon measure η in R n ×R. Moreover if we define 3 ν := π # (|η|)
, where π : R n ×R → R n is the projection on the first factor, then ν ≪ H n-1 . Indeed by Lemma 3.

1 |η k | ≪ H n-1 so that if H n-1 (A) = 0 then ν(A) ≤ R |η k |(A)dk = 0.
Step 2 (Kinetic formulation). The function (k, z) → χ(k, u(z)) := sign(u(z)k) is a solution of the kinetic equation, see [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF] (

) div z [χ(k, u)∂ v A(z, k)] -∂ k [χ(k, u) div a z A(z, k)] = -∂ k η in D ′ (R n+1 ), 22
where η(k, A) := η k (A). Indeed, let us consider in equation ( 20) a test function of the form Φ(k, z)

:= ϕ(z)∂ k ψ(k).
Recalling the definition of the measure η and of χ(k, u), integrating in k we get

- ∇ϕ(z)∂ k ψ(k)χ(k, u)[A(z, u) -A(z, k)] dz dk + ϕ(z)∂ k ψ(k)χ(k, u) div a z A(z, k) dz dk = ϕ(z)∂ k ψ(k) dη(k, z), so that ∇ϕ(z)ψ(k)∂ k χ(k, u)[A(z, u) -A(z, k)] dz dk - ϕ(z)ψ(k)∂ k χ(k, u) div a z A(z, k) dz dk = -ϕ(z)ψ(k) d∂ k η(k, z). Since the function k → χ(k, u)[A(z, u) -A(z, k)] is Lipschitz, it is straightforward to check that ∂ k χ(k, u)[A(z, u) -A(z, k)] = -χ(k, u)∂ v A(z, k), hence (22) holds.
Step 3 (Blow-up). Let η(k, z) = ν(z) ⊗ λ z (k) be the disintegration of the measure η with respect to ν, see [5, Sect. 2.5]. Since H n-1 N is σ-finite by the Radon-Nikodym Theorem we can write

(23) ν = h H n-1 N + ν (R n \ N ) with h ∈ L 1 (H n-1 N ).
Let us now fix a point z 0 ∈ N and for r > 0 let us consider the following rescalings in the variable z:

(24) u r (z) := u(z 0 + rz), A r (z, v) := A(z 0 + rz, v), η k,r (V ) := η k (z 0 + rV ) r n-1 , η r (U × V ) := η(U × (z 0 + rV )) r n-1 , U ⊂ R, V ⊂ R n Borel.
3 Recall that given a Borel measure η on a space X and a Borel map π : X → Y the measure π # η on Y is defined as π # η(U ) = η(π -1 (U )) for every Borel set U ⊂ Y .

Recall the proof of Lemma 3.1: for H n-1 almost every z 0 in N we have

(25) h(z 0 + rz)H n-1 J -z 0 r * ⇀ h(z 0 )H n-1 {ν(z 0 ) • z = 0}.
We now claim that for H n-1 almost every such z 0 and for every k ∈ R (26)

A r (z, k) → A z0 (z, k) := A + (z 0 , k)1 H + (z) + A -(z 0 , k)1 H -(z), ∂ v A r (z, k) → ∂ v A z0 (z, k) := ∂ v A + (z 0 , k)1 H + (z) + ∂ v A -(z 0 , k)1 H -(z), div a z A r (z, k) → 0 , locally in L 1 (R n ), with H ± = {z : ±z • ν(z 0 ) > 0}.
Indeed the first two equations follow directly from the hypotheses on A, see [4, Proposition 3.2], while the last limit in ( 26) is a consequence of the fact that sup k | div a A(z, k)| ≤ σ a (z) and that

lim r→0 1 r n-1 Br (z0)
|σ a (z)| dz = 0 , for H n-1 almost every point in N , see [5, Equation 2.41]. We now prove that, up to H n-1negligible subset of z 0 ∈ N it holds:

(27) η r * ⇀ h(z 0 ) λ z0 (k) ⊗ H n-1 ∂H + .
To this end observe that by [5, Equation 2.41], for

H n-1 almost every z 0 ∈ N , lim r→0 |ν (R n \ N )|(B r (z 0 )) r n-1 = 0.
Now it is easy to see that, up to negligible sets, Step 4 (Limiting equation and existence of traces). Let us take a point z 0 such that (26) and ( 27) hold true. According to Lemma 3.2 below, the sequence (u r ) r is relatively compact in L 1 loc (R n ). Let us now compute the equation satisfied by any cluster point u ∞ of (u r ) r . To this end, note that u r solves

(28) z ∈ N : 0 < lim sup r→0 |ν N |(B r (z)) r n-1 < ∞ = {z ∈ N : h(z) > 0}. Since H n-1 (N ∩ {h > 0}) ≪ ν N , H n-1 almost every z 0 ∈ N ∩ {h > 0}
div z sign(u r -k)∂ v [A r (z, u r ) -A r (z, k)] + sign(u r -k) div a z A r (z, k) = η k,r
. Let (r j ) be a sequence converging to 0 such that u rj → u ∞ in L 1 (B 1 ). Passing to the limit in the kinetic equation satisfied by the function (k, z) → χ(k, u rj (z)),

div z χ(k, u rj )∂ v A rj (z, k) -∂ k χ(k, u rj ) div a z A rj (z, k) = -∂ k η rj in D ′ (R n+1
) , and taking into account ( 26) and ( 27), we obtain

(29) div z χ(k, u ∞ )∂ v A z0 (z, k) = -∂ k h(z 0 ) λ z0 (k) H n-1 ∂H ± in D ′ (R n+1 ).
In particular, due to the special form (26) of A z0 , in the half-space H + (resp. H -), equation ( 29) is a transport equation of the form

(30) a + (k) • ∇ z χ(k, u ∞ ) = 0 (resp. a -(k) • ∇ z χ(k, u ∞ ) = 0), where a ± (k) := ∂ v A ± (z 0 , k).
Since, by (GLN), these vector fields are genuinely nonlinear, we conclude that u ∞ must be constant on H + and on H -, i.e. there exist u -, u + ∈ R such that (31)

u ∞ = u + 1 H + + u -1 H - compare [13, Proposition 7(b)]. Indeed let z ∈ H + be a Lebesgue point of u ∞ and -u ∞ -1 < k < u ∞ (z) such that L n ({u ∞ = k}) = 0.
Fix τ > 0 and convolve with a nonnegative smooth kernel δ ε supported in B ε : for ε < ε(τ, z) sufficiently small

δ ε * χ( k, u ∞ )(z) ≥ 1 -τ.
Thanks to (GLN) we can choose n values k 1 , . . . , k n (depending on τ , ε and k) with |k n -k| sufficiently small and such that k

< k 1 < • • • < k n , {a + (k i )} are linearly independent and
(32)

δ ε * χ(k n , u ∞ )(z) ≥ 1 -2τ.
For every z the function k → χ(k, u ∞ (z)) is decreasing, and so it remains when we convolve it with δ ε : in particular 30), which holds also for δ ε χ(k, u ∞ ), implies that δ ε * χ(k i , u ∞ ) is constant along lines parallel to a + (k i ). Since the {a + (k i )} are linearly independent, starting from (32) and exploiting (33) we obtain

(33) δ ε * χ( k, u ∞ )(z) ≥ δ ε * χ(k 1 , u ∞ )(z) ≥ • • • ≥ δ ε * χ(k n , u ∞ )(z) ∀z ∈ H + ε := {z • ν(z 0 ) > ε}. Equation (
ρ ε * χ( k, u ∞ ) ≥ 1 -2τ in H + ε . Letting τ ↓ 0 we get χ( k, u ∞ ) ≥ 1 in H + .
Since k can be taken arbitrarily close to u ∞ (z), u ∞ is constantly equal to u ∞ (z). A completely analogous argument holds for H -. In particular Γ u,N (z 0 ) = ∅.

Step 5 (Characterization of traces). By (29) and the special form (31) of u ∞ , we deduce that (34)

χ(k, u + ) a + (k) • ν(z 0 ) -χ(k, u -) a -(k) • ν(z 0 ) = -∂ k h(z 0 ) λ y (k) in D ′ (R).
Let us now show as the above equality uniquely determines u ± whenever u + = u -, in particular they do not depend on the choice of the subsequence (r j ). To this end, let (u ρj ) be another converging subsequence of (u r ): by Step 4 we have

u ρj → v ∞ := v + 1 H + + v -1 H - in L 1 (B 1 ),
so that the pair (v -, v + ) also satisfies (34). Subtracting the equation satisfied by the pair (u -, u + ) we get for almost every k ∈ R,

[χ(k, u + ) -χ(k, v + )] a + (k) • ν(z 0 ) = [χ(k, u -) -χ(k, v -)] a -(k) • ν(z 0 ) that is sign(u + -v + )1 (u + ,v + ) (k) a + (k) • ν(z 0 ) = sign(u --v -)1 (u -,v -) (k) a -(k) • ν(z 0 ).
Since, again by the assumption (GNL) of genuine nonlinearity, the functions

k → a ± (k) • ν(z 0 )
cannot vanish on any interval, the two intervals I(u -, u + ) and I(v -, v + ) must coincide 4 . If u -= u + , the condition I(u -, u + ) = I(v -, v + ) can be satisfied either in the case v -= u -, v + = u + or in the case v -= u + , v + = u -. On the other hand, this second possibility is excluded by the fact that the map r → u(y + rz) is continuous from (0, 1] to L 1 (B 1 ). Indeed, since

u rj → u ∞ = u + 1 H + + u -1 H -, 4 Here, I(a, b) denotes the interval [a, b] if a ≤ b or the interval [b, a] if b < a. u ρj → v ∞ = u -1 H + + u + 1 H -,
we have

B1 |u rj -u ∞ | → 0, B1 |u ρj -u ∞ | → B1 |v ∞ -u ∞ | =: m = 0.
By the continuity of the map

(0, 1] ∋ r → B1 u r ,
and the relative compactness of the family (u r ) r , we can find a third sequence (u sj ) such that

u sj → w ∞ := w + 1 H + + w -1 H -, B1 |w ∞ -u ∞ | = m 2 ≤ B1 |w ∞ -v ∞ |, (35)
But then we must have I(w -, w + ) = I(u -, u + ) = I(v -, v + ), so that either w -= u -and w + = u + , or w -= u + and w + = u -, and in each case we get a contradiction with (35).

In conclusion, if u -= u + then all subsequences of (u r ) must converge to the same function u ∞ , hence the traces are uniquely determined.

In the case u -= u + , reasoning as above we can always conclude that w -= w + for every (w -, w + ) ∈ Γ u,N . Moreover exploiting again the continuity of the map r → B1 u r we get that Γ u,N is a compact connected set. Finally the Rankine-Hugoniot condition follows from Lemma 3.1, thus concluding the proof.

The following Lemma has been used in the proof of Theorem 2.5. Lemma 3.2 (Strong pre-compactness of blow-ups). The family (u r ) defined in (24) is precompact in L 1 (B 1 ).

Proof of Lemma 3.2. For every r > 0, the function u r is a solution to

div z A r (z, u r (z)) = 0, hence div z A z0 (z, u r (z)) = -div z A r (z, u r (z)) -A z0 (u r (z)) .
We claim that the family of functions

q r (z) := A r (z, u r (z)) -A z0 (z, u r (z))
is pre-compact in L 2 (B 1 ), so that (div z A z0 (z, u r (z))) r is pre-compact in the negative Sobolev space W -1,2 (B 1 ). If this condition is satisfied, then by [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF]Thm. 6] we can conclude that (u r ) is pre-compact in the strong L 1 (B 1 ) topology.

Let us consider the functions

f r,z (v) := |A r (z, v) -A z0 (z, v)|, r > 0, z ∈ B 1 , v ∈ R. By (26) (36) lim r↓0 f r,z (v) = 0 for every z ∈ B 1 \ D 0 and ∀v ∈ R, where D 0 ⊂ B 1 is a set of Lebesgue measure 0. Moreover |f r,z (v) -f r,z (v ′ )| ≤ |A r (z, v) -A r (z, v ′ )| + |A z0 (z, v) -A z0 (z, v ′ )| ≤ 2 ∂ v A ∞ |v -v ′ |,
hence (f r,z ) r is an equi-Lipschitz family of functions converging pointwise to 0 for every z ∈

B 1 \ D 0 . Let L = u ∞ and let (v k ) ⊂ [-L, L] be a countable dense set in [-L, L].
Using a diagonal argument, we can construct a sequence (r j ) converging to 0 such that

lim j→+∞ f rj,z (v k ) = 0 ∀z ∈ B 1 \ D, ∀k ∈ N,
where D ⊇ D 0 is a set of Lebesgue measure 0.

Using the classical argument in the proof of the Ascoli-Arzelà compactness theorem, we have that, for every z ∈ B 1 \ D, the sequence (f rj ,z ) j converges uniformly to 0 in [-L, L]. In other words,

g j (z) := sup |v|≤L |A rj (z, v) -A z0 (z, v)| → 0. ∀z ∈ B 1 \ D.
Since the functions g j are equi-bounded, they converge to 0 in L 2 (B 1 ). Moreover,

|q rj (z)| 2 := |A r (z, u rj (z)) -A z0 (z, u rj (z))| 2 ≤ g j (z) 2 ,
so that the sequence (q rj ) j converges to 0 in L 2 (B 1 ) and the claim is proved.

Proof of Theorem 2.6

In this section we prove Theorem 2.6. To this end we will need two technical lemmas: the first one is a slight generalization of classical arguments used in [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]. The second one allows to study the limiting behavior of the incremental quotient of A in spirit of [START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF]Thm. 2.4] and [15, Lemma II.1] and it is crucial in the proof of Theorem 2.6. For the sake of exposition we postpone the proofs of both lemmas at the end of the section. Lemma 4.1. Let f : R n × R → R m satisfy the following assumptions:

• z → sup v |f (z, v)| ∈ L 1 loc (R n ); • |f (z, v) -f (z, v ′ )| ≤ g(z)ω(|v -v ′ |) for some g ∈ L 1
loc and some modulus of continuity ω.

Then for every u, v ∈ L ∞ loc (R n ) |f (z + τ, u(z)) -f (z, u(z))| → 0 sign(u(z + τ ) -v(z))[f (z + τ, u(z + τ )) -f (z, v(z))] → sign(u(z) -v(z))[f (z, u(z)) -f (z, v(z))]
in L 1 loc as τ → 0. Lemma 4.2 (Uniform differential quotients). Let A satisfy (H1)-(H5) and let w ∈ R n . Then there exists a measurable set D = D w ⊂ R n , with L n (D) = 0, such that the difference quotients for A can be canonically written as

A(z + εw, v) -A(z, v) ε = A 1 ε (z, v) + A 2 ε (z, v)
where A 1 ε and A 2 ε satisfy the following properties:

(i) lim ε↓0 A 1 ε (z, v) = ∇ z A(z, v) • w, ∀v ∈ R and z ∈ R n \ D;
(ii) The family of functions h ε : R n → R defined by

h ε (z) := |w| sup v∈R A 1 ε (z, v) is equi-integrable; (iii) For every compact set K ⊂ R n we have K sup v∈R A 2 ε (z, v) dz ≤ σ s (K ε|w| )|w|,
where K τ := K + B τ (0).

Proof Theorem 2.6. We divide the proof into several steps:

Step 1: Doubling of variables. We follow the classical technique of Kruzhkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]. Let u(z) and v(z ′ ) be WES: let us set k = v(z ′ ) in [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF] for u and k = u(z) in [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF] for v. Let us also choose a test function Φ(z, z ′ ) = ϕ(z + z ′ )δ ε (zz ′ ) where ϕ ∈ C 1 c (R n ) is nonnegative and δ ε is the usual smooth approximation of the identity in 0:

δ ε (ζ) = ε n ψ ζ/ε ψ ∈ C 1 c (B 1 ), ψ = 1, ψ(z) = ψ(-z).
Multiplying both equations by Φ, integrating in z and z ′ and subtracting the corresponding inequalities we obtain

δ ε (z -z ′ )∇ϕ(z + z ′ ) + ϕ(z + z ′ )∇δ ε (z -z ′ ) sign u(z) -v(z ′ ) A(z, u(z)) -A(z, v(z ′ )) -sign u(z) -v(z ′ ) div a z A(z, v(z ′ ))ϕ(z + z ′ )δ ε (z -z ′ ) + δ ε (z -z ′ )∇ϕ(z + z ′ ) -ϕ(z + z ′ )∇δ ε (z -z ′ ) sign v(z ′ ) -u(z) A(z ′ , v(z ′ )) -A(z ′ , u(z)) -sign v(z ′ ) -u(z) div a z ′ A(z ′ , u(z))ϕ(z + z ′ )δ ε (z -z ′ )dzdz ′ ≥ -2 δ ε (z -z ′ )ϕ(z + z ′ )dz ′ dµ(z).
This can be written as (37)

I ε 1 -I ε 2 + I ε 3 ≥ -2 δ ε (z -z ′ )ϕ(z + z ′ )dz ′ dµ(z)
where

I ε 1 = ψ(w)∇ϕ(2z -εw) sign (u(z) -v(z -εw) × A(z, u(z)) + A(z -εw, u(z)) -A(z, v(z -εw)) -A(z -εw, v(z -εw)) dwdz, I ε 2 = ϕ(2z -εw) sign u(z) -v(z -εw)
× ∇ψ(w) A(zεw, u(z)) -A(z, u(z)) ε ψ(w) div a z A(zεw, u(z)) dwdz,

I ε 3 = ϕ(2z + εw) sign u(z + εw) -v(z) × ∇ψ(w) A(z, v(z)) -A(z + εw, v(z)) ε -ψ(w) div a z A(z + εw, v(z)) dwdz. Regarding I ε 1 , Lemma 4.1 implies that (38) I ε 1 → 2 ∇ϕ(2z) sign (u(z) -v(z) dz A(z, u(z)) -A(z, v(z)) .
We will now show that

(39) lim sup ε→0 |I ε 2 -I ε 3 | ≤ C ϕ ∞ |σ s |(spt ϕ).
This, together with (37) and (38), will then give that, in the sense of distributions,

(40) div z sign (u(z) -v(z) A(z, u(z)) -A(z, v(z)) ≤ 2µ + C|σ s | =: β where (µ + C|σ s |)(R n \ N ) = 0.
In turn the left hand side of (40) is a signed measure, which we denote by α, for which:

α ≤ α + = α + N = (α N ) + ≤ β. Since the map (u, v) → sign(u -v A(z, u) -A(z, v) ,
is Lipschitz and A(z 0 + εz, v) → A ± (z 0 , v) in L 1 loc for every v ∈ R, by arguing as in Lemma 4.1 the traces of the vector field

z → sign (u(z) -v(z) A(z, u(z)) -A(z, v(z))
exist for H n-1 almost every z ∈ N and are given by sign (u

± (z) -v ± (z) A ± (z, u ± (z)) -A ± (z, v ± (z)) .
A direct application of Lemma 3.1 yields the desired representation [START_REF] Karlsen | L 1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients[END_REF].

To show uniqueness of the traces at points where w(z) = 0 we note that we only have to discuss the case when (say) v -(z) = v + (z) = v and u -(z) = u + (z), otherwise either the traces are unique by Theorem 2.5 or w = 0. The Rankine-Hugoniot condition gives (41)

w(z) = [sign(u + (z) -v) -sign(u -(z) -v)][A + (z, u + (z)) -A + (z, v)] • ν(z).
Moreover we know by Theorem 2.5 that if Γ v,N is not a singleton it contains pairs (v ′ , v ′ ) with v ′ ranging in a non trivial interval [a, b]. Being w uniquely determined and non zero we have that (41) holds for any such v ′ ∈ [a, b] and that v ′ ∈ I(u -, u + ). This implies that

[A + (z, v ′ ) -A + (z, v)] • ν(z) = 0 ∀ v, v ′ ∈ [a, b] ,
contradicting the genuine nonlinearity assumption (GLN).

In order to conclude the proof of the Theorem we only have to show the validity of (39). According to Lemma 4.2 above we can write

A(z -εw, u(z)) -A(z, u(z)) ε = A 1 ε,w (z) + A 2 ε,w (z)
where

A 1 ε,w (z) L 1 loc -→ -∇A(z, u(z)) • w and dz|A 2 ε,w (z)|ϕ(z) ≤ |w| ϕ ∞ |σ s |((spt ϕ) ε|w| )
. Hence, by also using Lemma 4.1, we obtain that

I ε 2 = ϕ(2z -εw) sign u(z) -v(z -εw) × -∇ψ(w)∇A(z, u(z)) • w -ψ(w) div a z A(z, u(z)) dwdz + R ε 1 + R ε 2 , where lim sup ε→0 |R ε 1 | ≤ C(ψ) ϕ ∞ |σ s |(spt ϕ) and lim ε→0 |R ε 2 | = 0.
By applying the same decomposition to I ε 3 we obtain, after a change of variable, that lim sup

ε→0 |I ε 2 -I ε 3 | ≤ 2C(ψ) ϕ ∞ |σ s |(spt ϕ) + lim sup ε→0 ϕ(2z -εw) sign u(z) -v(z -εw) × ∇ψ(w) ∇A(z, u(z)) • w -∇A(z -εw, v(z -εw)) • w + ψ(w) div a z A(z, u(z)) -div a z A(z -εw, v(z -εw)) dwdz .
By Lemma 4.1 the latter integral converges to

ϕ(2z) sign u(z) -v(z) ∇ψ(w)∇A(z, u(z)) • w + ψ(w) div a z A(z, u(z)) dwdz - ϕ(2z) sign u(z) -v(z) ∇ψ(w)∇A(z, v(z)) • w + ψ(w) div a z A(z, v(z)) dwdz.
Integrating by parts with respect to the w variable, we get that both integrals are zero, thus concluding the proof of (39).

We conclude the Section by proving Lemma 4.1 and Lemma 4.2.

Proof of Lemma 4.1. Let Q ⊂ R be a countable dense set: by the continuity of translations in

L 1 |f (z + τ, u) -f (z, u)| → 0 in L 1 loc for every u ∈ Q. If now u ∈ L ∞ , there exists u k = N k i=1 u i k 1 A i k with u i k ∈ Q and such that u -u k ∞ → 0. Hence for every compact set K ⊂ R n K |f (z + τ, u(z)) -f (z, u(z))|dz ≤ ω( u -u k ∞ ) K (g(z) + g(z + τ ))dz + K |f (z + τ, u k (z)) -f (z, u k (z))|dz ≤ o k (1) + N k i=1 K∩A i k |f (z + τ, u i k ) -f (z, u i k )|dz
where o k (1) → 0 independently on τ as k → ∞. Passing to the limit first on τ and then on k proves the first claim. To prove the second claim note that thanks to what we have proved it is enough to show that

sign(u(z + τ ) -v(z))[f (z, u(z + τ )) -f (z, v(z))] → sign(u(z) -v(z))[f (z, u(z)) -f (z, v(z))] in L 1 loc as τ → 0. Since the map (u, v) → sign(u -v)[f (z, u) -f (z, v)]
has modulus of continuity 2ω independently on z this plainly follows by the continuity of translations in L 1 .

Proof of Lemma 4.2. Up to dilating and rotating we can assume that w = e n . We will write

z = (z ′ , z n ) with z ′ ∈ R n-1 and z n ∈ R. Let Q = (v j ) ⊂ R be a countable dense set in R.
By slicing theory for BV functions, see [5, Chapter 3], for every j ∈ N there exists a set D j ⊂ R n with L n (D j ) = 0, such that, for every z ∈ R n \ D j , the function t → A(z ′ , z n + t, v j ) belongs to BV (R) and the absolutely continuous part of its derivative, denoted by ∂A ∂zn (z ′ , z n + t, v j ), coincides with ∇ z A(z ′ , z n + t, v j ) • e n . Hence for j ∈ N and z ∈ R n \ D j we define

A 1 ε (z ′ , z n , v j ) = 1 0 ∂A ∂z n (z ′ , z n + εt, v j ) dt = 1 0 ∇ z A(z ′ , z n + εt, v j ) • e n dt .
From [2, Thm. 2.4] there exists a measurable set

D ⊂ R n , with D ⊃ C A ∪ j D j and L n (D) = 0, such that (42) lim ε↓0 A 1 ε (z, v j ) = ∇ z A(z, v j ) • e n , ∀j ∈ N and z ∈ R n \ D.
Moreover, up to add to D a set of Lebesgue measure zero, we can assume that every z in R n \ D is a Lebesgue point for the function g appearing in (H4) and that

(43) G ε (z) := 1 0 g(z ′ , z n + εt) dt → g(z)
as ε ↓ 0.

Let us now fix z ∈ R n \ D and j, k ∈ N: by (H4) we have that

A 1 ε (z, v j ) -A 1 ε (z, v k ) ≤ 1 0 |∇ z A(z ′ , z n + εt, v j ) -∇ z A(z ′ , z n + εt, v k )| dt ≤ G ε (z) ω(|v j -v k |). (44)
Let us now take v ∈ R and v j ∈ Q with v j → v. By (44) A 1 ε (z, v j ) j is a Cauchy sequence, hence it converges to a unique limit ℓ ε (z, v). Let us define for v ∈ R and z ∈ R n \ D A 1 ε (z, v) = ℓ ε (z, v) and

A 2 ε (z, v) = A(z + εw, v) -A(z, v) ε -A 1 ε (z, v
). We now verify the validity of (i)-(iii). First of all (44) implies (45)

A 1 ε (z, v) -A 1 ε (z, v ′ ) ≤ G ε (z) ω(|v -v ′ |) ∀v, v ′ ∈ R.
Moreover, according to [4, Lemma 3.4], we can add to D a set of measure zero outside which ∇A(z, v) is well defined and continuous in v. Hence for z ∈ R n \ D and v ∈ R, by ( 45) and (H4), we have Since (v j ) in dense in R, we conclude that (i) holds.

A 1 ε (z, v) -∇ z A(z, v) • e n ≤ A 1 ε (z, v) -A 1 ε (z, v j ) + A 1 ε (z, v j ) -∇ z A(z, v j ) • e n + |∇ z A(z,
Let us prove (ii). For almost every z ∈ R n we have By the Dunford-Pettis compactness criterion we conclude that the family (h ε ) is equi-integrable in L 1 (R n ), hence (ii) is proved.

h ε (z) = sup j∈N A 1 ε (z, v j ) ≤
To conclude (iii), for every j ∈ N we let ϕ j z ′ (z n ) := A(z ′ , z n , v j ) and we note that for almost every z n A(z + εe n , v j ) -A(z, v j )

ε = 1 ε Dϕ j z ′ ([z n , z n + ε]) = 1 0 ∇ z A(z + tεe n , v j ) • e n dt + 1 ε D s ϕ j z ′ ([z n , z n + ε]), hence A 2 ε (z, v j ) ≤ 1 ε |D s ϕ j z ′ |([z n , z n + ε]). If K is a compact subset of R n we get K A 2 ε (z, v j ) dz ≤ R n-
d|D s ϕ j z ′ |(t) R dz n 1 ε 1 [zn,zn+ε] (t) ≤ |D s A(•, v j )|(K ε ) ≤ σ s (K ε ). ( 46
)
Let now v ∈ R. From (45), ( 46) and (H1) we get

K A 2 ε (z, v) dz ≤ K A 2 ε (z, v) -A 2 ε (z, v j ) dz + K A 2 ε (z, v j ) dz ≤ K
A(z + εw, v) -A(z, v) ε -A(z + εw, v j ) -A(z, v j ) ε dz

+ K A 1 ε (z, v) -A 1 ε (z, v j ) dz + K A 2 ε (z, v j ) dz ≤ 2 ε M L n (K ε ) |v -v j | + K G ε (z) dz ω(|v -v j |) + σ s (K ε ).
Exploiting the density of (v j ), we get (iii).

5. Proofs of Theorems 2.9 and 2.11

In this Section we briefly sketch the proofs of Theorems 2.9 and 2.11.

Proof of Theorem 2.9. By Theorem 2.6 we have that any two G-entropy solutions u, v satisfy the Generalized Kato Inequality [START_REF] Garavello | Conservation laws with discontinuous flux[END_REF]. By the usual test function argument, see [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] we then obtain that for every T > 0 and every R > 0 where w(t, x) is given by [START_REF] Karlsen | L 1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients[END_REF]. Since u, v are G-entropy solutions w ≤ 0, from which uniqueness immediately follows.

Proof of Theorm 2.11. If F (•, u) belongs to W 1,1 it easily follows from the definition of supremum of measures that σ s = 0 which implies that H N (N ) = 0. Theorem 2.6 then gives that any two entropy solutions satisfy a true Kato inequality:

∂ t |u -v| + div x sign(uv)[F (t, x, u) -F (t, x, v)] ≤ 0, from which the validity of the L 1 contraction inequality is then straightforward.

  is a Lebesgue point for the measure valued map z → λ z with respect to ν. By combining this with (25) one can argue as in Lemma 3.1 to deduce (27) on {h > 0}, see for instance[START_REF] De Lellis | Structure of entropy solutions for multi-dimensional scalar conservation laws[END_REF] Proposition 9]. Finally by (28) we have that H n-1 almost every z 0 ∈ N satisfies (27), since this convergence trivially holds for H n-1 almost every z 0 ∈ {h = 0} ∩ N .

  v j ) • e n -∇ z A(z, v) • e n | ≤ G ε (z) ω(|vv j |) + A 1 ε (z, v j ) -∇ z A(z, v j ) • e n + g(z) ω(|vv j |) .Taking the limsup as ε ↓ 0 and taking into account (42) and (43) we get lim supε↓0 A 1 ε (z, v) -∇ z A(z, v) • e n ≤ 2g(z) ω(|vv j |).

1 0σ 1 0 1 0 R R n- 1 ψ

 1111 a (z ′ , z n + εt) dt.Since σ a ∈ L 1 (R n ) there exists a superlinear, convex, increasing function ψ : [0, +∞) → [0, +∞) such that R n ψ(σ a (z)) dz < +∞.Then, by Jensen's inequality,R n ψ(h ε (z)) dz ≤ R n-1 R ψ σ a (z ′ , z n + εt) dt dz n dz ′ ≤ (σ a (z ′ , z n + εt))dz ′ dz n dt = R nψ(σ a (z)) dz < +∞ .

BR

  |u(T, x)v(T, x)| dx ≤ BR+V T |u(0, x)v(0, x)| dx + N ∩([0,T ]×BR+V T ) w(t, x) dH N (t, x) ,

Recall that a set N ⊂ R n is said H n-1 -rectifiable (shortened: rectifiable) if there are countably many C 1 submanifolds M i of dimension n -1 such that H n-1 (N \ i M i ) = 0.