Electromagnetic Optimization of Microwave Filters using Adjoint Sensitivities

Matthias Caenepeel, Fabien Seyfert, Martine Olivi, and Yves Rolain, Fellow, IEEE Abstract-This paper introduces a novel computer-aided tuning (CAT) method for coupled-resonator microwave bandpass filters. The method is based on the estimation of the Jacobian of the function that relates the physical filter design parameters to the extracted coupling parameters. Lately commercial full-wave electromagnetic (EM) simulators provide the adjoint sensitivities of the S-parameters with respect to the geometrical parameters. This information leads to an efficient estimation of the Jacobian since it no longer requires finite difference based evaluation. The tuning method first extracts the physically implemented coupling matrix and estimates the corresponding Jacobian. Next it compares the extracted coupling matrix to the target coupling matrix (golden goal). Using the difference between the coupling matrices and the pseudo-inverse of the estimated Jacobian, a correction that brings the design parameters closer to the golden goal is obtained. This process is repeated iteratively until the correction becomes sufficiently small with respect to a user specified goal.

In the case of coupling structures with multiple solutions, the Jacobian is calculated for each admissible solution. This paper presents a criterion to identify the physical solution among the different solutions.

The CAT method is applied to the design of a cascaded triplet (CT) filter implemented in a microstrip technology. This filter is a well-known examples of a non-canonical coupling structure.

Index Terms-Coupling Matrix, Filter Tuning, Parameter Extraction, Jacobian Estimation, Adjoint Sensitivity

I. INTRODUCTION

T HE design of coupled-resonator microwave bandpass filters is widely based on coupling matrix theory [START_REF] Cameron | Microwave filters for communication systems[END_REF]. The design starts from a rational scattering (S-) matrix that fulfills the filter specifications. The next step synthesizes an equivalent low-pass circuit (Fig. 1) that realizes these Sparameters [START_REF] Cameron | General coupling matrix synthesis methods for chebyshev filtering functions[END_REF]. The low-pass circuit is also represented by a coupling matrix. The coupling parameters (circuital elements) are implemented physically by correctly dimensioning an adequate physical layout. The initial dimensioning of the filter uses empirical design curves relating the coupling parameters to the physical parameters [START_REF] Hong | Microstrip filters for RF/microwave applications[END_REF]. These empirical curves neglect the more complex parasitic interactions between resonators such as loading effects of adjacent resonators. Resulting initial coupling values must hence be optimized or tuned in order to meet for the filter its specifications.

Over the last years several computer-aided tuning (CAT) techniques have been developed to optimize the physical design parameters in various ways [START_REF] Harscher | Automated filter tuning using generalized low-pass prototype networks and gradient-based parameter extraction[END_REF]- [START_REF] Hu | An adaptive computer-aided design-and-tuning scheme for a predistorted filter with uneven-q effect[END_REF]. These tuning procedures extract a coupling matrix starting from simulated or measured scattering parameters of the initial design and compare it to the target coupling matrix (golden goal). The corrected physical parameters are then used to tune the filter. As this extracted coupling matrix is an approximation, extraction errors will inevitably be present. Even when they are small, their presence can lead to incoherent corrections, possibly destroying the tuning process. This problem becomes central when the coupling topology admits several solutions: in this case the extracted coupling matrix and the implemented coupling matrix may differ totally [START_REF] Caenepeel | Microwave filter design based on coupling topologies with multiple solutions[END_REF]. Ensuring that the extraction process yields the physically implemented coupling matrix when starting from the simulated S-parameters is therefore crucial.

The extraction techniques can roughly be divided in two categories. The first one optimizes the electrical parameters of the low-pass equivalent circuit to fit the measured response [START_REF] Harscher | Automated filter tuning using generalized low-pass prototype networks and gradient-based parameter extraction[END_REF], [START_REF] Harscher | Em-simulator based parameter extraction and optimization technique for microwave and millimeter wave filters[END_REF]. These methods yield only the solution closest to the initial guess by construction. Unfortunately, this solution does not necessarily correspond to the physically implemented one. (The solution strongly depends on the initial values for the coupling parameters specified by the user.) The second set [START_REF] Bila | Direct electromagnetic optimisation method for microwave filter design[END_REF]- [START_REF] Hu | An adaptive computer-aided design-and-tuning scheme for a predistorted filter with uneven-q effect[END_REF] first identifies the characteristic polynomials or a rational form representing the scattering parameters. A coupling matrix is then obtained for the identified model. Both classes of methods yield results with a low residual error, but suffer from the same problem: they only handle circuits with a canonical coupling topology. The coupling topology represents the way that the resonators are coupled to each other. Certain filters such as cascaded triplet (CT), cascaded quartet (CQ) and extended-box filters, have a non-canonical coupling topology [START_REF] Cameron | Microwave filters for communication systems[END_REF]. For such filters several circuits that share the same coupling structure, but have different values for the coupling parameters, realize the same response. The extraction method in this paper systematically derives all possible circuital realizations corresponding to the implemented coupling topology. The main difference with the extraction method used in [START_REF] Caenepeel | Microwave filter design based on coupling topologies with multiple solutions[END_REF], [START_REF] Caenepeel | Parametric modeling of the coupling parameters of planar coupled-resonator microwave filters[END_REF] is that here a single simulation suffices to identify the physically implemented coupling matrix.

The method estimates the Jacobian matrix of the functional model that relates the circuit parameters to the physical parameters. It thereto combines the adjoint sensitivities of the S-parameters with respect to the circuit parameters on the one hand and the adjoint sensitivities of the S-parameters with respect to the geometrical parameters on the other hand. Formal expressions exist to calculate the adjoint sensitivities of the S-parameters with respect to the circuit parameters using only the values of the extracted coupling parameters. Lately commercial EM-simulators, such as CST Microwave Studio [START_REF]CST Microwave Studio[END_REF], provide the adjoint sensitivities of the S-parameters with respect to the geometrical parameters. One simulation Fig. 1: The equivalent network models a filter consisting of N multi-coupled resonators. hence suffices to estimate the Jacobian matrix. This drastically reduces the number of required EM-simulations, by avoidance of finite difference approximations. It also allows to identify the physically implemented coupling matrix in the case of non-canonical topologies. One simulation suffices to estimate the Jacobian for each admissible solution. To use a Jacobian to tune the physical parameters in the case of a coupling topology with multiple solutions, it is necessary to identify the coupling matrix corresponding to the physically implemented couplings.The use of variational information based on adjoint sensitives computations is here of great help. The idea is that in a physical design, a coupling parameter is mainly affected by a specific set of design parameters. Thus the partial derivatives of the coupling parameter with respect to these parameters are larger than the partial derivatives with respect to the other design parameters having a second order effect at most. This assumption allows to predict which elements of the Jacobian are dominant in the case of a physical relation. Selecting the correct coupling matrix and its associated Jacobian boils down to finding the Jacobian for which those elements are dominant.

The main novelty of this paper is to provide a method to estimate the Jacobian based on adjoint sensitivities and to determine a criterion to select the physically implemented solution for topologies with multiple solutions. Section II describes the equivalent low-pass network to model the filter behavior and derives formal expressions for the adjoint sensitivity of the S-parameters with respect to the circuit parameters. Section III details the coupling matrix extraction method. Section IV combines the adjoint sensitivities to estimate the Jacobian matrix. Section V presents the optimization method as is used to tune the filter response. Finally section VI applies the method to a cascaded trisection (CT) filter implemented in a microstrip technology. This filter is a typical example of a non-canonical coupling structures.

II. EQUIVALENT LOW-PASS CIRCUIT

The equivalent network used to model the filters behavior consists of N coupled low-pass resonators (Fig. 1). The lowpass resonator contains a shunt capacitor C i , a frequency invariant reactance (FIR) M ii [START_REF] Cameron | Microwave filters for communication systems[END_REF] and an admittance G i . The FIR M ii models a frequency shift of the individual resonators with respect to the center frequency f c of the filter and the admittance G i models the loss. The capacitors as well as the input and output loads are all normalized to unity. The resonators are coupled through frequency invariant symmetric couplings M ik = M ki . The resonators are also coupled to the source (input) and the load (output) by a frequency invariant coupling labeled M Si and M Li respectively. Note that the same network is used in [START_REF] Cameron | Microwave filters for communication systems[END_REF], [START_REF] Martinez-Mendoza | Formal expression of sensitivity and energy relationship in the context of the coupling matrix[END_REF]. Here the losses are also taken into account and are modeled by G i . Solving the nodal equations of the network yields following solution:

[G + jM + jω Ĩ][u] = [A][u] = [i] (1)
where

• G ∈ R (N +2)×(N +2) is a diagonal matrix with [G] ii = G i and [G] 11 = [G] N +2,N +2 = 1. • Ĩ ∈ R (N +2)×(N +2) is an identity matrix with elements [ Ĩ] 11 = [ Ĩ] N +2,N +2 = 0. • M ∈ R (N +2)×(N +2
) is the coupling matrix containing the couplings and FIRs.

• ω is the normalized low-pass frequency.

• i = [i S , . . . , 0] t is the excitation vector of currents of size (N + 2) × 1 • u = [u S , u 1 , . . . , u N , u L ] t is the voltage vector of size (N + 2) × 1 • A = [G + jM + jωI] ∈ C (N +2)×(N +2) • j 2 = -1
where [A] kl denotes the element at position (k, l) of the matrix A. Writing the S-parameters as a function of A results in [START_REF] Martinez-Mendoza | Formal expression of sensitivity and energy relationship in the context of the coupling matrix[END_REF]:

S 11 = 2[A -1 ] 11 -1 S 21 = 2[A -1 ] N +2,1 S 22 = 2[A -1 ] N +2,N +2 -1 (2)
We now derive a formal expression for ∂Sij ∂M k,l (ω) which is the sensitivity of a scattering parameter S ij with respect to a coupling parameter M k,l at a certain frequency ω. In section IV these sensitivities are used to estimate the Jacobian of the functional relation between the physical filter parameters and the circuit parameters. The sensitivities can be expressed as a function of A -1 . Remember that the partial derivative of the inverse of a matrix A can be written as:

∂A -1 ∂x = -A -1 ∂A ∂x A -1 (3)
Applying (3) to the expressions given in (2) yields:

For l = k : ∂S 11 ∂M l,k = -4j([A -1 ] 1,l [A -1 ] 1,k ) ∂S 21 ∂M l,k = -2j([A -1 ] N +2,l [A -1 ] k,1 + [A -1 ] 1,k [A -1 ] l,N +2 ) ∂S 22 ∂M l,k = -4j([A -1 ] N +2,l [A -1 ] N +2,k ) For l = k : ∂S 11 ∂M l,l = -2j[A -1 ] 2 1,l ∂S 21 ∂M l,l = -2j[A -1 ] 1,l [A -1 ] l,N +2 ∂S 22 ∂M l,l = -2j[A -1 ] 2 N +2,l (4)
Remark that these expressions are equivalent to those found in [START_REF] Amari | Sensitivity analysis of coupled resonator filters[END_REF] where the dual circuit based on impedance inverters and series inductors is used.

III. PARAMETER EXTRACTION

The parameter extraction method synthesizes a canonical coupling matrix starting from a rational approximation of the S-parameters. This coupling matrix is then reconfigured to match the implemented coupling topology. In the case of multiple solutions, all admissible solutions are determined.

A. Rational Approximation

The extraction method first estimates a stable commondenominator rational matrix of prescribed McMillan degree N for the simulated S-parameters of the filter. The rational approximation of the data is done using the software PRESTO-HF [START_REF] Seyfert | PRESTO-HF : a matlab toolbox dedicated to the identification problem of low pass coupling parameters of band pass microwave filters[END_REF]. This software is based on analytic and rational H 2 approximation [START_REF] Olivi | Identification of microwave filters by analytic and rational h2 approximation[END_REF]. The scattering data is transformed to the low-pass domain [START_REF] Cameron | Microwave filters for communication systems[END_REF]. The software identifies the following mathematical model for the low-pass domain data:

e j α 2 ω 0 0 e j β 2 ω R(jω) e j α 2 ω 0 0 e j β 2 ω ( 5
)
where R is the rational matrix and α and β are the delays introduced by the access lines that are assumed to be ideal lines. The McMillan degree of the matrix R corresponds to the number of resonators N that are present in the filter.

B. Coupling Matrix Synthesis

A canonical arrow form coupling matrix M is synthesized starting from the rational matrix R. The coupling structure imposed by the arrow form does not necessarily correspond to the implemented coupling structure. Therefore M must be reconfigured to match the actual coupling structure of the filter by performing a series of Givens rotations. In the case of non-canonical circuits several solutions exist for this reduction problem. Using a Groebner basis and homotopy based approach is is possible to determine all admissible coupling matrices. This requires solving a related non-linear system of equations [START_REF] Cameron | Exhaustive approach to the coupling matrix synthesis problem and application to the design of high degree asymmetric filters[END_REF]. This technique has been implemented in the Matlab toolbox DEDALE-HF [START_REF] Seyfert | DEDALE-HF : a matlab toolbox dedicated to the equivalent network synthesis for microwave filters[END_REF]. Whenever multiple solutions exist, all admissible solutions are determined. In the next section we show how the Jacobian is estimated for each of these solutions and give a criterion to determine which one corresponds to the physical solution.

IV. ESTIMATION OF THE JACOBIAN MATRIX J

The tuning procedure is based on the relation of the geometrical design parameters to the extracted circuit parameters describing the physical implementation of the filter. Assume that there are N g geometrical parameters g and N c circuital parameters m. The coupling matrix extraction process formally relates the geometrical design parameters with the extracted electrical parameters. We will first suppose that this relation is single-valued, that is that we are dealing with a topology admitting a single solution to the synthesis process, and we denote it by:

f (g) = m ( 6
)
where g is a vector of size N g ×1 containing the geometrical parameters and m is a vector of size N c × 1 containing the extracted circuital parameters. The Jacobian matrix J of f is estimated by combining the adjoint sensitivities of the Sparameters.

Commercial EM-simulators, such as Computer Simulation Technology (CST) [START_REF] Technology | [END_REF], provide the adjoint sensitivities of the S-parameters with respect to the geometrical parameters as a function of the frequency. This information together with the adjoint sensitivities of the S-parameters with respect to the circuital parameters (calculated using (4)) is used to estimate the Jacobian of f .

Using the chain rule of differential calculus, the sensitivity of S ij with respect to a geometrical parameter g s can be written as

∂S ij ∂g s (ω) = Nc k=1 ∂S ij ∂m k (ω) ∂m k ∂g s (7)
where ∂Sij ∂m k is the adjoint sensitivity of S ij with respect to the k th circuital parameter of m and ∂m k ∂gs is element [J ] k,s of the Jacobian J . From formulas (4) it is possible to calculate the sensitivity of S ij with respect to each extracted circuit parameter at each simulated frequency. The simulated frequencies which are transformed to the normalized low-pass domain and are ordered in a vector Ω = [ω 1 , ..., ω N f ] t , where N f is the number of frequencies. The column wise ordering these sensitivities yields a matrix of size 4N f × N c :

S dM =                            ∂S11 ∂m1 (ω 1 ) . . . ∂S11
) . . . ∂S22 ∂m k (ω n F ) . . . ∂S22 ∂mn c (ω n F )                            (8)
Similarly, the sensitivity of S ij with respect to the geometrical parameters provided by the simulator are ordered columnwise in a matrix of size

4N f × N g S dG =                              ∂S11 ∂g1 (ω 1 ) . . . ∂S11 ∂g k (ω 1 ) . . . ∂S11 ∂g Ng (ω 1 ) . . . . . . . . . ∂S11 ∂g1 (ω n F ) . . . ∂S11 ∂g k (ω n F ) . . . ∂S11 ∂g Ng (ω n F ) ∂S12 ∂g1 (ω 1 ) . . . ∂S12
) . . . ∂S22 ∂g k (ω n F ) . . . ∂S22 ∂g Ng (ω n F )                              (9)
S dG can be written as a function of S dM using the Jacobian J = [ ∂m k ∂gs ]:

S dG = S dM J (10)
This overdetermined equation can be solved for the Jacobian J in the least squares sense:

J = (S * dM S dM ) -1 S * dM S dG (11)
where S * dM is the conjugate transpose of S dM . Note that we have implicitly supposed that the coupling topologies we are dealing with are non-redundant [START_REF] Cameron | Exhaustive approach to the coupling matrix synthesis problem and application to the design of high degree asymmetric filters[END_REF], meaning that locally no variation of a coupling can be compensated by the re-tuning of others. This property can be shown to mathematically imply the invertibility of the matrix S * dM S dM , provided the number N f of sampled frequencies is big enough. Determination of the physical coupling matrix:

In the case of non-canonical coupling topologies, several equivalent circuits having the same coupling structure but different coupling values exist. Assume that the number of equivalent solutions is N s . This means that there are N s equivalent functions that relate the design parameters g to the coupling parameters m. Among them only one function f p relates g to the physically implemented coupling parameters m p . To identify f p among all others, we assume that in a physical design a coupling is mainly determined by a physical parameter and that it is possible to determine which physical parameters influence the couplings the most. This assumption is also implicitly made during the initial dimensioning of the filter since also then second order effects, such as loading of the resonators, are neglected. One can therefore predict in advance which partial derivatives ∂m k ∂gs are dominant for the physical design. These derivatives correspond to the dominant elements of the Jacobian matrix J p of f p . Let V denote the set of indices corresponding to these elements. For each estimated J , the relative influence of these predicted elements is calculated:

c = k,l∈V [J ] 2 l,k Nc m=1 Ng n=1 [J ] 2 m,n (12)
In the set of equivalent Jacobian matrices one selects the matrix for which c is maximal. This matrix corresponds to the solution for which the design parameters have the expected main effects on the coupling parameters. This solution can then safely be assumed to be the physical solution.

V. OPTIMIZATION METHOD

A. Correction of the Design Parameters

The optimization method updates the design parameters g such that the corresponding circuital parameters f (g) = m coincide maximally with the target coupling parameters m of the golden goal. Evaluating the function f for the initial design parameters g 0 yields the difference between the circuital parameters of the initial design m 0 and the target parameters m which allows to evaluate the design error:

∆m 0 = m -f (g 0 ) (13)
The corrections of g 0 necessary to improve the filter response are therefore readily obtained as:

∆g 0 = g 1 -g 0 = f -1 (m o + ∆m 0 ) -g 0 ( 14
)
where g 1 are the design parameters of the improved filter. A linear approximation of f using the estimated Jacobian J 0 in g 0 yields:

m 1 = f (g 1 ) ≈ J 0 ∆g 0 + g 0 (15)
Expression [START_REF] Seyfert | PRESTO-HF : a matlab toolbox dedicated to the identification problem of low pass coupling parameters of band pass microwave filters[END_REF] approximates the necessary correction of the design parameters using the pseudo-inverse of the Jacobian J -1 0 :

∆g 0 ≈ J -1 0 ∆m 0 ( 16
)
A simulation of the updated filters response yields the coupling parameters m 1 and the Jacobian J 1 in the same way as above. Repeating the process above then allows to refine the correction until the necessary corrections become sufficiently small:

∆g k ∞ = max l [∆g k ] l < δ ( 17
)
where δ is a user-defined value. Since it has no use to model the filter for geometrical parameters more precisely than the fabrication tolerance, a valid choice for δ is readily found to be the accuracy of the fabrication process. For all of the considered examples, the optimization method converged after a number of iterations smaller than the number of design parameters to be tuned.

B. Optimization Algorithm

1) Generation of the ideal rational scattering matrix that fulfills the specifications. 2) Synthesis of the target coupling matrix M (which are ordered in a vector m). In the case of multiple solutions, one solution is chosen as the target matrix. 3) Computation of the initial design parameters g k of the filters physical implementation, where k = 0. 4) Full-wave EM-simulation of the filters S-parameters in the frequency band of interest and the adjoint sensitivities of the S-parameters with respect to the design parameters g k . 5) Extraction of the coupling parameters (Section III): a) Rational approximation of the simulated S-parameters. b) Coupling matrix synthesis. c) Determination of all the possible solutions corresponding to the implemented coupling topology. 6) Estimation of the Jacobian for each solution and determination of the physically implemented coupling matrix using expression (12) (Section IV). 7) Computation of the error of the circuit parameters ∆m k . 8) Estimation of the correction for the design paramters ∆g k ≈ J -1 k ∆m k . 9) Termination of the optimization process if ∆g k ∞ < δ. 10) Update of the design parameters: g k+1 = g k + ∆g k and return to step 4.

VI. EXAMPLES A. Cascaded Triplet (CT) Filter

The optimization method is applied to the design of a 6 th order CT filter. The filter is designed to have center frequency f c = 1.2 GHz, a fractional bandwidth F BW = 0.04, a minimum return loss RL of 20 dB and 2 finite transmission zeros (TZs) at ω = 1.05 and ω = 2.1. The target coupling matrix is synthesized using Dededale-HF [START_REF] Seyfert | DEDALE-HF : a matlab toolbox dedicated to the equivalent network synthesis for microwave filters[END_REF]. There are 2 solutions to the reduction problem of which one is chosen (arbitrarily). In the chosen solution, the first triplet realizes the TZ at ω 1 = 2.1 and the second one the TZ at ω 2 = 1.05. The filter is implemented on a RO4360 substrate with a thickness of 1.016 mm and ε r = 6.15. To implement the λ 2 -resonators we use a variation of the square open-loop (SOLR) resonator, which is not square but rectangular. Figure 2 shows the topview of the layout of the CT filter. Table I contains the initial dimensions of the filter. There are 15 design parameters which are the spacing d kl between the resonators k and l and the distance between the ends g i of the resonator i. Note that the positions of the feeding lines t i are also design parameters. They are however not included in the optimization method, since CST is not capable to calculate the adjoint sensitivity with respect to a parameter that influences the position of a port. Therefore these parameters have to be tuned manually. Figure 3 shows the S-parameters of the initial design. Note that the initial design is far away from the target design. We did not use design curves to obtain initial values for the mixed couplings M 12 , M 23 , M 45 and M 56 but we have chosen values within the range of values found for the other spacings using visual inspection.

The coupling matrices are extracted using the method described in Section III. There are 2 solutions to the reconfiguration problem and thus there are 2 possible Jacobian matrices too. The parameters are ordered such that dominant elements should be positioned on the main diagonal of the matrix. When we calculate the relative influence of these dominant elements over the others for the first iteration, we find the following values for c p as defined in (12):

This indicates that the first solution corresponds to the physically implemented coupling matrix. Table II contains the coupling parameters associated to the initial design parameters.

The value of δ corr was set to 0.025 mm and after 4 iterations (5 EM-simulations) we found a ∆g k ∞ = 0.024 mm. On average, an EM-simulation takes 1 hour 5 min (with adjoint sensitivities included). Note that the adaptive meshing is included in this time and this already takes 40 minutes. An EM-simulation without adjoint sensitivities takes 45 minutes on average, indicating that the extra time that is needed to calculate the adjoint sensitivities is acceptable. Figure 4 shows the evolution of the inter-resonator couplings and of the selfcoupling during the tuning process. The figure shows that after 2 iterations, the coupling parameters are already very close to the target values. The last 2 iterations can be seen as fine tuning. Note that for clarity of the figure, we have not taken into account the sign of the inter-resonator couplings. Figure 5a shows the evolution of the 2-norm and the ∞norm of the design error ∆m. Note that ∞-norm is the smallest after iteration 2 (at simulation 3), meaning that the maximum difference with respect to the target value is the lowest at that iteration. However the 2-norm of the vector ∆m is clearly minimal after the 4 th iteration. Figure 5b shows the evolution of the 2-norm and the ∞-norm of the correction vector ∆g during the tuning process. The maximum correction proposed for the second iteration is larger than for the first, for the following iteration it clearly decreases. Note that these corrections are in mm. Finally remark that the TZs do not perfectly coincide. This is due to the presence of the unwanted parasitic couplings M 16 , M 26 and M 36 (shown in Table II). We have not re-distributed the parasitic couplings, so note that they are not necessarily physically located at those positions. Figure 6 shows the response that is obtained from the ideal coupling matrix when the parasitic couplings are added. It is clear that then the TZs now coincide. Note however that the parasitic couplings are small, we therefore choose not to re-optimize the target matrix.

VII. CONCLUSION

This paper introduces a novel CAT method which optimizes the coupling parameters. The method estimates the Jacobian of the function relating the physical design parameters of the filter to the implemented coupling parameters of the filter. The estimation combines the adjoint sensitivities of the S-parameters with respect to the coupling parameters and adjoint sensitivities of the S-parameters with respect to the design parameters. Formal expressions to calculate the adjoint sensitivities of the S-parameters with respect to the coupling parameters are given in this paper. Commercial EMsimulators are able to calculate adjoint sensitivities of the S-parameters with respect to the design parameters without drastically increasing simulation time. This makes that Jacobian is numerically cheap to estimate and only requires 1 EM simulation. Another major advantage is that the Jacobian provides a criterion to determine the implemented solution in the case of non-canonical topologies. The novel method is applied to a design having multiple solutions: a CT filter implemented in a microstrip technology. The example proves that the number of required EM-simulations is very low (less than half of the number of the design parameters).

Fig. 2 :

 2 Fig. 2: Top-view of the layout of a sixth order cascaded triplet filter.

Fig. 3 :

 3 Fig. 3: |S 11 | and |S 21 | for the initial design, the final loss-less design and the ideal response of the CT filter.

  Evolution of the inter-resonator couplings during the tuning process. Evolution of the self-couplings during the tuning process.

Fig. 4 :

 4 Fig. 4: Evolution of the coupling parameters during the tuning process. The full lines mark the target values of the first triplet and the dashed lines mark the target values of the second triplet.

  The 2-norm of the error vector ∆m 2 (-o) and the maximal error ∆m ∞ (-x).

  The 2-norm of the correction vector ∆g 2 (-o) and the maximal correction ∆g ∞ (-x) (mm).

Fig. 5 :

 5 Fig. 5: Evolution of the 2-norm and ∞-norm of the error vector ∆m and the correction vector ∆g.

Fig. 6 :

 6 Fig. 6: |S 21 | (dB) of the final (-) design and response obtained from the ideal coupling matrix (---) when the parasitic couplings are added versus the ideal response (---).

  ∂m k (ω 1 ) . . . ∂S11 ∂m k (ω n F ) . . .

	. . .	. . .	∂S11 ∂mn c . . .	(ω 1 )
	∂S12 ∂m1 (ω 1 ) . . . ∂S12 ∂m k (ω 1 ) . . . . . . . . .	∂S11 ∂mn c ∂S12 ∂mn c (ω n F ) (ω 1 ) . . .
	∂S12 ∂m1 (ω n F ) . . . ∂S12 ∂m k (ω n F ) . . . ∂S12 ∂mn c ∂S21 ∂m1 (ω 1 ) . . . ∂S21 ∂m k (ω 1 ) . . . ∂S21 ∂mn c (ω n F ) (ω 1 ) . . . . . . . . .
	∂S21 ∂m1 (ω n F ) . . . ∂S21 ∂m k (ω n F ) . . . ∂S21 ∂mn c ∂S22 ∂m1 (ω 1 ) . . . ∂S22 ∂m k (ω 1 ) . . . ∂S22 ∂mn c (ω n F ) (ω 1 ) . . . . . . . . .
	∂S22 ∂m1 (ω n F

∂S11

∂m1 (ω n F ) . . .

TABLE I :

 I Initial and final values for the physical design parameters of the CT filter.

TABLE II :

 II Extracted and target coupling parameters of the SOLR CT filter.


Boris Haspot
email: haspot@ceremade.dauphine.fr

On the Oseen vortices in dimension N = 2 for the inhomogeneous Navier-Stokes equations for radial initial density

This paper is dedicated to the proof of the existence of Lamb Oseen vortex for the inhomogeneous Navier-Stokes equation when N = 2. We restrict our study to the case of radial initial data ρ 0 which belongs to C α (R 2 ) with 0 < α < 1. To do this we recall the construction of fundamental solution for reaction diffusion equations. We point out that when ρ 0 = 1, our Lamb Oseen solution are not self similar. We prove also that the vorticity of inhomogeneous Navier-Stokes equations (when curlu 0 is radial and in L 1 (R 2 )) converges asymptotically in time to the Oseen solution of Navier Stokes equation provided that the fluctuation of the initial density is sufficiently small.

Introduction

In this paper, we are concerned with the following model of incompressible viscous fluid with variable density:

         ∂ t ρ + div(ρu) = 0,
∂ t (ρu) + div(ρu ⊗ u) -µ∆u + ∇Π = 0, divu = 0, (ρ, u) /t=0 = (ρ 0 , u 0 ). (1.1) Here u = u(t, x) ∈ R N stands for the velocity field and ρ = ρ(t, x) ∈ R + is the density, Du = 1 2 (∇u + t ∇u) is the strain tensor. We denote by µ the viscosity coefficients of the fluid, which is assumed to satisfy µ > 0. The term ∇Π (namely the gradient of the pressure) may be seen as the Lagrange multiplier associated to the constraint divu = 0. We supplement the problem with initial condition (ρ 0 , u 0 ) and an outer force f . Throughout the paper, we assume that the space variable x ∈ R N or to the periodic box T N a with period a i , in the i-th direction. We restrict ourselves to the case N = 2. The equations (1.1) are invariant under the sealing transformation: ρ(t, x) → ρ(λ 2 t, λx); u(t, x) → λu(λ 2 t, λx) and curlu(t, x) → λ 2 u(λ 2 t, λx), λ > 0.

(1.2)

1
We observe that the vorticity equation can be rewritten as follows:

             ∂ t ρ + div(ρu) = 0, ∂ t curlu + u • ∇curlu -µdiv( 1 ρ ∇curlu) + ∇ ⊥ ( 1 ρ ) • ∇Π = 0, divu = 0,
(ρ, u) /t=0 = (ρ 0 , u 0 ), (1.3) with ∇ ⊥ = (-∂ 2 , ∂ 1 ). Let us recall now some classical results for the Navier-Stokes equations in dimension N = 2.

Classical results of global strong solution for the Navier-Stokes equations

For the Navier Stokes equation, it is well known that there exists global self similar solution when N = 2. In order to obtain such solution, the main difficulty consists in proving the existence of global strong solution for initial vorticity in a functional space X which is invariant for the norm by the scaling of the equation (1.2). In addition X must be sufficiently large such that he admits homogeneous initial velocity of order -1.

A natural candidate is the space M(R 2 ) of all finite measures on R 2 , equipped with the total variation norm. The existence of global solution with initial data was first proved by Cottet [START_REF] Carpio | Asymptotic behavior for the vorticity equations in dimensions two and three[END_REF] and independently by Giga, Miyakawa and Osada [START_REF] Giga | Two-dimensional Navier-Stokes flow with measures as initial vorticity[END_REF]. In [START_REF] Giga | Two-dimensional Navier-Stokes flow with measures as initial vorticity[END_REF], the authors proved also the uniqueness when the atomic part of the initial vorticity is sufficiently small. Let us mention that the Navier-Stokes equations are well-posed for any large initial vorticity when curlu 0 ∈ L 1 (R 2 ) (see [START_REF] Ben-Artzi | Global solutions of two-dimensional Navier-Stokes and Euler equations[END_REF]). All the previous results use in a crucial way the bound of the L 1 norm of the vorticity for t > 0, this is a consequence in some sense of the maximum principle.

In the particular case where µ = αδ 0 for α ∈ R and (ρ = 1 the Navier Stokes equation) there exists a particular self similar solution, the so called Lamb-Oseen vortex given by:

curlu(t, x) = α t G( x √ t ), u(t, x) = α √ t v G ( x √ t ), x ∈ R 2 , t > 0, (1.4)
where:

G(ξ) = 1 4π e -|ξ| 2 4 , v G (ξ) = 1 2π ξ ⊥ |ξ| 2 (1 -e -|ξ| 2 4 ), ξ ∈ R 2 .
In [START_REF] Gallay | Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation[END_REF], Gallay and Wayne showed the uniqueness of the Lamb-Oseen solution by rewriting the system in self similar variables and using Lyapounov functions for the Fokker-Planck equations combined with the so called log Sobolev inequalities and Csiszár-Kullback inequalities (see [START_REF] Villani | A review of mathematical topics in collisional kinetic theory In: Handbook of mathematical fluid dynamics[END_REF]). We refer also to [START_REF] Gallagher | On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity[END_REF] for an another proof using symetrization techniques for parabolic equations The uniqueness for any curlu 0 ∈ M(R 2 ) is proved in [START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF], it allows to obtain the existence and the uniqueness of global self similar solution when N = 2.

Results of global strong solution for the non homogenous Navier-Stokes equations

Compared with the homogeneous Navier-Stokes equation (ρ = 1), to obtain self similar solutions seems more tricky. Indeed it is not clear due to the term

∇ ⊥ ( 1 ρ ) • ∇Π in (1.3)
that the L 1 norm of the vorticity curlu remains bound in L 1 norm all along the time.

Similarly if we try to show the existence of global self similar solution with

u 0 ∈ B N p -1 p,∞
small enough (which corresponds to the results obtained in [START_REF] Cannone | Solutions auto-similaires des équations de Navier-Stokes. Séminaire sur les équations aux dérivées partielles[END_REF] for the Navier-Stokes equation) and

ρ 0 -1 ∈ B N p
p,∞ ∩L ∞ , then it seems difficult to propagate the regularity

B N p p,∞
on the density via the mass equation. Indeed the velocity u will be a priori log Lipschitz and it will involve a loss of derivative on the density (see [START_REF] Haspot | Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity[END_REF] for more development on this question, where we prove the existence of global strong solution for initial data small enough in (B

N p +ε p,∞ ∩ L ∞ ) × B N p -1
p,∞ ). Danchin in [START_REF] Danchin | Density-dependent incompressible viscous uids in critical spaces[END_REF] proved the local wellposedness of the system (1.3) if the initial data

(ρ 0 -1, u 0 ) are in the critical Besov space B N p p,1 × B N p -1 p,1 for 1 ≤ p ≤ N . Furthermore the solution is global if the initial data are sufficiently small in B N p p,1 × B N p -1
p,1 . The smallness assumption for the initial density in [START_REF] Danchin | Density-dependent incompressible viscous uids in critical spaces[END_REF] the smallness assumption on the fluctuation of the initial density was removed in [START_REF] Abidi | On the decay and stability of global solutions to the 3D inhomogeneous Navier-Stokes equations[END_REF][START_REF] Abidi | On the wellposedness of 3D inhomogeneous Navier-Stokes equations in the critical spaces[END_REF], and in [START_REF] Danchin | A Lagrangian approach for the incompressible Navier-Stokes equations with variable density[END_REF] the authors extends the result of [START_REF] Danchin | Density-dependent incompressible viscous uids in critical spaces[END_REF] to the case 1 ≤ p < 2N . In [START_REF] Haspot | Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity[END_REF], we generalize the result of [START_REF] Cannone | Solutions auto-similaires des équations de Navier-Stokes. Séminaire sur les équations aux dérivées partielles[END_REF] to the case of the inhomogeneous Navier-Stokes equation by allowing to the initial velocity to belong to

B N p -1 p,r
with 1 ≤ r ≤ +∞ and p well chosen. By opposite we need to require to be slightly subcritical in terms of regularity for the fluctuateion of the density which must be in B N p +ε p,r ∩ L ∞ with ε > 0. Danchin and Mucha in [START_REF] Danchin | A Lagrangian approach for the incompressible Navier-Stokes equations with variable density[END_REF] introduced the Lagrangian approach and proved the existence global strong solution for small initial data with

(ρ 0 -1, u 0 ) ∈ M(B N p -1 p,1 ) × B N p p,1 ( M(B N p -1 p,1
) corresponds here to the multiplier set of B N p -1 p,1 ). This result is very interesting since it allows to the authors to deal with initial density which are not necessary continuous. We refer also to [START_REF] Danchin | Incompressible flows with piecewise constant density[END_REF][START_REF] Germain | Strong solutions and weak-strong uniqueness for the nonhomogeneous Navier-Stokes system[END_REF][START_REF] Chen | Fujita-Kato theorem for the 3d inhomogeneous Navier-Stokes equations[END_REF] for some extensions of this previous results. To finish let us recall that (1.1) is globally well-posed in dimension N = 2 (see [START_REF] Antontsev | Boundary value problems in mechanics of nonhomogeneous fluids[END_REF], see also more recently [START_REF] Chen | Fujita-Kato theorem for the 3d inhomogeneous Navier-Stokes equations[END_REF]), the question remains open for general viscosity coefficients depending on the density.

Fundamental solution for parabolic system and Parametrix

In the present paper we wish to exhibit Lamb Oseen solution for the non homogeneous Navier-Stokes equation when N = 2. It means solutions such that the initial vorticity corresponds to αδ 0 with α ∈ R. As we explained previously the term

∇ ⊥ ( 1 ρ ) • ∇Π in (1.
3) prevents any L 1 control of the voracity all along the time. It is then natural to search solution such that ∇ ⊥ ( 1 ρ ) • ∇Π = 0. To do this, we can assume geometrical condition on the velocity and the density, more precisely we are going to consider rotationally invariant initial data of the form:

ρ 0 (x) = ρ 0 (|x|) and u 0 (x) = ∇ ⊥ ψ(|x|), (1.5
)

with 1 ρ 0 ∈ L ∞ (R 2 ).
It is important to observe that this choice of initial density does not allow to obtain the existence of global self similar solution since ρ 0 should be homogeneous of degree 0. Let us search now solution of (1.1) verifying the following property:

ρ(t, x) = ρ(t, |x|) and u(t, x) = ∇ ⊥ ψ 1 (t, |x|) = x ⊥ |x| ψ 1 (t, |x|), (1.6)
with ψ 1 a radial function. We observe then at least heuristically that via the mass equation, the density is stationary and verifies for all t > 0:

ρ(t, |x|) = ρ 0 (|x|). (1.7)
Furthermore the velocity and the vorticity u and curlu verify:

     ρ 0 ∂ t u -µ∆u = 0, divu = 0, u(0, •) = u 0 .        ∂ t curlu -µdiv( 1 ρ 0 ∇curlu) = 0 divu = 0, curlu(0, •) = curlu 0 . (1.8)
We observe in particular that for such initial data curlu verifies a linear heat equation and conserve the mass L 1 at least if curlu 0 is positive (it suffices to apply the maximum principle). More generally we have for all t > 0:

curlu(t, •) L 1 (R 2 ) ≤ curlu 0 L 1 . (1.9)
Now we are interested in constructing Lamb Oseen solution which corresponds to the case where curlu 0 = αδ 0 with δ 0 the Dirac measure.

To do this, we are interested in constructing the fundamental solution associated to the equations (1.8). We assume now that ρ 0 verifies the following conditions:

                         ρ 0 (x) = ρ 0 (|x|) ∀x ∈ R 2 , 1 ρ 0 ∈ L ∞ (R 2 ), ρ 0 ∈ L ∞ , ∃C > 0, ∀(x, y) ∈ R 2 | 1 ρ 0 (x) - 1 ρ 0 (y) | ≤ C|x -y| α with 0 < α < 1, for i ∈ {1, 2} ∃C i > 0, ∀(x, y) ∈ R 2 | x i ρ 0 (|x|) ρ 2 0 (x) - y i ρ 0 (|y|) ρ 2 0 (y) | ≤ C i |x -y| α with 0 < α < 1, (1.10)
We are going now to recall some results due to A. Friedman and Ladyzenskaya et al (see [START_REF] Friedman | Partial differential equations of parabolic type[END_REF][START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF]) on the construction of the fundamental solution for linear parabolic equations. More precisely we wish to solve the following equation:

∂ t Z(x, ξ, t, τ ) -µdiv( 1 ρ 0 ∇Z)(x, ξ, t, τ ) = δ (x-ξ) δ (t-τ ) . (1.11)
with Z the fundamental solution of (1.8). We have in particular for any continuous function f with compact support:

lim t→τ R 2 Z(x, ξ, t, τ )f (ξ)dξ = 0. (1.12)
This implies that the initial data associated to Z(x, ξ, t, 0) is δ 0 .

We define now a parametric (this method is due to E. E. Levi (see [START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF]):

Z 0 (x -ξ, ξ, t, τ ) = ρ 0 (|ξ|) 4µπ(t -τ ) exp - ρ 0 (|ξ|) 4µ(t -τ ) |x -ξ| 2 , (1.13)
with Z 0 (x -ξ, ξ, t, τ ) = 0 when t < τ . We know in particular that:

∂ t [Z 0 (x -ξ, ξ, t, τ )] - µ ρ 0 (|ξ|) ∆ x Z 0 (x -ξ, ξ, t, τ ) = δ(x -ξ)δ(t -τ ). (1.14)
We define now the following kernel:

K(x, ξ, t, τ ) = µ( 1 ρ 0 (|ξ|) - 1 ρ 0 (|x|) )∆ x Z 0 (x -ξ, ξ, t, τ ) + L 1 (x, t, ∂ ∂x )Z 0 (x -ξ, ξ, t, τ ),
(1.15) with:

L 1 (x, t, ∂ ∂x )Z 0 (x -ξ, ξ, t, τ ) = -µ∇( 1 ρ 0 )(x, t) • ∇ x Z 0 (x -ξ, ξ, t, τ ). (1.16)
We deduce that:

∂ t [Z 0 (x-ξ, ξ, t, τ )]-µdiv x ( 1 ρ 0 ∇ x [Z 0 (x-ξ, ξ, t, τ )]) = δ(x-ξ)δ(t-τ )+K(x, ξ, t, τ ). (1.17)
We search now a solution Z(x, ξ, t, τ ) of (1.11) under the following form:

Z(x, ξ, t, τ ) = Z 0 (x -ξ, ξ, t, τ ) + t τ dλ R 2 Z 0 (x -y, y, t, λ)Q(y, ξ, λ, τ )dy. (1.18)
Assume that Q is a function satisfying a Hölder condition in (y, λ) then we have when we set Z (x, ξ, t, τ ) = t τ dλ R 2 Z 0 (x -y, y, t, λ)Q(y, ξ, λ, τ )dy:

∂ t [Z (x, ξ, t, τ )]-µdiv x ( 1 ρ 0 ∇ x Z (x, ξ, t, τ )) = Q(x, ξ, t, τ )+ t τ dλ R 2
K(x, y, t, λ)Q(y, ξ, λ, τ )dy.

(1.19) From (1.17), (1.18), (1.19) we deduce that (1.11) is verified if Q is solution of the following Volterra equation:

Q(x, ξ, t, τ ) + t τ dλ R 2 K(x, y, t, λ)Q(y, ξ, λ, τ )dy + K(x, ξ, t, τ ) = 0.
(1.20)

Proposition 1.1 There exists a solution Q of (1.20) with:

Q(x, ξ, t, τ ) = +∞ m=1 (-1) m K m (x, ξ, t, τ ), (1.21)
where K m is defined as follows:

K m (x, ξ, t, τ ) = t τ dλ R 2 K(x, y, t, λ)K m-1 (y, ξ, λ, τ )dy. (1.22)
In addition there exists c, C > 0 such that for any (x, ξ, t, τ ): .23) proof: We refer to [START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF], where the authors prove that there exists c, C > 0 such that for any m ≥ 1:

|Q(x, ξ, t, τ )| ≤ c(t -τ ) -4-α 2 exp(-C |x -ξ| 2 t -τ ). ( 1
|K m (x, ξ, t, τ )| ≤ c m ( π C ) m-1 Γ m ( α 2 ) Γ( mα 2 ) (t -τ ) mα-4 2 exp(-C |x -ξ| 2 t -τ ).
(1.24)

The series in (1.21) is then uniformly convergent and we obtain (1.1).

Let us now consider the Oseen solution u α,ρ 0 of (1.8) which verifies the heat equation:

   ∂ t curlu α,ρ 0 -µdiv( 1 ρ 0 ∇curlu α,ρ 0 ) = 0 curlu(0, •) = αδ 0 .
(1.25)

This solution corresponds to the Oseen tourbillon for the non homogeneous Navier Stokes equation with initial density ρ 0 (|x|) and initial tourbillon αδ 0 . We recover u by using the Biot Savart law.

We observe with the previous notation that for any t > 0:

curlu α,ρ 0 (t, •) = Z(x, 0, t, 0). (1.26)
We have the following property (see [START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF]).

Proposition 1.2 We have:

• |D r t D s x curlu α,ρ 0 (t, x)| ≤ Ct -N +2r+s 2 exp(-C 1 |x| 2 t ) (1.27)
where 2r + s ≤ 2, t > τ .

• |D r t D s x curlu α,ρ 0 (t, x) -D r t D s x curlu α,ρ 0 (x , t)| ≤ C |x -x | γ t -N +2+γ 2 + |x -x | β t -N +2-α+β 2 exp(-C 1 |x| 2 t ), (1.28)
where 2r + s = 2 (i.e r = 0, s = 2 and r = 1, s = 0), 0 ≤ γ, 0 ≤ β ≤ α, t > 0.

• |D r t D s x curlu α,ρ 0 (t, x) -D r t D s x curlu α,ρ 0 (t , x)| ≤ C (t -t )(t ) -N +2r+s+2 2 + (t -t ) 2-2r-s+α 2 (t ) -N +2 2 exp(-C 1 |x| 2 t ), (1.29)
where 2r + s = 1, 2 and t > t > 0.

To simplify the notation we assume now that µ = 1. Let us state our main result.

Theorem 1.1 Let N = 2, assume that ρ 0 verifies the assumptions (1.10) and u 0 (x) = ∇ ⊥ ψ(|x|) with curlu 0 ∈ L 1 (R 2 ). Then there exists a unique global strong solution curlu for the system (1.8) and furthermore for any p ∈ [1, +∞] there exists C p > 0 such that for all t > 0:

curlu(t, •) L p ≤ C p t 1-1 p curlu 0 L 1 ∀t > 0.
(1.30) Furthermore (ρ 0 , curlu(t, •)) is the unique solution for the non homogeneous Navier Stokes equation (1.3) in the sense of the mild solution.

Let us assume now in addition that 1 ρ 0 -1 ∈ L 2 (R 2 ) and that there exists ε > 0 sufficiently small and C > 0 such that for t 0 > 0:

R 2 |curlu 0 (x)| 2 exp( |x -x 0 | 2 4t 0 )dx ≤ C. (1.31)
and:

1 ρ 0 -1 L ∞ ≤ ε. (1.32)
Then for any α ∈ R If R 2 curlu o (x)dx = α then we have for a constant γ with 0 < γ < 1 2 and some constant C p depending only on p ∈ [1, +∞]:

(t + t 0 ) 1-1 p curlu(t) - α t + t 0 G( x -x 0 √ t + t 0 ) L p ≤ C p 1 (1 + t t 0 ) γ .
(1.33)

In addition there exists Lamb Oseen solution of (1.3) for initial data (ρ(0, •), curlu(0, •)) = (ρ 0 , αδ 0 ) with α ∈ R and (ρ(t, •), curlu(t, •)) = (ρ 0 , Z(•, 0, t, 0)) with Z defined by (1.26).

Remark 1

The uniqueness (for the equation (1.1) of the Lamb Oseen solution is true if α is small enough and if (ρ 0 -1) ∈ B 2+α 1,∞ , this is provided from [START_REF] Haspot | Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity[END_REF]. However the uniqueness of the Lamb Oseen solution for large α remains open. Let us also mention that in [START_REF] Haspot | Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity[END_REF], the existence of strong solution in finite time is proved

for initial data u 0 ∈ C ∞ c ∩ B 1 1,∞ B 1
1,∞ , it does not cover the case of vorticity curlu 0 which are finite measure. In this sense the existence of Oseen vortices extend the results of [START_REF] Haspot | Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity[END_REF]. The uniqueness (for (1.1 with (ρ 0 -1) ∈ B 2+α 1,∞ ) of the solution for curlu 0 ∈ L 1 (R 2 ) with the condition of the previous theorem provides of [START_REF] Haspot | Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity[END_REF]

since C ∞ c ∩ B 0 ∞,1 B 0 1,∞ is embedded in L 1 (R 2 )
and the the uniqueness is a local problem.

Remark 2

The problem concerning the existence or not of self similar solution with ρ 0 = 1 remains actually open.

Remark 3 We would like to point out that a vorticity in L 1 (R 2 ) does not imply a velocity field in R 2 . Indeed if u 0 ∈ L 2 (R 2 ) and curlu 0 ∈ L 1 (R 2 ) then we can verify that R 2 curlu 0 (x)dx = 0. Since the integral of curlu is conserved, it implies that if R 2 curlu 0 (x)dx = 0 then curlu 0 will be never in L 2 (R 2 ).

Remark 4

In [START_REF] Gallay | Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation[END_REF][START_REF] Gallay | Sur le temps de vie de la turbulence bidimensionnelle[END_REF] the authors estimate the time asymptotic rate of convergence to 0 of curlu(t)

-α t G( • √ t ) L 1 when R 2 w 0 (x)dx = α and w 0 ∈ L 1 (R 2 )
for the Navier-Stokes equations. We extend this result to the case of non homogeneous Navier-Stokes equation, indeed it is easy to observe that (1.33) implies that curlu(t) -α t G( • √ t ) L 1 converges to 0 when t goes to +∞. In particular we generalize the results of [START_REF] Rodrigues | Asymptotic stability of Oseen vortices for a density-dependent incompressible viscous fluid[END_REF] since we do not assume any smallness assumption on curlu(1, •) -αG L 1 .

We wish now to weaken the conditions (1.10) on the initial density ρ 0 . Indeed in the previous theorem, we need to assume that ∇ 1 ρ 0 belongs to C α (R 2 ) for 0 < α < 1. This is essentially due to the fact that we consider the solution of the equation (1.3) on curlu. Let us deal now with the equation (1.1), we have then the following result. Theorem 1.2 Let N = 2, assume that ρ 0 is radial and 1 ρ 0 belongs to C α (R 2 ) with 0 < α < 1. In addition we have 0 < c < ρ 0 ≤ M < +∞ and u 0 (x) = ∇ ⊥ ψ(|x|) with u 0 ∈ L 2,∞ (R 2 ). Then there exists a unique global strong solution u for the system (1.8). Furthermore (ρ 0 , u) is solution for the system (1.1) in the sense of the mild solution.

Remark 5 The uniqueness of the solution u for the equation ( 1

.1) is true if u 0 is small enough in B N p -1 p,∞ with suitable p such that 1 ≤ p ≤ 2, this is provided from [21] (if in addition (ρ 0 -1) ∈ B N p p,∞ ). In particular we can choose u 0 = α x ⊥ |x| 2 which is in B 1 ∞,∞ ∩L 2
,∞ and such that curlu 0 = αδ. Indeed we can observe that u 0 (x) = ∇ ⊥ ln(|x|) and we have (since the Fourier transform of ln(|x|) is -C |x| 2 with C > 0 a universal constant:

F∆ l u(0)(ξ) = 1 |ξ| 2 ϕ( ξ 2 l ),
we refer to [START_REF] Haspot | Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity[END_REF] for the definition of Littlewood-Paley theory with ϕ ∈ C ∞ (R 2 ) and suppϕ ∈ C( 14 , 3 4 ). We deduce than using the inverse Fourier transform that:

∆ l u 0 (x) = Fψ(-2 l x) with ψ(z) = ϕ(z) |z| 2 .
We conclude that ln(|x|) ∈ B 2 1,∞ .

Let us give an other direct consequence of the form that take the solution u of (1.8) which is simply an heat equation.

Corollary 1 Let N = 2, assume that ρ 0 is radial and 0 < c < ρ 0 ≤ M < +∞.

Furthermore let us consider u

0 (x) = ∇ ⊥ ψ(|x|) with u 0 ∈ B N p -1
p,∞ with 1 ≤ p < +∞. Assume that there exists ε > 0 sufficiently small such that:

1 ρ 0 -1 M(B N p -1 p,∞ ) ≤ ε,
then there exists a unique solution u of (1.8). (ρ 0 , u) is solution of (1.1).

Remark 6

The proof follows the same idea than in [START_REF] Danchin | A Lagrangian approach for the incompressible Navier-Stokes equations with variable density[END_REF] and in your case is a simple fixed pointed for the solution of (1.8). It is a priori not clear to know if (ρ 0 , u) is the unique solution of (1.1) even for small initial velocity. Indeed the initial data is completely critical for the initial density and the velocity.

2 Proof of the theorem 1.1 and 1.2

The existence part of the theorem is a direct consequence of the construction of a fundamental solution for the equation (1.8). Indeed the solution reads as follows:

curlu(t, x) = R 2 Z(x, t, ξ, 0)curlu 0 (ξ)dξ. (2.34)
We observe now that since curlu 0 is radial curlu(t, •) is also radial for any t > 0. Indeed it provides from the fact that curlu(t, •) is a convolution between curlu 0 with a kernel K(x -ξ, t) = Z(x, ξ, t, 0) which is radial. The convolution preserves the radial property.

Using Biot Savart formula we deduce that u is solution of the non homogeneous Navier-Stokes equation since ρ 0 u • ∇u is a gradient of a radial function. Indeed we have for t > 0:

u(t, x) = 1 2π ∇ ⊥ x R 2
ln(|x -y|)curlu(t, y)dy, and we have that the initial data is verified in the sense that for any

ϕ ∈ C ∞ c (R 2 )
It implies that there exists a radial function F such that ρ 0 u • ∇u = ∇F (t, |x|). It shows in particular that u is solution of the equation (1.3). The proof of (1.30) is a direct consequence of (2.34) and (1.27).

Let us now prove the asymptotic time decay estimate (1.33). Following [START_REF] Gallay | Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation[END_REF][START_REF] Gallay | Sur le temps de vie de la turbulence bidimensionnelle[END_REF], we define the rescaled vorticity w 1 (τ, ξ) = curlu 1 (τ, ξ) and the density ρ 1 (τ, ξ) by:

w(t, x) = 1 (t + t 0 ) w 1 ( x -x 0 √ t + t 0 , log(1 + t t 0 )) and ρ(t, x) = ρ 1 ( x -x 0 √ t + t 0 , log(1 + t t 0 )), with ξ = x -x 0 √ t + t 0 and τ = log(1 + t t 0 ) ∈ [0, +∞[.
(2.35) Here we have defined w as follows w = curlu. We have in particular: w 1 (τ, ξ) = t 0 e τ w(t 0 (e τ -1), √ t 0 e τ 2 ξ + x 0 ) and ρ 1 (τ, ξ) = ρ(t 0 (e τ -1),

√ t 0 e τ 2 ξ + x 0 ) (2.
36) The equation verified by w 1 (τ, ξ) reads:

∂ τ w 1 -div( 1 ρ 1 ∇w 1 ) - 1 2 ξ • ∇w 1 -w 1 = 0.
(2.37)

We now define L as follows:

Lf = ∆f + 1 2 ξ • ∇f + f,
which is the so called Fokker-Planck operator. We observe that KerL = Vect G with G defined as follows:

G(ξ) = 1 4π e -|ξ| 2 4 .
G in kinetic theory is called the Maxwellian. We are going now to estimate w = w 1 -αG which satisfies:

∂ τ w -∆ w - 1 2 ξ • ∇ w -w = div(( 1 ρ 1 -1)∇w 1 ). (2.38)
The idea is classical now for the Fokker Planck equations, we are going to estimate the time asymptotic convergence of a solution w 1 of (2.37) to the Maxwellian G. To do this, the idea consists (see also [START_REF] Rodrigues | Asymptotic stability of Oseen vortices for a density-dependent incompressible viscous fluid[END_REF]) in multiplying the equation (2.38) by G -1 w and estimate the norm | w| G,2 with:

| w| G,2 = G -1 2 w L 2 (R 2 ) .
(2.39)

The idea is to observe that if

L = G -1 2 LG 1 2 then L = -∆ + |ξ| 2 16 -1
2 is a harmonic oscillator with spectrum {0, 1 2 , 1, 3 2 , • • • } ( see [START_REF] Rodrigues | Asymptotic stability of Oseen vortices for a density-dependent incompressible viscous fluid[END_REF]). Moreover 0 is a simple eigenvalue with eigenvector G

1 2 . In particular if f is in the domain of L with R 2 G 1 2 f = f, G 1 2 = 0, then: R 2 f Lf dx ≥ 1 2 f 2 L 2 .
(2.40)

Coming back to L, we obtain if G -1 2 w belongs to the domain of L with R 2 wdx = 0, then for any 0 < γ < 1 2 taking w = G 1 2 f we have:

R 2 G -1 wL w = -(1 -γ) R 2 f Lf + γ R 2 G -1 wL w ≤ - 1 2 (1 -γ)| w| 2 G,2 + γ R 2 G -1 wL w
Now using integration by parts on the formula of L, we have:

R 2 G -1 wL w ≤ - 1 2 (1 -2γ)| w| 2 G,2 -γ( ∇(G -1 2 w) 2 L 2 + | |ξ| 4 w| 2 G,2 ).
From Young inequality we obtain:

R 2 G -1 wL w ≤ - 1 2 (1 -2γ)| w| 2 G,2 -γ( 1 3 |∇ w| 2 G,2 + 1 2 | |ξ| 4 w| 2 G,2 ).
Next we have:

R 2 div(( 1 ρ 1 -1)∇w 1 ) w G -1 dξ = R 2 div(( 1 ρ 1 -1)∇ w) w G -1 dξ + α R 2 div(( 1 ρ 1 -1)∇G) w G -1 dξ = - R 2 ( 1 ρ 1 -1)|∇ w| 2 G -1 dξ - 1 2 R 2 ( 1 ρ 1 -1)∇ w • ξG -1 wdξ + α 2 R 2 ( 1 ρ 1 -1)ξ • ∇ wdξ + α 4 R 2 ( 1 ρ 1 -1) w|ξ| 2 dξ
Multiplying the momentum equation (2.38) by G -1 w and combining all the previous estimates, we obtain:

1 2 d dt | w| 2 G,2 + 1 2 (1 -2γ)| w| 2 G,2 + γ( 1 3 |∇ w| 2 G,2 + 1 2 | |ξ| 4 w| 2 G,2 ) ≤ - R 2 ( 1 ρ 1 -1)|∇ w| 2 G -1 dξ - 1 2 R 2 ( 1 ρ 1 -1)∇ w • ξG -1 wdξ + α 2 R 2 ( 1 ρ 1 -1)ξ • ∇ wdξ + α 4 R 2 ( 1 ρ 1 -1) w|ξ| 2 dξ.
(2.41)

Using Young inequality we have:

1 2 d dt | w| 2 G,2 + 1 2 (1 -2γ)| w| 2 G,2 + γ( 1 3 |∇ w| 2 G,2 + 1 2 | |ξ| 4 w| 2 G,2 ) ≤ 1 ρ 1 -1 L ∞ |∇ w| 2 G,2 + ε 4 1 ρ 1 -1 L ∞ |∇ w| 2 G,2 + 1 4ε 1 ρ 1 -1 L ∞ ||ξ| 2 w| 2 G,2 + |α| 2 ( 1 ρ 1 -1)G 1 2 L 2 |∇ w| G,2 + |α| 4 ( 1 ρ 1 -1)G 1 2 L 2 ||ξ| w| G,2 .
(2.42)

We observe now that:

( 1 ρ 1 (τ, •) -1)G 1 2 L 2 ≤ 1 2 √ πt 0 e -τ 2 1 ρ 0 -1 L 2 .
(2.43) From (2.42) and (2.43), we deduce from Young inequality that for ε , ε 1 > 0:

1 2 d dτ | w(τ )| 2 G,2 + 1 2 (1 -2γ)| w(τ )| 2 G,2 + γ( 1 3 |∇ w(τ )| 2 G,2 + 1 2 | |ξ| 4 w(τ )| 2 G,2 ) ≤ |∇ w(τ )| 2 G,2 ( 1 ρ 1 (τ ) -1 L ∞ + ε 4 1 ρ 1 (τ ) -1 L ∞ + |α|ε 8 √ πt 0 1 ρ 0 -1 L 2 ) + |α| 8ε √ πt 0 e -τ 2 1 ρ 0 -1 L 2 e -τ + ||ξ| 2 w(τ )| 2 G,2 ( 1 4ε 1 ρ 1 -1 L ∞ + ε 1 |α| 16 √ πt 0 1 ρ 0 -1 L 2 ) + |α| 16ε 1 √ πt 0 1 ρ 0 -1 L 2 e -τ ≤ |∇ w(τ )| 2 G,2 ( 1 ρ 0 -1 L ∞ + ε 4 1 ρ 0 -1 L ∞ + |α|ε 8 √ πt 0 1 ρ 0 -1 L 2 ) + |α| 8ε √ πt 0 e -τ 2 1 ρ 0 -1 L 2 e -τ + ||ξ| 2 w(τ )| 2 G,2 ( 1 4ε 1 ρ 0 -1 L ∞ + ε 1 |α| 16 √ πt 0 1 ρ 0 -1 L 2 ) + |α| 16ε 1 √ πt 0 1 ρ 0 -1 L 2 e -τ
(2.44) Since 1 ρ 0 -1 L ∞ is sufficiently small, we deduce by bootstrap that there exists 0 < C <

Proof of the theorem 1.2

It suffices to construct as previously a fundamental solution of the equation:

ρ 0 ∂ t ∆u -∆u = 0.
We have then an explicit solution which is also solution of (1.1).

  +∞, we deduce that there exists C 2 > 0 depending on D such that:| w(τ )| G,2 ≤ C 2 e -Cτ ,(2.47) withe 0 < C < 1 2 . It implies in particular that setting:R 2 |t 0 e τ w(t 0 (e τ -1), √ t 0 e τ 2 ξ + x 0 ) -αG(ξ)| 2 G -1 (ξ)dξ ≤ C 2 2 e -2CτFrom Höder inequality we deduce that for 1 ≤ p ≤ 2 since by definition w = curlu:(t 0 + t)By duality we prove the same estimate for 2 ≤ p ≤ +∞. It concludes the proof of the theorem 1.1.

	and C 1 > 0 such that: From Grönwall lemma, we have for any τ > 0: 1 2 d dτ | w(τ )| 2 G,2 + C| w(τ )| 2 G,2 ≤ C 1 e -τ . | w(τ )| 2 G,2 ≤ e -2Cτ | w(0)| 2 G,2 + 2C 1 e -τ . Now since we have D = R 2 |w 0 (x)| 2 exp( |x -x 0 | 2 4t 0 )dx = R 2 |w 1 (0, ξ)| 2 exp( |ξ| 2 4 |t 0 e τ w(t 0 (e τ -1), z) -αG( z -x 0 √ t 0 e τ 2 )| 2 G -1 ( z -x 0 √ t 0 e τ 2 )dξ ≤ C 2 2 e -2Cτ R 2 |(t 0 + t)w(t, z) -αG( z -x 0 √ t 0 + t )| 2 dξ ≤ C 2 2 1 (1 + t t 0 ) 2C √ t 0 + t w(t, •) -α t + t 0 G( • -x 0 √ t 0 + t ) G -1 ( z -x 0 √ t 0 e τ 2 ) L 2 ≤ C 2 1 (1 + t t 0 ) C )dξ ≤ C < R 2 1-1 p curlu(t, •) -α t + t 0 G( • -x 0 √ t 0 + t L 2 ≤ C 2,p (1 + t t 0 ) C .	1 2 (2.45) (2.46)

Acknowledgements

The author has been partially funded by the ANR project INFAMIE ANR-15-CE40-0011.


Jean Louet
email: jean.louet@math.u-psud.fr

Some results on Sobolev spaces with respect to a measure and applications to a new transport problem

We recall some known and present several new results about Sobolev spaces defined with respect to a measure µ, in particular a precise pointwise description of the tangent space to µ in dimension 1. This allows to obtain an interesting, original compactness result which stays open in R d , d > 1, and can be applied to a new transport problem, with gradient penalization.

Introduction

Let us consider variational problems, consisting in the minimization of

J : u → Ω L(x, u(x), ∇u(x)) dµ(x)
where the usual Lebesgue measure is replaced by a generic Borel measure µ, under possible µ-a.e. or boundary constraints. In calculus of variations, the direct method consists in extracting a converging subsequence (in a suitable sense) from a minimizing sequence, thanks to a compactness result on the set of admissible functions, and to conclude by semi-continuity of the functional J. For our functional, two problems appear:

• which functional space should we consider in order to give a sense to the gradient ∇u? More precisely, if µ is the Lebesgue measure or has a density f bounded from above and below, we can work in the classical Sobolev space H 1 (Ω) (which is exactly the set of functions u ∈ L 2 µ having weak derivatives in L 2 µ ), but it is not so clear if this assumption on f does not hold or if µ has a singular part.

• does there exist a compactness result which allows to extract from a minimizing sequence a subsequence converging, in a suitable sense, to an admissible function? For instance, in the classical Sobolev space H 1 (Ω), the Rellich theorem allows to extract from any bounded sequence a stronglyconvergent subsequence in L 2 (Ω) which is a.e. convergent on Ω.

Let us fix more precisely the notations. Let Ω be a bounded open set of R d and f a measurable and a.e. positive function on Ω. If we assume f to be bounded from below and above, it is obvious that the set {u ∈ L 2 µ (Ω) : ∇u exists in the weak sense and belongs to L 2 µ (Ω) d } is exactly the classical Sobolev space H 1 (Ω), since the L 2 µ -norm is equivalent to the usual L 2 -norm on Ω. If f is only assumed to be positive, for u ∈ L 2 (Ω), the Cauchy-Schwarz inequality gives

Ω |u(x)| dx = Ω |u(x)| f (x) dx f (x) ≤ Ω |u(x)| 2 f (x) dx 1/2 Ω dx f (x) 1/2
thus, under the assumption

1/f ∈ L 1 (Ω) (1)
we have the continuous embedding

L 2 f (Ω) := u : Ω |u(x)| 2 f (x) dx < +∞ → L 1 (Ω).
In this case, any function f ∈ L 2 f (Ω) has a gradient ∇u in the weak sense, since it is locally integrable on Ω, and we can define the weighted Sobolev space with respect to f

H 1 f (Ω) = u ∈ L 2 f (Ω) : ∇u ∈ L 2 f (Ω) .
More generally, if p ∈ ]1, +∞[ , the assumption

(1/f ) 1/(p-1) ∈ L 1 loc (Ω)
is a well-known sufficient condition to define the weighted Sobolev space W 1,p (Ω) with respect to f (see [START_REF] Kufner | How to define reasonably weighted Sobolev spaces[END_REF] for more details). For our problem, if the Lagrangian functional is quadratic with respect to the gradient, for instance

J(u) = Ω |∇u| 2 dµ,
it means that the set of admissible functions is well-defined if µ has a density f such that 1/f is integrable: it is the set of the elements of the weighted Sobolev space H 1 f satisfying the constraints. If µ is absolutely continuous with density f , the weighted Lebesgue space with respect to f is exactly the space L p µ , so that the space H 1 f can be seen as a Sobolev space with respect to the measure µ. A natural generalization consists in defining the Sobolev space with respect to the measure µ, without condition on its density or when µ is not anymore assumed to be absolutely continuous with respect to L d . There exists some general definitions of the Sobolev space in a generic metric measure space (X, d, µ) (see [START_REF] Hajłasz | Sobolev spaces on metric-measure spaces[END_REF]), but we will not enter to the details of this notions in this paper and we prefer to focus on the case of an open set of R d .

We begin this paper by an overview of the definitions and already known results about this Sobolev spaces [START_REF] Bouchitté | Energies with respect to a measure and applications to low dimensional structures[END_REF][START_REF] Fragalà | Tangential calculus and variational integrals with respect to a measure[END_REF][START_REF] Fragalà | On some notions of tangent space to a measure[END_REF][START_REF] Preiss | Geometry of measures on R n : distribution, rectifiability and densities[END_REF][START_REF] Zhikov | On an extension of the method of two-scale convergence and its applications[END_REF][START_REF] Zhikov | Homogenization of elasticity problems on singluar structures[END_REF], and present several new results: in particular, we give a precise description of the tangent space to any measure µ on the real line. As a corollary of this result, we show a compactness result in H 1 µ , which states precisely that any bounded sequence admits a pointwise µ-a.e. convergent subsequence on the set of points where the tangent space is not null (this result is already known in any dimension under strong conditions on the measure µ, when the compact embedding of the Sobolev space W 1,p µ with respect to µ into the Lebesgue space L p µ still holds; see [START_REF] Bouchitté | Second-order energies on thin structures: variational theory and nonlocal effects[END_REF][START_REF] Hajłasz | Sobolev met Poincaré[END_REF]). This is applied to a variational problem coming from optimal transportation: we consider the minimization of the functional

J(T ) = Ω L(x, T (x), D µ T (x)) dµ(x)
among all the maps T : Ω → R d which admit a Jacobian matrix D µ T with respect to µ and under a constraint on the image measure T # µ (it corresponds to the classical Monge-Kantorovich optimal transportation problem [START_REF] Villani | Optimal transport: Old and New[END_REF] if L does not depend on its third variable, and is linked to minimization problems under volume-preservation or area-preservation constraints [START_REF] Angenent | On area preserving maps of minimal distorsion[END_REF]). In the one-dimensional case, we get the existence of a solution for any measure µ (the optimal map is known if µ is assumed to be the uniform measure on the interval, see [START_REF] Louet | A sharp inequality for transport maps in W 1,p (R) via approximation[END_REF] for details). However, we are not able to give a precise description of the tangent space and to obtain the existence of solution to this transport problem in the most general case in any dimension.

1 Sobolev spaces with respect to a measure

This section is devoted to an overview of the definitions and already known results about tangent spaces to a generic Borel measure µ and Sobolev spaces associated to this measure. First, let us recall that there exist some notions of Sobolev spaces in arbitrary metric measure spaces (X, d, µ), for instance in the papers by Shanmugalingam [START_REF] Shanmugalingam | Newtonian spaces: an extension of Sobolev spaces to metric measure spaces[END_REF], Hajłasz [START_REF] Hajłasz | Sobolev Spaces on an Arbitrary Metric Space[END_REF] or Hajłasz and Koskela [START_REF] Hajłasz | Sobolev met Poincaré[END_REF] (see [START_REF] Hajłasz | Sobolev spaces on metric-measure spaces[END_REF] for a global summary of this notions). In our case, a usual method consists in defining the tangent space to µ (which is a function defined µ-a.e. on R d and taking values in the set of linear subspaces of R d ), and the gradient with respect to µ for a regular function u through

∇ µ u(x) = p Tµ(x) (∇u(x)) for µ-a.e. x ∈ R d ,
where p Tµ(x) is the orthogonal projection on T µ (x) in R d . Then we consider for the Sobolev space

H 1 µ the closure of C ∞ Ω for the norm u ∈ C ∞ Ω → ||u|| L p µ + ||∇ µ u|| L p µ .
There exist several ways to define the tangent space of a generic measure µ. Preiss [START_REF] Preiss | Geometry of measures on R n : distribution, rectifiability and densities[END_REF] gives a method based on the idea of blow-up: a k-dimensional subspace P µ is said to be an approximate tangent space of µ at x if we have, for some θ > 0, the following convergence in the vague topology of measure when ρ goes to 0:

µ(x + ρ • ) θH k | Pµ .
In order to examine variational problems, Bouchitté et al. [START_REF] Bouchitté | Energies with respect to a measure and applications to low dimensional structures[END_REF] have introduced a dual-formulation of the tangent space: it is the µ-ess. union (see [START_REF] Castaing | Convex analysis and measurable multifunctions[END_REF] or later) x → Q µ (x) of the tangent fields, i.e. the vector fields belonging to

X p µ = {ϕ ∈ (L p µ ) d : div(µϕ) ∈ L p µ }
, where the operator div(µv) is defined in the distributional sense. Fragalà and Mantegazza [START_REF] Fragalà | On some notions of tangent space to a measure[END_REF] have noticed that, with this notation, we have the inclusion Q µ (x) ⊆ P µ (x) for µ-a.e. of R d (see the PhD. thesis [START_REF] Fragalà | Tangential calculus and variational integrals with respect to a measure[END_REF] for a complete overview and more details about these definitions).

We are interested in another way to define tangent and Sobolev spaces, introduced by Zhikov [START_REF] Zhikov | On an extension of the method of two-scale convergence and its applications[END_REF][START_REF] Zhikov | Homogenization of elasticity problems on singluar structures[END_REF]. Let Ω be a bounded open set of R d and µ a finite positive measure on Ω We will say that u ∈ L 2 µ belongs to the space H 1 µ if it can be approximated by a sequence of regular functions whose gradients have a limit in the space L 2 µ :

u ∈ H 1 µ ⇐⇒ ∃(u n ) n ∈ C ∞ (Ω), v ∈ (L 2 µ ) d : u n → u v n → v for the L 2 µ -norm.
The set of these limits v is denoted by Γ(u), and its elements are called gradients of u. In general, u can have many gradients (see below the example of a measure supported on a segment of R 2 ), and it is obvious that Γ(u) is a closed affine subspace of (L 2 µ ) d with direction Γ(0). The projection of 0 onto this subspace (in the Hilbert space (L 2 µ ) d ) is thus the unique element of Γ(u) with minimal L 2 µ -norm: we call it tangential gradient of u with respect to µ. Pointwise description of ∇ µ u and tangent space to µ. We define the tangent space to µ as follows: the space Γ(0) can be seen as the set of vector-valued functions which are pointwise orthogonal to the measure µ. Let us denote by (e 1 , . . . , e d ) the canonical basis of R d , and set

ξ i = p Γ(0) (e i )
where the projection is taken in the Hilbert space L 2 µ (here e i is seen as a constant function on Ω). For x ∈ Ω, we denote by

T µ (x) = (Vect(ξ 1 (x), . . . , ξ d (x)))
⊥ and call T µ (x) (which is defined for µ-a.e. x ∈ Ω) the tangent space to µ at x. Then, the following equivalence holds:

v ∈ Γ(0) ⇐⇒ for µ-a.e. x ∈ Ω, v(x) ⊥ T µ (x).
This result, combined to the orthogonality property of ∇ µ u in L 2 µ , implies a pointwise description of the tangential gradient:

Proposition 1.1. Let u ∈ H 1 µ . Then, for v ∈ Γ(0), the function x ∈ Ω → p Tµ(x) (v(x))
is independent of the function v and only depends on u, and we have

∇ µ u(x) = p Tµ(x)
for µ-a.e. x ∈ Ω.

Some natural examples. We can see that the words "tangential gradients" are quite natural in the following cases:

• if µ is the Lebesgue measure L 1 concentrated on the segment I = [0, 1] × {0} × • • • × {0}, then T µ is the line R × {0} × • • • × {0} a.
e. on I and

H 1 µ = u ∈ L 2 µ : ∂u ∂x 1 ∈ L 2 µ and ∇ µ u = ∂u ∂x 1 , 0, . . . , 0 ;
• more generally, if µ is the uniform Hausdorff measure supported on a k-dimensional manifold M , then T µ is the tangent space to M in the sense of the differential geometry.

Let us remark that, if v is a tangent field as defined above, i.e. the operator div(ϕµ) is continuous for the L 2 µ -norm on D(Ω), we have for any sequence (u n ) n of smooth functions having 0 for limit in L 2 µ :

Ω ∇u n • ϕ dµ ≤ C ||u n || L 2 µ → 0. Then, if v ∈ Γ(0), we have v • ϕ in L 2 µ for any element ϕ ∈ X 2 µ .
We deduce that, with the above notations, the space Q µ (x) is pointwise orthogonal to F µ (x): it means that, up to a µ-negligible set, we have the inclusion between tangent spaces

Q µ (x) ⊆ T µ (x).
We are not able for the moment to prove the inverse inclusion, but the equality between this linear spaces holds for all the examples that we have studied.

2 Precise description and compactness result in one dimension

The main results

Let us now give a precise pointwise description of the tangent space T µ (x) when d = 1 and Ω is a bounded interval of R (which we denote by I). In this case, there are only two options for T µ (x) which are of course {0} and R, and the definitions of the tangent space give the following characterizations: Fact. Let B ⊆ I be a Borel set with µ(B) > 0. We have the following implications:

1. if any v ∈ Γ(0) is µ-a.e. null on B, then T µ = R µ-a.e. on B;
2. if, for any u ∈ H 1 µ , there exists a gradient of u which is µ-a.e. null on B, then T µ = 0 µ-a.e. on B;

3. if there exists a gradient of 0 which is positive µ-a.e. on B, then T µ = 0 µ-a.e. on B.

Notations. We denote by:

• µ = µ a + µ s , where µ a and µ s are respectively the absolutely continuous and the singular part of µ with respect to the Lebesgue measure;

• A a Lebesgue-negligible set on which is concentrated µ s ;

• f the density of µ a , and

M = x ∈ I : ∀ε > 0, I∩B(x,ε) dt f (t) = +∞ which is a closed set of I verifying 1/f ∈ L 1 loc (I \ M ).
Notice that if µ is absolutely continuous with respect to L 1 , the Sobolev space with respect to µ (thus, to f ) is well-defined exactly "outside of the set M ". In our case, we find an analogous result:

Theorem 2.1. For µ-a.e. x ∈ I, the tangent space is given by

T µ (x) = {0} if x ∈ M ∪ A R otherwise.
Let us give a short comment of this result. Saying that the tangent space is R on a set B means exactly that, if u ∈ H 1 µ is given, all the gradients of u are equals on B. In our case, let us denote by

V = I \ (M ∪ A) and U = I \ M . Notice that U is an open subset of I coinciding with V up to the L 1 - negligible set A. Let us fix u ∈ H 1
µ . We will prove that the distributional derivative of u| U is well-defined, belongs to L 2 f and that, if v ∈ Γ(u), u = v µ-a.e. on V ; therefore, u | V is the only gradient of u on the set V .

First, let us recall that if u is an element of L

) → (u, v) in L 2 µ , testing v -u against ϕ gives | < v -u , ϕ > D (U ),D(U ) | = lim n→+∞ I (u n -u)ϕ ≤ ||u n -u|| L 2 f I (ϕ ) 2 f 1/2
where the last inequality comes from the Hö lder inequality, and the last term is finite since ϕ is bounded and 1/f integrable on the support of ϕ. This proves that u n → v in the sense of distributions on U . Then v| U is the weak derivative of u on this set, and we know that v| U ∈ L 2 f (U ). Finally, any element u ∈ H 1 µ gives by restriction an element of the weighted Sobolev space H 1 f (U ) and, on U , ∇ µ u and u are coinciding a.e. for the regular part f L d of µ. To summarize, we have just proved the following:

Proposition 2.1. We denote by V = I \ (M ∪ A). Let us recall that A is Lebesgue-negligible and that V ∪ A is open; we still denote by H 1 f (V ) the weighted Sobolev space H 1 f (V ∪ A).
Then, a measurable function u belongs to the Sobolev space H 1 µ (I) if and only if the two following conditions are satisfied:

u ∈ L 2 µ (I) and u| V ∈ H 1 f (V )
and in this case, its µ-Sobolev norm is given by

||u|| 2 H 1 µ (I) = ||u|| 2 L 2 µ (I) + ||u || 2 L 2 f (V )
where u is the weak derivative of u| V .

Compactness result in H 1 µ (I). In order to examine variational problems in this Sobolev spaces, the following compactness result is useful (it is already known in the case of the Lebesgue measure, as a consequence of the Rellich theorem):

Proposition 2.2. Let (u n ) n be a bounded sequence of H 1 µ (I).
Then there exists a subsequence (u n k ) k which admits a pointwise limit u on µ-a.e. every point on which T µ is R.

Proof. We know that V is exactly (up to a µ-negligible set) the set of points where T µ is R. We still denote by U = I \ M . U is an open set and we have U = V ∪ A. We will show that (u n ) n admits a subsequence which is pointwise convergent on L 1 -a.e. any point of U : it will be enough to conclude that this subsequence is µ-a.e. convergent on V , since µ| V is absolutely continuous with respect to the Lebesgue measure.

The sequence (u n ) is bounded in H 1 µ (I), thus the sequence (u n | U ) n is bounded in the weighted Sobolev space H 1 f . But since U is exactly the set of points around which 1/f is integrable, we know that L 2 f (U ) → L 1 loc (U ); this implies that the sequence of the weak derivatives of u n (which are functions of L 2 f (U )) is bounded in L 1 loc (U ). Then (u n ) n is bounded in the Sobolev space W 1,1 loc (U ), and admits a subsequence which is strongly convergent in L 1 (K), for any compact subset K of I. We can again extract a subsequence which is pointwise convergent on µ-a.e. point of I; the proof is complete.

First part of the proof: the regular part, outside of the critical set

First, let us prove that T µ = R outside of M ∪ A. Using the first characterization of the tangent space, we take an element g of Γ(0) and we want to show that g = 0 µ-a.e. outside of M ∪ A; by definition of A, it is enough to show that g = 0 L 1 -a.e. on U . As in the above remark, taking a sequence of regular functions u n → 0 with u n → g and a test function ϕ such that 1/f is integrable on the support of ϕ, we obtain

U u n ϕ = U u n ϕ ≤ U u n f ϕ √ f ≤ U u 2 n f 1 2 U ϕ 2 f 1 2
which goes to 0 as n → +∞. The same computation gives U u n ϕ → U gϕ. We deduce that g = 0 L 1 -a.e.

Second part: the singular part of the measure

Second, we prove that T µ = {0} for the singular part of µ. We use the third characterization of the tangent space and build a sequence of C 1 functions (u n ) n such that

u n → 0 and u n → 1 A in L 2 µ .
where 1 A is the characteristic function of the set A; this will prove that 1 A ∈ Γ(0) and imply the result.

For n ∈ N, let Ω n be an open set such that A ⊆ Ω n and µ(Ω n \ A) + L 1 (Ω n ) ≤ 1/n. By Lusin theorem, there exists a continuous function v n with 0 ≤ v n ≤ 1 on I and

(µ + L 1 )({x ∈ I : v n (x) = 1 Ωn (x)}) ≤ 1/n Let us consider u(x) = x a v n (x) dx,
where a is the lower bound of I. Then we have:

• for any x ∈ I, |u n (x)| ≤ I (|v n -1 Ωn |(t) + 1 Ωn (t)) dt ≤ L 1 ({v n = 1 Ωn }) + L 1 (Ω n ) ≤ 2/n
thus (u n ) n goes to 0 uniformly, and also in the space L 2 µ ;

• on the other hand, since u n = v n coincides with 1 A outside of a set E n such that µ(E n ) ≤ 1/n, we have

I |u n (x) -1 A (x)| 2 dµ(x) ≤ ||v n -1 A || 2 ∞ µ(E n ) ≤ 4/n thus u n → 1 A in L 2 µ (I).
We obtain that 1 A ∈ Γ(0), which guarantees that T µ = 0 on A.

Third part: the critical set

This part is more difficult. Given a function u ∈ C 1 I , we build a sequence (u n ) n of regular functions (say, C 1 ) such that u n → u and u n → v for the L 2 µ -norm, with v = 0 on M . The strategy is the following:

• given a set Ω n which is "almost" M , we start from a function u n which coincides with u outside of Ω n and is piecewise constant on Ω n (so that its derivative is null on M );

• then, using the fact that the discontinuity points of u n belong to M , we regularize u n around this points so that its derivative stays small for the L 2 µ -norm.

First, we build our set Ω n :

Lemma 2.1. Let us denote by (x n ) n a sequence containing all the atoms of µ. For n ∈ N, there exists a set Ω n such that:

• Ω n = pn i=1 ]a i , b i [ , with b i < a i+1 for each i, and ]a i , b i [ ∩M = ∅; • Ω n ⊇ M \ {x 1 , . . . , x n }; • µ(Ω n \ (M \ {x 1 , . . . , x n })) ≤ 1/n.
Proof. Let U n be an open set such that M ⊆ Ω n and µ(U n \ M ) ≤ 1/n (such a set exists since µ is regular from above); U n is a union of open intervals, and since M is compact we can assume this union to be finite. We denote by Ω n = U n \ {x 1 , . . . , x n }. It is still a finite union of open intervals, containing M \ {x 1 , . . . , x n } and with µ(Ω n \ (M \ {x 1 , . . . , x n })) ≤ 1/n. Moreover, we may assume that all these intervals contain an element of M : it is enough to remove from Ω n the intervals which do not contain any element of M (if after that we obtain Ω n = ∅, it means that M ⊆ A and we already know that T µ = {0} on A, so there is nothing to prove).

Let us thus take a sequence (g n ) n of piecewise constant functions such that g n → u in L 2 µ (it is possible since u is continuous, thus can be approximated uniformly on I by a sequence of piecewise functions) and ||g n || ∞ ≤ C, where C only depends on u; we replace g n by u outside of the set Ω n (the new function will still be called g n ), so that we have now

• g n → u in L 2 µ ;
• g n coincides with u outside of Ω n ;

• g n coincides on Ω n with a piecewise constant function.

We begin by regularizing g n around the endpoints of the intervals forming Ω n . Let ε n > 0 be small enough so that: • w n (a i ) = u(a i ), w n (a i ) = u (a i ) and w n = 0 on a (small) open interval having a i for upper bound;

• a i + ε n < b i -ε n , for each i (we will set a i = a i + ε n and b i = b i -ε n ); • ]a i , b i [ contains at least an element of M , for each i; • on ]a i , b i [ , g n has not any discontinuity point outside ]a i , b i [ ; • if we denote by Ω n the union of the intervals ]a i , b i [ , we have µ(Ω n \ Ω n ) ≤ 1/n.
• w n (b i ) = u(b i ), w n (b i ) = u (b i ) and w n = 0 on a (small) open interval having b i for lower bound. Proof. It is enough to replace g n on the interval ]a i , a i + ε n [ by the function x → Q(a i + x)
where

Q(t) = - u (a i ) 2ε n t 2 + u (a i )t + u(a i ) (so that Q(0) = u(a i ), Q (0) = Q (a i ) and Q (ε n ) = 0), to scale the new function on the interval ]a i , a i [ by replacing it by x → w n (a i + 2(x -a i )) if a i ≤ x ≤ a i + ε n /2 w n (a i -) otherwise
and to make a similar construction on the interval

]b i -ε n , b i [.
Since g n and w n are bounded uniformly in n and coincide outside of the set Ω n \ Ω n , whose measure is at most 1/n, the sequence (w n ) n still converges to u in L 2 µ ; moreover, we have

||w n || L 2 µ (Ωn\Ω n ) ≤ (2/n)||u || ∞
and for any discontinuity point y of w n , w n is piecewise constant on a (small) neighborhood of y. We now have to regularize w n around its discontinuity points, which belong to Ω n ; this is possible with a small cost only if these points belong to the set M . For this reason we are interested by the following "displacement" procedure of the discontinuity points:

Lemma 2.3. For any n, there exists a function v n such that

• v n = w n outside of Ω n ;
• v n is still piecewise constant on Ω n ;

• any discontinuity point of v n belongs to M ;

• v n → u in L 2 µ .
Proof. We have to modify w n only on each interval ]a i , b i [ . On this interval, the number of discontinuity points of w n is finite; we denote these points by

a i ≤ x 1 < • • • < x n = b i .
We make the following construction:

• Let m = inf([a i , b i ] ∩ M ). We define v n on the interval [a i , m[ (if it is nonempty) by setting v n = w n (a i + ).
• Then we reiterate the construction starting from m: if m = m < x n , then we denote by j the smallest index such that x j > m, we set v n = w n on [m, x j [ and we reiterate this construction starting from x j ;

-if ]m, b i [ ∩M = ∅, we set v n = w n (b i -) on
finally, if m < m < x n , we set v n = w n on [m, m [ and we reiterate this construction starting from m .

With this construction, w n -v n = 0 only on Ω n and outside of the set M . Since µ(Ω n \ M ) ≤ 1/n and w n , v n are uniformly bounded, we get ||v n -w n || L 2 µ ≤ C/n, and we thus still have v n → u. Moreover, by construction, v n is still piecewise constant on the set Ω n and all its discontinuity points belong to M .

To finish, we have to modify v n around each discontinuity point, so that the new function u n is regular and admits a derivative which is small for the L 2 µ -norm. This is possible thanks to the following result about embeddings between functional spaces: Lemma 2.4. Let J a bounded interval of R, and µ a finite measure on J with density f > 0. The following assertions are equivalent:

1. The function 1/f belongs to L 1 (J) 2. The space L 2 µ (J) is continuously embedded into L 1 (J)
Proof. The direct implication is obvious and comes directly from the Cauchy-Schwarz inequality. For the converse one, let us assume that J 1/f = +∞ and set

E n = t ∈ J : 1 n + 1 ≤ f (t) < 1 n and l n = L 1 (E n ).
We know that n l n < +∞ (it is the length of J) and

n nl n = n J n 1 {n-1≤1/f ≤n} ≥ n J 1 f 1 {n-1≤1/f ≤n} ≥ J 1 f = +∞ thus n nl n = +∞.
We will build a function U which is constant on each set E n , belongs to L 2 µ and does not belong to L 1 . If we denote by u n the value of U on E n , it is equivalent to find a sequence (u n ) n verifying End of the proof of Theorem 2.1. Thanks to the two last lemmas, we are now able to transform the function w n into a C 1 function u n , which will provide us our approximation of u. Let us recall that v n coincides with u outside Ω n , is piecewise constant on Ω n , all its discontinuity points are located in M and each of this points admits a neighborhood where v n is null. Denoting by y 1 < • • • < y p the discontinuity points of v n , we find ε n such that, for each j, v n is constant on ]y j -ε n , y j [ and ]y j , y j + ε n [ . Moreover, we have:

p j=1 µ( ]y j -ε n , y j + ε n [ ) ≤ 2pε n + p j=1
µ({y j }).

We take ε n small enough so that 2pε n ≤ 1/n. On the other hand, since each y j does not belong to the set {x 1 , . . . , x n } of the "big atoms" of µ, we have

p j=1 µ({y j }) ≤ k≥n µ({x k }).
Therefore,

µ   p j=1 ]y j -ε n , y j + ε n [   -----→ n→+∞ 0.
On the interval ]y j -ε n , y j + ε n [ , thanks to Lemma 2.4, L 2 µ is not embedded into L 1 , thus we can find a regular function g j such that yj +εn yj -εn

g j = v n (y + j ) -v n (y - j ) and yj +εn yj -εn g 2 j dµ ≤ 1 nq .
Then, we set

u n (x) = ṽn (y j -ε n ) + x yj -εn g j if y j -ε n ≤ x ≤ y j + ε n v n (x)
otherwise.

3 Application to a transport problem with gradient penalization

Problem statement, and the easiest case

We investigate the following problem, which is somehow intermediate between optimal transportation and elasticity theory:

inf Ω (|T (x) -x| 2 + |∇T (x)| 2 ) dµ(x) ,
where the infimum is taken among all maps T : Ω → R d with prescribed image measure T # µ = ν and admitting a Jacobian matrix ∇T in a suitable sense. Contrary to the Monge-Kantorovich optimal transport problem, if µ has a density f bounded from above and below, then the existence of a solution is obvious and comes from the direct method of the calculus of variations; more precisely:

Proposition 3.1. Let f : Ω → R d a measurable function such that 0 < c < f < C < +∞ for some constants c, C > 0. Let ν ∈ P(R d ).
We assume that there exists at least one Sobolev transport map between dµ = f • dL d and ν. Then the problem inf

Ω (|T (x) -x| 2 + |∇T (x)| 2 )f (x) dx : T ∈ H 1 (Ω), T # µ = ν
admits at least one solution.

Proof. Let (T n ) n be a minimizing sequence. We can extract from (T n ) n a sequence having, thanks to the Rellich theorem, a strong limit T in L 2 , and we also can assume that T n → T L 1 -a.e. on Ω, thus µ-a.e. on Ω. Then for any function ϕ

∈ C b (R d ) we have ∀n ∈ N Ω ϕ(T n (x)) dµ(x) = R d ϕ(y) dν(y).
Thanks to the pointwise µ-a.e. convergence of (T n ) n , we can pass to the limit in the left-hand-side of this equality, which gives

∀ϕ ∈ C b (R d ) Ω ϕ(T (x)) dµ(x) = R d ϕ(y) dν(y)
and T satisfies the constraint on the image measure. Moreover, the functional that we consider is of course lower semicontinuous with respect to the weak convergence in H 1 (Ω), and T minimizes our problem.

The general formulation, and the one-dimensional case

If µ is a generic Borel measure, we replace the term with the jacobian matrix of T by ∇ µ T , so that our problem is now written

inf Ω (|T (x) -x| 2 + |∇ µ T (x)| 2 ) dµ(x) : T ∈ H 1 µ (Ω) . (2)
The existence of solutions is not clear in general. In the case of the classical Sobolev space H 1 (Ω), we have seen that the key point to prove the existence is the following: from any minimizing sequence (T n ) n we can extract a sequence which converges L d -a.e. on Ω, and this is enough to obtain that the limit is admissible. This is not possible in general, since we don't have any equivalent of Rellich compactness theorem for the Sobolev spaces with respect to a generic measure µ.

In the one-dimensional case, if µ is the Lebesgue measure, it is known that the monotone transport map between L 1 and ν is optimal for the problem (2) (see [START_REF] Louet | A sharp inequality for transport maps in W 1,p (R) via approximation[END_REF]). This result does not hold if we do not make any assumption of µ, but we can get an existence result thanks to the µ-a.e. compactness result of the second section: Theorem 3.1. In dimension 1, the problem (2) admits at least one solution.

Proof. Let us begin by rewriting precisely the functional that we consider in this case: we know that T µ = {0} on M ∪ A and R on V , so that we are now minimizing

J : U ∈ H 1 µ (I) -→ V ((U (x) -x) 2 + U (x) 2 )f (x) dx + M ∪A (U (x) -x) 2 dµ(x).
• Outside of the set

M = x ∈ Ω : ∀ε > 0, Ω∩B(x,ε) dx f (x) = +∞
we have T µ = R d , a.e. for the regular part of µ. The proof is identical to the one-dimensional case, based on the Cauchy-Schwarz inequality and the embedding L 2 f (Ω \ M ) → L 1 loc . • Of course, we don't have anymore T µ = 0 for the singular part of µ: for instance, if µ is uniform and supported on a segment, then µ is singular, and we know that dim T µ = 1 on any point. However, the tangent space on the atoms of µ is known:

Proposition 3.2. If x 0 is an atom of µ, then T µ (x 0 ) = {0}.
Proof. Let us prove it if x 0 = 0. We want to build a sequence of functions (u n ) n such that

u n → 0 and ∇u n → e 1 {0}
where e is an arbitrary unit vector (this shows that any unit vector belongs to the space T µ (0) ⊥ , thus T µ (0) = {0}). For this goal, let us consider a smooth cutoff function χ such that 0 ≤ χ ≤ 1 and χ n (x) = 1 if 0 ≤ |x| ≤ 1 and χ n (x) = 0 if |x| ≥ 2 and we denote by χ n (x) = χ(nx). We then set u n (x) = x, e χ n (x) and show that (u n ) n is the function that we are looking for. First, noting that u n (0) = 0, that u n is null outside of B(0, 2/n) and that 0 ≤ u n ≤ 1 for any n, we have where C is an upper bound of χ and ∇χ. Thus, (∇u n ) n is uniformly bounded by a positive constant, and ∇u n -e 1 {0} is supported on the set B(0, 2/n) \ {0} whose measure µ goes to 0. This completes the proof.

• Finally, we can prove that there exists absolutely continuous measure µ such that T µ is neither {0} nor R n on any point of Ω. We provide an explicit example: Proposition 3.3. Let be g : ]0, 1[ → ]0, +∞[ such that J g = +∞ for any open interval J ⊆ ]0, 1[ . Let Ω = ]0, 1[ 2 , f : (x, y) ∈ Ω → g(x), and µ the measure with density f . Then the tangent space is the vertical line R • e 2 on µ-a.e. point of Ω.

Proof. We first show that T µ (x) is at most one-dimensional on µ-a.e. x ∈ Ω. Let be u(x, y) = x.

Since the tangent space of the measure g(x) • L 1 on ]0, 1[ is {0}, we can find a sequence of functions (w n ) n such that and we denote by u n (x, y) = w n (x). It is clear that ∇u n → 0 and u n → u in L 2 µ . Then, ∇u = (1, 0) but ∇ µ u = (0, 0) µ-a.e. on Ω. This would impossible if T µ was R 2 on a non-negligible set of Ω.

We now have to show that R • e 2 is included to T µ (x) for µ-a.e. x. For this, we prove that any element v = (v 1 , v 2 ) of Γ(0) satisfies v 2 = 0. Indeed, we have v 2 = lim ∂ 2 u n with u n → 0. For any test function ϕ, integrating by parts with respect to y (since the density of µ depends only on x) gives µ . This gives Ω v 2 ϕ dµ = 0 for any ϕ ∈ D(Ω), thus v 2 = 0, and the proof is complete.

Lemma 2 . 2 .

 22 There exists a function w n coinciding with g n outside of Ω n \ Ω n , and such that, on each interval ]a i , a i [ and ]b i , b i [ , • w n and w n are bounded by constants depending only on u and u ;

  this interval, and we are done; otherwise, let m = inf( ]m, b i [ ∩M ). We have naturally m ≥ m. If m ≥ x n , then we set v n = w n (b i -) on ]m , b i [ ,w n on [m, m [ and we are done;

n u 2 n

 2 (nl n ) < +∞ and n |u n | l n = +∞ To summarize, we want to prove the following statement: for any sequence (l n ) n of positive numbers such that n nl n = +∞ and n l n < +∞, there exists a sequence (u n ) n of positive numbers such that n u 2 n (nl n ) < +∞ and n nu n = +∞. By contraposition, it is equivalent to the following: for any sequence (l n ) n of positive numbers such that n l n < +∞, if the following implication holds: n u 2 n (nl n ) < +∞ ⇒ n l n |u n | < +∞ then we have n nl n < +∞. This result can be seen as a corollary of the Banach-Steinhaus theorem. Denoting by 2 nln the space of sequences (u n ) n such that n u 2 n (nl n ) < +∞, the operator T N : u ∈ 2 nln about (l n ) n is equivalent to ∀u ∈ 2 nln sup N ∈N |T N (u)| < +∞. By Banach-Steinhaus theorem, we get sup N ∈N ||T N || < +∞ and n∈N nl n < +∞; the proof is complete.

  ||u n || L 2 µ = ||u n || L 2 µ(Ω\x0) ≤ µ(B(0, 2/n)) -µ({0}) which goes to 0 as n → +∞; this gives us u n → 0 in L 2 µ .Second, for any n, we have ∇u n (x) = χ n (x) e + e, x ∇χ n (x) = χ n (x) e + e, x n∇χ(nx).Let us notice that ∇u n (0) = e for any n, thus it is enough to prove that ||∇u n || L 2 µ (Ω\{0}) → 0. But χ n and ∇χ n are null outside of B(0, 2/n) and if 0 < |x| ≤ 2/n we have |∇u n (x)| ≤ |e||χ(nx)| + | e, x |n|∇χ(x)| ≤ C(1 + n|x|) ≤ 3C

1 0 1 0

 11 |w n (x) -x| 2 g(x) dx → 0 and |w n (x)| 2 g(x) dx → 0

Ω ∂ 2

 2 u n ϕ dµ = -Ω u n ∂ 2 ϕ dµ which goes to 0 since u n → 0 in L 2

This shows that we cannot hope to obtain a compactness result analogous to the one-dimensional case, where any bounded sequence in H 1 µ has a subsequence which converges on µ-a.e. x such that T µ (x) = {0}: it is enough to take a sequence of functions (u n ) n depending only on x and non-compact for the a.e. convergence.

This functions u n form our desired approximation of u: Proposition 2.3. This sequence (u n ) n satisfies u n → u and u n → v in the space L 2 µ , where v(x) = u (x) if x / ∈ M or is an atom of µ 0 otherwise.

Consequently, T µ = {0} µ-a.e. on M .

Proof. We know that v n → u, thus u n → u in the space L 2 µ outside of the intervals ]y j -ε n , y j + ε n [ . But since the total mass of these intervals goes to 0 and (u n ) n is uniformly bounded, we get u n → u. For the derivative, since u n = u outside of Ω n , we have

where the first term goes to 0 (since (u n ) n is uniformly bounded and µ(Ω n \ M ) goes to 0); for the second one, we have v = 0 on M , thus it is enough to prove that u n goes to 0 for the L 2 µ -norm on M \{x 1 , . . . , x n }; this term is bounded by

Since (u n ) n is uniformly bounded, we know that the second term goes to 0, and since u n is constant outside of the intervals ]y j -ε n , y j + ε n [ the first one is equal to

which, by definition of g j , is smaller than 1/n. This completes the proof.

Let (U n ) n be a minimizing sequence. On the set V , which is exactly the set where T µ is R, we can extract from (U n ) n a µ-a.e. (which means L 1 -a.e. wherever f = 0) pointwise convergent subsequence, whose limit is denoted by U ; let us remark that U is the weak limit of (U n ) n (up to a subsequence) in the space H 1 f , and by semicontinuity, we have

Moreover, let us set, for n ∈ N, ν n = (U n ) # (µ| M ∪A ) and Ũn the optimal transport map for the Monge-Kantorovich quadratic cost between the measures µ| M ∪A and ν n . It is well-known that Ũn is the unique nondecreasing transport map between µ| M ∪A and ν n ; because of compactness properties of nondecreasing maps, we can assume that ( Ũn ) n admits, for the µ-a.e. convergence, a limit Ũ . For any n, thanks to the optimality of Ũn , we have

and by semicontinuity

Thus, if we denote by

we have T n → T µ-a.e. on I, and

where (U n ) n is a minimizing sequence for J on the set of H 1 µ transport maps between µ and ν. Thus, it is enough to prove that T satisfies the constraint on image measure to conclude. But for each n, by construction, (T n ) # µ = ν and the µ-a.e. convergence allows to obtain the same for the limit T ; the proof is complete.

Remark. This result can be generalized to any functional J : U → Ω (L 1 (x, U (x))+L 2 (∇ µ U (x))) dµ(x), where L 1 and L 2 have one of the following forms:

• L 1 is a transport cost such that the nondecreasing map is optimal for the Monge-Kantorovich problem: it is the case if L 1 (x, u) = h(|x -u|), where h is a convex function. Let us notice that in particular the statement holds if we study the problem of minimization of the norm of the gradient among all Sobolev transport maps (this corresponds to L 1 = 0). Of course we need to assume that the class

is nonempty (to guarantee that J ≡ +∞ on the set of admissible functions). Thanks to the quadratic structure of H 1 µ , this is automatically the case if L 1 is the quadratic cost and there exists a Sobolev transport map.

• L 2 is "quadratic", so that the space where we study the problem is actually the Sobolev space

, where I d is the identity matrix (in this last case, we can consider the functional U → ||U -id|| H 1 µ , which is a Sobolev version of the quadratic transport problem where we minimize ||U -id|| L 2 µ ).

Difficulties and partial results in any dimension

As we said in the second section of this paper, we don't have a precise pointwise description of the µ-Sobolev space H 1 µ (Ω) if Ω is an open set of R d , which was the key point for the compactness result. More precisely, the following results still hold in any dimension: