We extend a classical approximation result of harmonic functions in planar domains due to Bernstein and Walsch to the setting of harmonic functions in Riemann surfaces. This result gives an exact characterization of the rate at which a harmonic function in a subdomain of a compact Riemann surface may be approached by globally defined harmonic functions with prescribed poles. We illustrate the effectiveness and the impact of the method solving general boundary value Laplace problems in subdomains of the surface; we lay the groundwork for this numerical method in Riemann surfaces represented by a gluing of hyperbolic polygons. In particular, we give a general approximation procedure that computes this basis efficiently with arbitrary precision.

This article revolves around shape and topology optimization, in the applicative context where the objective and constraint functionals depend on the solution to a physical boundary value problem posed on the optimized domain. We introduce a novel numerical framework based on modern concepts from computational geometry, optimal transport and numerical analysis. Its pivotal feature is a representation of the optimized shape by the cells of an adapted version of a Laguerre diagram. Although such objects are originally described by a collection of seed points and weights, recent results from optimal transport theory suggest a more intuitive parametrization in terms of the seed points and measures of the associated cells. The discrete polygonal mesh of the shape induced by this diagram is a convenient support for the deployment of the Virtual Element Method for the numerical solution of the boundary value problem characterizing the physics at play and for the evaluation of the functionals of the domain involved in the optimization problem. The sensitivities of the latter are derived next; at first, we calculate their derivatives with respect to the positions of the vertices of the Laguerre diagram by shape calculus techniques; a suitable adjoint methodology is then developed to express them in terms of the seed points and cell measures of the diagram. The evolution of the shape through the optimization process is realized by first updating the design variables according to these sensitivities and then reconstructing the diagram thanks to efficient algorithms from computational geometry. Our shape and topology optimization strategy is versatile: it can be applied to a whole gammut of physical situations (such as thermal and structural mechanics) and optimal design settings (including single- and multi-phase problems). It is Lagrangian and body-fitted by essence, and it thereby benefits from all the assets of an explicit, meshed representation of the shape at each stage of the process. Yet, it naturally handles dramatic motions of the optimized shape including topological changes, in a very robust fashion. These features, among others, are illustrated by a series of 2d numerical examples.

Identifying Blaschke-Santalo diagrams is an important topic that essentially consists in determining the image Y = F(X) of a map F : X → Rd, where the dimension of the source space X is much larger than the one of the target space. In some cases, that occur for instance in shape optimization problems, X can even be a subset of an infinite-dimensional space. The usual Monte-Carlo method, consisting in randomly choosing a number N of points x1,..., xN in X and plotting them in the target space Rd, produces in many cases areas in Y of very high and very low concentration leading to a rather rough numerical identification of the image set. On the contrary, our goal is to choose the points xi in an appropriate way that produces a uniform distribution in the target space. In this way we may obtain a good representation of the image set Y by a relatively small number N of samples which is very useful when the dimension of the source space X is large (or even infinite) and the evaluation of F (xi) is costly. Our method consists in a suitable use of Centroidal Voronoi Tessellations which provides efficient numerical results. Simulations for two and three dimensional examples are shown in the paper.

In this paper, we prove a Logarithmic Conjugation Theorem on finitely-connected tori. The theorem states that a harmonic function can be written as the real part of a function whose derivative is analytic and a finite sum of terms involving the logarithm of the modulus of a modified Weierstrass sigma function. We implement the method using arbitrary precision and use the result to find approximate solutions to the Laplace problem and Steklov eigenvalue problem. Using a posteriori estimation, we show that the solution of the Laplace problem on a torus with a few holes has error less than 10−100 using a few hundred degrees of freedom and the Steklov eigenvalues have similar error.

This paper is motivated by the maximization of the k-th eigenvalue of the Laplace operator with Neumann boundary conditions among domains of ℝ𝑁 with prescribed measure. We relax the problem to the class of (possibly degenerate) densities in ℝ𝑁 with prescribed mass and prove the existence of an optimal density. For 𝑘=1,2, the two problems are equivalent and the maximizers are known to be one and two equal balls, respectively. For 𝑘≥3 this question remains open, except in one dimension of the space, where we prove that the maximal densities correspond to a union of k equal segments. This result provides sharp upper bounds for Sturm-Liouville eigenvalues and proves the validity of the Pólya conjecture in the class of densities in ℝ. Based on the relaxed formulation, we provide numerical approximations of optimal densities for 𝑘=1,…,8 in ℝ2.

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally. In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σj(Σ,g) L(∂Σ,g) over a class of smooth metrics, g, where σj(Σ,g) is the j-th nonzero Steklov eigenvalue and L(∂Σ,g) is the length of ∂Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus γ=0 and b=2,…,9,12,15,20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b=2 boundary components and for the third and fifth eigenvalues with b=3 boundary components.

This article provides numerical evidence that under volume constraint the ball is the set which maximizes the perimeter of the least-perimeter partition into cells with prescribed areas. We introduce a numerical maximization algorithm which performs multiple optimizations steps at each iteration to approximate minimal partitions. Using these partitions we compute perturbations of the domain which increase the minimal perimeter. The initialization of the optimal partitioning algorithm uses capacity-constrained Voronoi diagrams. A new algorithm is proposed to identify such diagrams, by computing the gradients of areas and perimeters for the Voronoi cells with respect to the Voronoi points..

We study a family of gradient obstacle problems on a compact Riemannian manifold. We prove that the solutions of these free boundary problems are uniformly semiconcave and, as a consequence, we obtain some fine convergence results for the solutions and their free boundaries. Precisely, we show that the elastic and the λ-elastic sets of the solutions Hausdorff converge to the cut locus and the λ-cut locus of the manifold.

We propose a new method for the numerical computation of the cut locus of a compact submanifold of R3 without boundary. This method is based on a convex variational problem with conic constraints, with proven convergence. We illustrate the versatility of our approach by the approximation of Voronoi cells on embedded surfaces of R3.

This article deals with the optimization of the shape of the regions assigned to different types of boundary conditions in the definition of a ‘physical’ partial differential equation. At first, we analyze a model situation involving the solution uΩ to a Laplace equation in a domain Ω; the boundary ∂Ω is divided into three parts ΓD, Γ and ΓN, supporting respectively homogeneous Dirichlet homogeneous Neumann and inhomogeneous Neumann boundary conditions. The shape derivative J′(Ω)(θ) of a general objective function J(Ω) of the domain is calculated in the framework of Hadamard’s method when the considered deformations θ are allowed to modify the geometry of ΓD, Γ and ΓN (i.e. θ does not vanish on the boundary of these regions). The structure of this shape derivative turns out to depend very much on the regularity of uΩ near the boundaries of the regions ΓD, Γ and ΓN. For this reason, in particular, J′(Ω) is difficult to calculate and to evaluate numerically when the transition ΓD ∩Γ between homogeneous Dirichlet and homogeneous Neumann boundary conditions is subject to optimization. To overcome this difficulty, an approximation method is proposed, in which the considered ‘exact’ Laplace equation with mixed boundary conditions is replaced with a ‘smoothed’ version, featuring Robin boundary conditions on the whole boundary ∂Ω with coefficients depending on a small parameter ε. We prove the consistency of this approach in our model context: the approximate objective function Jε(Ω) and its shape derivative converge to their exact counterparts as ε vanishes. Although it is rigorously justified only in a model problem, this approximation methodology may be adapted to many more complex situations, for example in three space dimensions, or in the context of the linearized elasticity system. Various numerical examples are eventually presented in order to appraise the efficiency of the proposed approximation process.

Shape optimization techniques were first developed in the context of mechanical engineering and, more recently, applied to photonic components for data communication. Here, motivated by the growing application potential of mid-infrared photonics driven by chemical sensing and spectroscopy, we present the design by shape optimization of passive components operating in this wavelength range. A focus is placed on the creation of designs that are fabricable and robust to manufacturing uncertainties.

In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert–Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in Rn . Following the the analysis for the planar case presented in [4], we provide a variational approximation through Ginzburg–Landau type energies proving a Γ-convergence result for n larger than 3.

In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a network (connected one-dimensional structure), that has to be found in a suitable admissible class. We show the existence of an optimal solution that may present multiplicities, that is regions where the optimal structure overlaps. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal network when their total length becomes large.

We propose a framework to represent a partition that evolves under mean curvature flows and volume constraints. Its principle follows a phase-field representation for each region of the partition, as well as classical Allen-Cahn equations for its evolution. We focus on the evolution and on the optimization of problems involving high resolution data with many regions in the partition.

This article deals with the optimal design of the individual nanophotonic components of photonic integrated circuits. In the mathematical setting of the three-dimensional, time-harmonic Maxwell equations, we propose a shape and topology optimization algorithm combining Hadamard’s boundary variation method with a level set representation of shapes and their evolution. A particular attention is devoted to the robustness of the optimized devices with respect to small uncertainties over the physical or geometrical data of the problem. In this respect, we rely on a simple multi-objective formulation to deal with the two main sources of uncertainties plaguing nanophotonic devices, namely uncertainties over the incoming wavelength, and geometric uncertainties entailed by the lithography and etching fabrication process. Several numerical examples are presented and discussed to assess the efficiency of our methodology.

We describe a convex relaxation for the Gilbert–Steiner problem both in Rd and on manifolds. We discuss its sharpness by means of calibration type arguments. The minimization of the resulting problem is then tackled numerically and we present results for an extensive set of examples. In particular we are able to address the Steiner tree problem on surfaces.

We consider the volume-constrained minimization of the sum of the perimeter and the Riesz potential. We add an external potential that provides the existence of a minimizer for any volume constraint, and we study the geometry of large volume minimizers.

In this paper we consider variational problems involving 1-dimensional connected sets in the euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a full Γ-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to n-dimensional euclidean space or to more general manifold ambients, as shown in the companion paper.

We develop the first numerical study in four dimension of optimal eigenmodes associated to the Dirichlet Laplacian. We describe an extension of the Method of Fundamental Solutions adapted to the four dimensional context. Based on our numerical simulation and a post processing adapted to the identification of relevant symmetries, we provide and discuss the numerical description of the 8th first optimal domains.

We study partitions on three dimensional manifolds which minimize the total geodesic perimeter. We propose a relaxed framework based on a Γ-convergence result and we show some numerical results. We compare our results to those already present in the literature in the case of the sphere. For general surfaces we provide an optimization algorithm on meshes which can give a good approximation of the optimal cost, starting from the results obtained using the relaxed formulation.

Full waveform inversion starts being used as a standard stage of the seismic imaging workflow, at the exploration scale, for the reconstruction of high resolution wave velocity models. However, its successful application still relies on the estimation of an accurate enough initial velocity model, as well as on the design of a suitable hierarchical workflow, allowing to feed the inversion process progressively with data. These two requirements are mandatory to avoid the cycle skipping or phase ambiguity problem when comparing observed and synthetic data. This difficulty is due to the definition of the full waveform inversion problem as the least-squares minimization of the data misfit. The resulting misfit function has local minima which corresponds to the interpretation of the seismic data up to one or several phase-shifts. In this article, we review an alternative formulation of full waveform inversion based on the optimal transport distance we have proposed in recent studies. We propose to use a particular instance of the optimal transport problem, which is adapted to the interpretation of real seismic data, and for which we design an efficient low-complexity numerical strategy. Numerical results in 2D and 3D configurations (BP 2004, Chevron 2014 benchmark model, SEG/EAGE overthrust model) show that this reformulation should yield a more convex misfit function, less prone to cycle skipping. In this study, we present a simple illustration on the Marmousi model which illustrates how this new distance strongly relaxes the requirement on the initial model design. Starting from a rather simplistic approximation of the initial model, the method is able to reconstruct a meaningful estimation of the Marmousi model, while the standard least-squares formulation is trapped into a local, meaningless minimum.

In this paper we propose a variational approach to the Steiner tree problem, which is based on calibrations in a suitable algebraic environment for polyhedral chains which represent our candidates. This approach turns out to be very efficient from numerical point of view and allows to establish whether a given Steiner tree is optimal. Several examples are provided.

We characterize the solution to the Newton minimal resistance problem in a specific class of hollow profiles, satisfying a q-concavity condition. We treat two-dimensional bodies and radially symmetric three-dimensional bodies.

The use of optimal transport distance has recently yielded significant progress in image processing for pattern recognition, shape identification, and histograms matching. In this study, the use of this distance is investigated for a seismic tomography problem exploiting the complete waveform; the full waveform inversion. In its conventional formulation, this high resolution seismic imaging method is based on the minimization of the L2 distance between predicted and observed data. Application of this method is generally hampered by the local minima of the associated L2 misfit function, which correspond to velocity models matching the data up to one or several phase shifts. Conversely, the optimal transport distance appears as a more suitable tool to compare the misfit between oscillatory signals, for its ability to detect shifted patterns. However, its application to full waveform inversion is not straightforward, as the mass conservation between the compared data can not be guaranteed, a crucial assumption for optimal transport. In this study, the use of a distance based on the Kantorovich-Rubinstein norm is introduced to overcome this difficulty.

Full waveform inversion using the conventional L2 distance to measure the misfit between seismograms is known to suffer from cycle skipping. An alternative strategy is proposed in this study, based on a measure of the misfit computed with an optimal transport distance. This measure allows to account for the lateral coherency of events within the seismograms, instead of considering each seismic trace independently, as is done generally in full waveform inversion. The computation of this optimal transport distance relies on a particular mathematical formulation allowing for the non-conservation of the total energy between seismograms.

A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered. We prove the existence of an optimal reinforcement θ and that it has some higher integrability properties. We also provide some numerical computations for θ and u.

We consider the problem of optimizing the k eigenvalue of Dirichlet Laplace operator under perimeter constraint. We provide a new method based on a Γ-convergence result for approximating the corresponding optimal shapes. We also give new optimality conditions in the case of multiple eigenvalues. We deduce from previous conditions the fact that optimal shapes never contain flat parts in their boundaries.

Gradient flows in the Wasserstein space have become a powerful tool in the analysis of diffusion equations, following the seminal work of Jordan, Kinderlehrer and Otto (JKO). The numerical applications of this formulation have been limited by the difficulty to compute the Wasserstein distance in dimension >= 2. One step of the JKO scheme is equivalent to a variational problem on the space of convex functions, which involves the Monge-Ampère operator. Convexity constraints are notably difficult to handle numerically, but in our setting the internal energy plays the role of a barrier for these constraints. This enables us to introduce a consistent discretization, which inherits convexity properties of the continuous variational problem. We show the effectiveness of our approach on nonlinear diffusion and crowd-motion models.

Equilibrium multi-population matching (matching for teams) is a problem from mathematical economics which is related to multi-marginal optimal transport. A special but important case is the Wasserstein barycenter problem, which has applications in image processing and statistics. Two algorithms are presented: a linear programming algorithm and an efficient nonsmooth optimization algorithm, which applies in the case of the Wasserstein barycenters. The measures are approximated by discrete measures: convergence of the approximation is proved. Numerical results are presented which illustrate the efficiency of the algorithms.

We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give precise estimates of the distance between the approximation space and the admissible set. This framework applies to the approximation of convex functions by piecewise linear functions on a mesh of the domain and by other finite-dimensional spaces such as tensor-product splines. We show how these discretizations are well suited for the numerical resolution of problems of calculus of variations under convexity constraints. Our implementation relies on proximal algorithms, and can be easily parallelized, thus making it applicable to large scale problems in dimension two and three. We illustrate the versatility and the efficiency of our approach on the numerical resolution of three problems in calculus of variation : 3D denoising, the principal agent problem, and optimization within the class of convex bodies.

Motivated by a geometric problem, we introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreasing and converges in a finite number of iterations to a local minimum of the relaxed objective function. Our method is applied to several clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations. The model has a semi-supervised extension and provides a natural representative for the clusters as well.

We discuss in this paper phase-field approximations of theWillmore functional and the associated L2-flow. After recollecting known results on the approximation of the Willmore energy and its L1 relaxation, we derive the expression of the flows associated with various approximations, and we show their behavior by formal arguments based on matched asymptotic expansions. We introduce an accurate numerical scheme, whose local convergence can be proved, to describe with more details the behavior of two flows, the classical and the flow associated with an approximation model due to Mugnai. We propose a series of numerical simulations in 2D and 3D to illustrate their behavior in both smooth and singular situations.

This article reviews the use of first order convex optimization schemes to solve the discretized dynamic optimal transport problem, initially proposed by Benamou and Brenier. We develop a staggered grid discretization that is well adapted to the computation of the L2 optimal transport geodesic between distributions defined on a uniform spatial grid. We show how proximal splitting schemes can be used to solve the resulting large scale convex optimization problem. A specific instantiation of this method on a centered grid corresponds to the initial algorithm developed by Benamou and Brenier. We also show how more general cost functions can be taken into account and how to extend the method to perform optimal transport on a Riemannian manifold.

The goal of this article is to introduce a new algorithm for solving discrete optimal transport, using ideas from non-smooth convex optimization. We provide some insights on the relation between the complexity of the discrete optimal transport problem and the geometry of the underlying measures. This discrete method is applied to investigate the regularity of optimal transport plan on oblate ellipsoids, and to solve Alexandrov's problem of reconstructing a convex set from its Gaussian measure.

In this paper, we present numerical methods to solve optimization problems among convex bodies which satisfy some width constraints. We propose two different numerical approaches to handle width equality and width inequality constraints. To illustrate the efficiency of our method, we use our algorithms to approximate optimal solution of Meissner''s problem and Heil's conjecture.

The Mα energy which is usually minimized in branched transport problems among singular 1-dimensional rectifiable vector measures with prescribed divergence is approximated by means of a sequence of elliptic energies defined on more regular vector fields. The procedure recalls the one of Modica-Mortola related to the approximation of the perimeter. In our context, the double-well potential is replaced by a concave term. The paper contains a proof of Γ-convergence and numerical simulations of optimal networks based on that previous result.

We introduce three new algorithms to regularize digital contours based on the minimization of Willmore energy: our first algorithm is based on tools coming from discrete geometry, the second is related to convex geometry while the third approach is a constrained phase field minimization. The three algorithms are described in details and the convergence of the phase field approach is investigated. We present a comparative evaluation of all three methods, in terms of the accuracy of curvature estimators and computation time.

In this paper we propose a numerical process to approximate optimal partitions in any dimension. The key idea of our method is to relax the problem into a functional framework based on the famous result of $\Gamma$-convergence obtained by Modica and Mortolla.

We introduce a new numerical method to approximate partitions of a domain minimizing the sum of Dirichlet-Laplacian eigenvalues of any order. First we prove the equivalence of the original problem and a relaxed formulation based on measures. Using this result, we build a numerical algorithm to approximate optimal configurations. We describe numerical experiments aimed at studying the asymptotic behavior of optimal partitions with large numbers of cells.

In this paper we study isoperimetric inequalities for the eigenvalues of the Laplace operator with constant and locally constant boundary conditions. Existence and stability results are presented in the frame of the gamma and weak gamma convergencies, together with identification of optimal sets by analytical and numerical methods.

In this paper we study isoperimetric inequalities for the eigenvalues of the Laplace operator with constant and locally constant boundary conditions. Existence and stability results are presented in the frame of the gamma and weak gamma convergencies, together with identification of optimal sets by analytical and numerical methods.

Starting from the work by Brenier and Benamou, where a dynamic formulation of mass transportation problems was given, we consider a more general framework, where different kind of cost functions are allowed. This seems relevant in some problems presenting congestion effects as for instance traffic on a highway, crowds moving in domains with obstacles, and in general in all cases where the transportation does not behave as in the classical Monge setting. We show some numerical computations obtained by generalizing to our framework the scheme introduced in Benamou and Brenier.

We present a complete analytic parametrization of constant width bodies in dimension 3 based on the median surface: more precisely, we define a bijection between some space of functions and constant width bodies. We compute simple geometrical quantities like the volume and the surface area in terms of those functions. As a corollary we give a new algebraic proof of Blaschke's formula. Finally, we derive weak optimality conditions for convex bodies which minimize the volume among constant width bodies.

A wedged configuration with Coulomb friction is a   nontrivial equilibrium state of a linear elastic body in a frictional unilateral contact with a rigid body  under vanishing external loads.  A supremal functional defined on the set of admissible normal displacement and tangential stresses is introduced. For the in-plane problem we discuss the link between the critical friction and the  smallest real eigenvalue which is related to the loss of uniqueness. The (WP) problem is stated in a discrete framework using a mixed finite element approach and the (discrete) critical friction coefficient is introduced as the minimum of a specific functional. A genetic algorithm is used for the global minimization problem involving this non differentiable and non-convex functional. Finally,  the  analysis is illustrated with some numerical experiments.

We give a number of characterizations of bodies of constant width in arbitrary dimension. As an application, we describe a way to construct a body of constant width in dimension n, one of its (n-1)-dimensional projection being given. We give a number of examples, like a four-dimensional body of constant width whose 3D-projection is the classical Meissner's body.

We present numerical methods to solve optimization problems on the space of convex functions or among convex bodies. Hence convexity is a constraint on the admissible objects, whereas the functionals are not required to be convex.To deal with, our method mix geometrical and numerical algorithms. We give several applications arising from classical problems in geometry and analysis: Alexandrov's problem of finding a convex body of prescribed surface function; Cheeger's problem of a subdomain minimizing the ratio surface area on volume; Newton's problem of the body of minimal resistance.  In particular for the latter application, the minimizers are still unknown, except in some particular classes. We give approximate solutions better than the theoretical known ones, hence demonstrating that the minimizers do not belong to these classes..

Vlasov simulations can for some situations be a valuable alternative to PIC simulations for the study of intense beam propagation. However, as they rely on a phase-space grid which is fixed for the whole simulation, important computing effort can be wasted in zones where no particles are present at some given time. In order to overcome this drawback, we introduce here a new method which makes use of a phase-space grid which is uniform at any given time, but moves in time according to the evolution of the envelope of the beam.

In this paper we introduce a numerical approach adapted to the minimization of the eigenmodes of a membrane with respect to the domain. This method is based on the combination of the Level Set method of J. Osher and J. Sethian with the relaxed approach. This algorithm enables both changing the topology and working on a fixed regular grid.

In this paper we are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. A natural candidate to be the optimum was the `stadium', the convex hull of two identical tangent disks (Troesch's conjecture). We refute this conjecture. Nevertheless, we prove that the optimum has two parallel segments in its boundary.

A Dirichlet region for an optimal mass transportation problem is, roughly speaking, a zone in which the transportation cost is vanishing. We study the optimal transportation problem with an unknown Dirichlet region S which varies in the class of closed connected subsets having prescribed 1-dimensional Hausdorff measure. We show the existence of an optimal Sopt and study some of its geometrical properties. We also present numerical computations which show the shape of Sopt in some model examples.

We are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. A natural candidate to be the optimum was the `stadium', the convex hull of two identical tangent disks (Troesch's conjecture). We refute this conjecture. Nevertheless, we prove that the optimum has two parallel segments in its boundary.

On souhaite dans cet article, donner un aperçu de quelques problèmes en théorie des nombres transcendants, autour du pivot que constitue le septième problème de Hilbert. On esquisse un historique du développement de la discipline antérieurement à l'énoncé des problèmes de Hilbert, puis on propose une preuve aussi élémentaire que possible et (presque) complète du théorème de Gel'fond-Schneider, qui répond à la question de Hilbert. Enfin on décrit l'importante postérité du septième problème dans les travaux mathématiques au vingtième siècle.

This work is devoted to the theoretical and numerical aspects of shape optimization. The first part (chapter I to IV) deals with optimization problems under convexity constraint or constant width constraint. We give several new results related to Newton's problem and Meissner's conjecture. The second part (chapter V to VII) deals with the numerical study of shape optimization problems where many shapes or phases are involved. Some new numerical methods are introduced to study optimal configurations of famous problems : Kelvin's problem and Caffarelli's conjecture. The last part (chapter VIII and IX) is devoted to optimal transportation problems and irrigation problems. More precisely, we introduce a general framework, where different kind of cost functions are allowed. This seems relevant in some problems presenting congestion effects as for instance traffic on a highway, crowds moving in domains with obstacles. In the last chapter we give preliminary results related to the numerical approximation of optimal irrigation networks.

Mon travail de thèse porte sur des aspects théoriques et numériques de l'optimisation de forme ainsi que sur la stabilisation de fonctions solutions d'équations aux dérivées partielles. Dans la première partie, on s'intéresse à la minimisation des valeurs propres du laplacien avec conditions aux limites de Dirichlet. On étudie plus particulièrement la minimisation de la seconde valeur propre du laplacien sous contraintes de volume et de convexité. Après avoir démontré certaines propriétés qualitatives d'un ouvert optimal (régularité minimale et maximale, description géométrique du bord), nous répondons à une question posée par Troesh en 1973: le stade (enveloppe convexe de deux disques tangents de mêmes rayons) n'est pas un ouvert optimal pour ce problème d'optimisation. Dans un deuxième chapitre, nous présentons différents résultats numériques ayant trait à la minimisation d'une valeur propre de rang donné. Dans un second temps, nous exposons certaines propriétés qualitatives d'un ensemble solution d'un problème de transport optimal. Là encore, ce travail est complété par des illustrations numériques obtenues à l'aide d'un algorithme de type stochastique. Le travail de la dernière partie est consacré à la stabilisation rapide de l'équation des ondes par des méthodes d'analyse non harmonique. Nous y présentons aussi un nouveau résultat de monotonie concernant des suites de zéros des dérivées de fonctions de Bessel.