We consider an incompressible fluid in a three-dimensional pipe, following the Navier-Stokes system with classical boundary conditions. We are interested in the following question: is there any optimal shape for the criterion "energy dissipated by the fluid"? Moreover, is the cylinder the optimal shape? We prove that there exists an optimal shape in a reasonable class of admissible domains, but the cylinder is not optimal. For that purpose, we explicit the first order optimality condition, thanks to adjoint state and we prove that it is impossible that the adjoint state be a solution of this over-determined system when the domain is the cylinder. At last, we show some numerical simulations for that problem.

This article considers a family of functionals $J$ to be maximized over the planar convex sets $K$ for which the perimeter and Steiner point have been fixed. Assuming that $J$ is the integral of a quadratic expression in the support function $h$, the maximizer is always either a triangle or a line segment (which can be considered as a collapsed triangle). Among the concrete consequences of the main theorem is the fact that, given any convex body $K_1$ of finite perimeter, the set in this class that is farthest away in the sense of the $L^2$ distance is always a line segment. The same property is proved for the Hausdorff distance.

In this paper we study the set of points, in the plane, defined by $\{(x,y)=(\lambda_1(\Omega),\lambda_2(\Omega)),\ |\Omega|=1\},$ where $(\lambda_1(\Omega),\lambda_2(\Omega))$ are either the two first eigenvalues of the Dirichlet-Laplacian, or the two first non trivial eigenvalues of the Neumann-Laplacian. We consider the case of general open sets together with the case of convex open domains. We give some qualitative properties of these sets, show some pictures obtained through numerical computations and state several open problems.

We look for the minimizers of the functional $J_\lambda(\Omega)=\lambda|\Omega|-P(\Omega)$ among planar convex domains constrained to lie into a given ring. We prove that, according to the values of the parameter $\lambda$, the solutions are either a disc or a polygon. In this last case, we describe completely the polygonal solutions by reducing the problem to a finite dimensional optimization problem. We recover classical inequalities for convex sets involving area, perimeter and inradius or circumradius and find a new one.

We explain why Donnelly's proof of the gap conjecture is not correct.

We prove an isoperimetric inequality of the Rayleigh-Faber-Krahn type for a nonlinear generalization of the first twisted Dirichlet eigenvalue. More precisely, we show that the minimizer among sets of given volume is the union of two equal balls.

We study a spectral problem related to a reaction-diffusion model where the preys and the predators do not live on the same area. We are interested in the optimal zone where the control should take place. First we prove existence of an optimal domain in a natural class. Then, it seems plausible that the optimal domain is localized in the intersection of the living areas of the two species. We prove this fact in one dimension for small size of domains.

We are interested in the properties of the 'minimal' $k$-partitions of  an open set $\Omega$ by $k$ disjoint open sets $D_i$ ($i=1,\dots,k$)  in $\Omega$. These partitions  are minimal in the sense that they  minimize  the maximum over $i=1,\dots,k$ of the lowest eigenvalues of the  Dirichlet realization of the Laplacian in $D_i$. The corresponding minimum is denoted by $\mathfrak L_k(\Omega)$. Such problems naturally appear in Biomathematics.  In particular, we would like to determine in which cases this minimal partition is actually the family of the  nodal domains of a given  eigenfunction of the  two-dimensional Dirichlet Laplacian in  $\Omega$. In the case of  $2$-partitions,  the answer is very simple because a variational characterization of  the second eigenvalue  of the Dirichlet Laplacian in $\Omega$ shows that a minimal  $2$-partition is always a nodal partition corresponding to the second eigenvalue. So the interesting questions start with $k=3$. Although general properties of these minimal partitions have been  proved in [HHOT] by  B. Helffer, T. Hoffmann-Ostenhof and S. Terracini, there are very few theoretical results (except for thin rectangles) for obtaining an explicit determination of minimal  partitions. This is already the case for $3$-partitions and for simple cases like  the square or the disk. For these reasons, it is particularly useful to mix  some theoretical remarks of [HHOT]  and still unpublished results of [HHO3] with efficient numerical  computations.

The spectral analysis of Aharonov-Bohm Hamiltonians with flux $\frac12$ leads surprisingly to a new insight on some questions of isospectrality appearing for example in [JLNP, LPP] and of minimal partitions [HHOT]. We will illustrate this  point of view by discussing  the question of  spectral minimal $3$-partitions for the rectangle $]-\frac a2,\frac a2[\times ] -\frac b2,\frac b2[\,$, with $0< a\leq b$. It has been observed in [HHOT] that when $0<\frac ab < \sqrt{\frac 38}$ the minimal $3$-partition is obtained by the three nodal domains of the third eigenfunction corresponding to the  three rectangles  $]-\frac a2,\frac a2[\times ] -\frac b2,-\frac b6[$,  $]-\frac a2,\frac a2[\times ] -\frac b6,\frac b6[$ and $]-\frac a2,\frac a2[\times ] \frac b6, \frac b2[$.  We will describe a possible mechanism of transition for increasing  $\frac ab$ between these  nodal minimal $3$-partitions and non nodal minimal $3$-partitions at the value $ \sqrt{\frac 38}$ and discuss the existence of symmetric candidates for giving minimal $3$-partitions when $ \sqrt{\frac 38}<\frac ab   \leq 1$.  Numerical analysis leads very naturally to nice questions  of isospectrality which are solved by introduction of  Aharonov-Bohm Hamiltonians or by going on the double covering of the punctured rectangle.

This paper is devoted to present  numerical simulations and interpret theoretically the results for determining the minimal $k$-partitions of a domain $\Omega$ as considered in [HHOT]. More precisely using the double covering approach introduced by B.~Helffer, M. and T. Hoffmann-Ostenhof, M. Owen and further developed for questions of isospectrality by the authors in collaboration with T. Hoffmann-Ostenhof and S. Terracini in [HHOT], we  analyze the variation of the eigenvalues of the one pole  Aharonov-Bohm Hamiltonian on the square and the nodal picture of the associated eigenfunctions as a function of the pole. This leads us to discover new candidates for minimal $k$-partitions of the square  with a specific topological type and without any symmetric assumption, contrary to our previous works [BHV10, BHHO10]. This illustrates also  recent results of B. Noris and S. Terracini, see [NT09]. This finally supports or disproves conjectures for the minimal $3$ and $5$-partitions on the square.

In this paper, we consider the well-known following shape optimization problem:$$\lambda_2(\Omega^*)=\min_{\stackrel{|\Omega|=V_0} {\Omega\textrm{ convex}}}\lambda_2(\Omega),$$ where $\lambda_2(\Omega)$ denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in $\Omega\subset\mathbb R^2$, and $|\Omega|$ is the area of $\Omega$. Our focus on that problem is motivated by the fact that both smooth strictly convex and flat parts appears in the boundary of an optimal shape. We therefore focus on the regularity of contact points between these two parts: we prove, under some technical assumptions, that any optimal shape $\Omega^*$ is $\mathcal{C}^{1,\frac{1}{2}}$ and is not $\mathcal C^{1,\alpha}$ for any $\alpha>\frac{1}{2}$. We also derive from our strategy some more general regularity results, in the framework of partially overdetermined boundary value problems, and we apply these results to some other shape optimization problems.

Motivated by a long-standing conjecture of Pólya and Szegö about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the Blaschke-concavity of variational functionals, including capacity. We then introduce a new algebraic structure on convex bodies, which allows to obtain global concavity and  indecomposability results, and we discuss their application to isoperimetric-like inequalities. As a byproduct of this approach we also obtain a quantitative version of the Kneser-Süss inequality. Finally, for a large class of functionals involving Dirichlet energies and the surface measure, we perform a local analysis of strictly convex portions of the boundary via second order shape derivatives. This allows in particular to exclude the presence of smooth regions with positive Gauss curvature in an optimal shape for Pólya-Szegö problem.

In the previous work [LN09], the first two authors derive some general optimality conditions in order to analyze solutions $\Omega_{0}$ to: $$J(\Omega_{0})=\min\{J(\Omega),\ \Omega\ \textrm{convex},\ \Omega\in\mathcal S_{ad}\},$$where $\mathcal S_{ad}$ is a set of 2-dimensional admissible shapes and $J:\mathcal{S}_{ad}\rightarrow\mathbb R$ is a shape functional. They use these optimality conditions to prove that minimizers of geometric-like functional satisfying a concavity property are polygons. Our main goal here is to obtain again qualitative properties of optimal shapes when functionals are no longer of geometric type, but may depend on a PDE on the set $\Omega$. We prove two main results:

• i) under a suitable convexity property of the functional $J$ we prove that $\Omega_0$ is a $W^{2,p}$-set, $p\in[1,\infty]$. This result applies when the shape functional can be written as $J(\Omega)=R(\Omega)+P(\Omega),$ where $R(\Omega)=F(|\Omega|,E_{f}(\Omega),\lambda_{1}(\Omega))$ involves the area $|\Omega|$, the Dirichlet energy $E_f(\Omega)$ or the first eigenvalue  of the Laplace operator $\lambda_1(\Omega)$ (with Dirichlet boundary conditions on $\partial\Omega$), and $P(\Omega)$ is the perimeter of $\Omega$. In such case $\Omega_0$ is a $\mathcal C^{1,1}$-set, that is to say that the curvature of $\partial\Omega_0$ is bounded;
• ii) under a suitable concavity assumption on the functional $J$, we prove that $\Omega_{0}$ is a polygon. This result applies when the functional is now written as $J(\Omega)=R(\Omega)-P(\Omega),$ with the same notations as above.

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We focus on the analysis of local minimizers of the Mahler volume, that is to say the local solutions to the problem$$\min\{ M(K):=|K||K^\circ|\;/\;K\subset\mathbb R^d\textrm{ open and convex}, K=-K\}, $$ where $K^\circ:=\{\xi\in\mathbb R^d ; \forall x\in K, x\cdot\xi<1\}$ is the polar body of $K$, and $|\cdot|$ denotes the volume in $\mathbb R^d$. According to a famous conjecture of Mahler the cube is expected to be a global minimizer for this problem.
We express the Mahler volume in terms of the support functional of the convex body, which allows us to compute first and second derivatives, and leads to a concavity property of the functional. As a consequence, we prove first that any local minimizer has a Gauss curvature that vanishes at any point where it is defined.  Going more deeply into the analysis in the two-dimensional case, we also prove that any local minimizer must be a parallelogram. We thereby retrieve and improve an original result of Mahler, who showed that parallelograms are global minimizers in dimension 2, and also the case of equality of Reisner, who proved that they are the only global minimizers.

In this paper, we show applications to isoperimetric inequalities of the approximation of $H^1_0(\Omega)$ functions  by functions which only depends on the distance to the boundary of $\Omega$: more precisely, we introduce the energy functional$$\mathcal J(u) =\int_{\Omega} \Big ( \frac{1}{p} |\nabla u| ^p - u \Big )\ ,$$ and the space $$\mathcal{W}(\Omega)=\{u\in W^{1,p}(\Omega);\,u(x)=u(d(x,\partial\Omega))\}\ .$$We introduce the so-called ($p$-)torsion and Web-($p$-)torsion (with $p\in(1,\infty)$):<$$D_p(\Omega):=-\min_{u\in W^{1,p}(\Omega)}\mathcal J(u)\ ,\qquad N_p(\Omega):=-\min_{u\in\mathcal{W}(\Omega)}\mathcal J(u)\ .$$In \cite{cfg} they study optimal bounds for the ratio $\frac{N_p}{D_p}$ among convex bodies. We use these optimal bounds to prove some refined isoperimetric inequalities involving the Torsion, the volume and the perimeter. These inequalities give a generalization of a result of Makai in the case $p=2$. An interesting corollary of these inequalities asserts that if a polygon has an isoperimetric ratio much bigger than the one of the regular polygon, then its ($p$-)torsion is smaller than the one of the regular polygon. This supports the famous conjecture that regular polygons maximize the torsion among polygons whose area and number of sides are given.

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We prove that a generic 2D example of shape changing body swimming in a potential flow can track approximately any given trajectory. These results where recently improved \cite{munnier2}. In this article, we consider a 3D shape changing body swimming in a potential flow. Under some symmetry assumptions (the swimmer is alone in the fluid and the fluid-swimmer system fills the whole space) we prove generic controllability results.

We consider a moving rigid solid immersed in a potential fluid. Our aim is to study the inverse problem  that consists in recovering the position and the velocity of the solid assuming that the potential function is known at a given time. We show that this problem is in general ill-posed by providing counterexamples for which the same potential corresponds to different positions and velocities of a same solid. However, it is also possible to find solids having a specific shape, like ellipses for instance, for which  the problem of detection admits a unique solution. We prove also that by performing continuous measurements of the fluid potential over a time interval, we can always track the position of the solid.

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In this paper, we are interested in the  study of shape optimizations problems with Stokes  constraints within the class of axisymmetric domains represented by the graph of a function. Existence results with weak assumptions on the regularity of the graph are provided.  We  strongly  use these assumptions to get some topological properties.  We formulate the  (shape) optimization  problem   using different constraints formulations: uniform bound constraints on the function and its derivative and/or  volume (global)  constraint. Writing the first order optimality conditions allows to provide quasi-explicit solutions in some particular cases and to give some hints for the treatment of the generic problem. Furthermore, we extend the (negative) result of [HPri2] dealing  with the non optimality of the cylinder.

We consider two inverse problems related to the tokamak Tore Supra through the study of the magnetostatic equation for the poloidal flux. The first one deals with the Cauchy issue of recovering in a two dimensional annular domain boundary magnetic values on the inner boundary, namely the limiter, from available overdetermined data on the outer boundary. Using tools from complex analysis and properties of genereralised Hardy spaces, we establish stability and existence properties. Secondly the inverse problem of recovering the shape of the plasma is addressed thank tools of shape optimization. Again results about existence and optimality are provided. They give rise to a fast algorithm of identification which is applied to several numerical simulations computing good results either for the classical harmonic case or for the data coming from Tore Supra.

A Bernoulli free boundary problem with geometrical constraints is studied. The domain $\Omega$ is constrained to lie in the half space determined by $x_1\geq 0$ and its boundary  to contain a segment of the hyperplane $\{x_1=0\}$ where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.

In this paper, we study the role of boundary conditions on the optimal shape of a dyadic tree in which flows a Newtonian fluid. Our optimization problem consists in finding the shape of the tree that minimizes the viscous energy dissipated by the fluid with a constrained volume, under the assumption that the total flow of the fluid is conserved throughout the structure. These hypotheses model situations where a fluid is transported from a source towards a 3D domain into which the transport network also spans. Such situations could be encountered in organs like for instance the lungs and the vascular networks.  Two fluid regimes are studied: (i) low flow regime (Poiseuille) in trees with an arbitrary number of generations using a matricial approach and (ii) non linear flow regime (Navier-Stokes, moderate regime with a Reynolds number $100$) in trees of two generations using shape derivatives in an augmented Lagrangian algorithm coupled with a 2D/3D finite elements code to solve Navier-Stokes equations. It relies on the study of a finite dimensional optimization problem in the case (i) and on a standard shape optimization problem in the case (ii). We show that the behaviours of both regimes are very similar and that the optimal shape is highly dependent on the boundary conditions of the fluid applied at the leaves of the tree.

We consider shape optimization problems with internal inclusion constraints, of the form$$\min\big\{J(\Omega)\ :\ D\subset\Omega\subset\R^d,\ |\Omega|=m\big\},$$ where the set $D$ is fixed, possibly unbounded, and $J$ depends on $\Omega$ via the spectrum of the Dirichlet Laplacian. We analyze the existence of a solution and its qualitative properties, and rise some open questions.

We consider a general formulation of gradient flow evolution for problems whose natural framework is the one of metric spaces. The applications we deal with are concerned with the evolution of capacitary measures with respect to the $\gamma$-convergence dissipation distance and with the evolution of domains in spectral optimization problems.

We consider the Steklov problem for the linear biharmonic equation. We survey existing results for the positivity preserving property to hold. These are connected with the first Steklov eigenvalue.We address the problem of minimizing this eigenvalue among suitable classes of domains. We prove the existence of an optimal convex domain of fixed measure.

Relying on the rugosity effect, we analyse the drag minimization problem in relation with the micro-structure of the surface of a given obstacle. We construct a mathematical framework for the optimization problem, prove the existence of an optimal solution by $\Gamma$-convergence arguments and analyse the stability of the drag with respect to the micro-structure. For Stokes flows we justify why rugosity does not decrease the drag, while for Navier-Stokes flows we give some numerical evidence supporting the thesis that adding rugosity on specific regions of the obstacle may contribute to decrease the drag.

For every positive regular Borel measure, possibly infinite valued, vanishing on all sets of $p$-capacity zero, we characterize the compactness of the embedding $W^{1,p}(\mathbb R^N)\cap L^p (\mathbb R^N,\mu)\hookrightarrow L^q(\mathbb R^N)$ in terms of the qualitative behavior of some characteristic PDE. This question is related to the well posedness of a class of geometric inequalities involving the torsional rigidity and the spectrum of the Dirichlet Laplacian introduced by Pólya and Szegö in 1951. In particular, we prove that finite torsional rigidity of an arbitrary domain (possibly with infinite measure), implies the compactness of the resolvent of the Laplacian.

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