arXiv subjects

Jimmy Lamboley

Publications and source records attributed to Jimmy Lamboley.

23 records · Page 2Linked to original sources

arXiv2010

About Hölder-regularity of the convex shape minimizing λ2

In this paper, we consider the well-known following shape optimization problem: $$λ_2(Ω^*)=\min_{\stackrel{|Ω|=V_0} {Ω\textrm{ convex}}} λ_2(Ω),$$ where $λ_2(\Om)$ denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in $\Om\subset\R^2$, and $|\Om|$ is the area of $\Om$. We prove, under some technical assumptions, that any optimal shape $Ω^*$ is $\mathcal{C}^{1,\frac{1}{2}}$ and is not $\C^{1,α}$ for any $α>\frac{1}{2}$. 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.

math.OC↗

arXiv2009

Polygons as optimal shapes with convexity constraint

In this paper, we focus on the following general shape optimization problem: $$ \min\{J(\Om), \Om convex, \Om\in\mathcal S_{ad}\}, $$ where $\mathcal S_{ad}$ is a set of 2-dimensional admissible shapes and $J:\mathcal{S}_{ad}\to\R$ is a shape functional. Using a specific parameterization of the set of convex domains, we derive some extremality conditions (first and second order) for this kind of problem. Moreover, we use these optimality conditions to prove that, for a large class of functionals (satisfying a concavity like property), any solution to this shape optimization problem is a polygon.

math.OC↗

arXiv2009

Counterexamples to Symmetry for Partially Overdetermined Elliptic Problems

We exhibit several counterexamples showing that the famous Serrin's symmetry result for semilinear elliptic overdetermined problems may not hold for partially overdetermined problems, that is when both Dirichlet and Neumann boundary conditions are prescribed only on part of the boundary. Our counterexamples enlighten subsequent positive symmetry results obtained by the first two authors for such partially overdetermined systems and justify their assumptions as well.

math.OC↗

arXiv2008

Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints

We consider the well-known following shape optimization problem: $$λ_1(Ω^*)=\min_{\stackrel{|Ω|=a} {Ω\subset{D}}} λ_1(Ω), $$ where $λ_1$ denotes the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary condition, and $D$ is an open bounded set (a box). It is well-known that the solution of this problem is the ball of volume $a$ if such a ball exists in the box $D$ (Faber-Krahn's theorem). In this paper, we prove regularity properties of the boundary of the optimal shapes $Ω^*$ in any case and in any dimension. Full regularity is obtained in dimension 2.

math.OC↗

arXiv2006

Structure of shape derivatives around irregular domains and applications

In this paper, we describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in $\R^N$. This structure allows us to define a useful notion of positivity of the shape derivative and we show it implies its continuity with respect to the uniform norm when the boundary is Lipschitz (this restriction is essentially optimal). We apply this idea to various cases including the perimeter-type functionals for convex and pseudo-convex shapes or the Dirichlet energy of an open set.

math.OC↗