On the honeycomb conjecture for Robin Laplacian eigenvalues

We prove that the optimal cluster problem for the sum of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity

Two optimization problems in thermal insulation

We consider two optimization problems in thermal insulation: in both cases the goal is to find a thin layer around the boundary of the thermal body which gives the best insulation. The total mass of the insulating material is prescribed.. The first problem deals with the case in which a given heat source is present, while in the second one there are no heat sources and the goal is to have the slowest decay of the temperature. In both cases an optimal distribution of the insulator around the thermal body exists; when the body has a circular symmetry, in the first case a constant heat source gives a constant thickness as the optimal solution, while surprisingly this is not the case in the second problem, where the circular symmetry of the optimal insulating layer depends on the total quantity of insulator at our disposal. A symmetry breaking occurs when this total quantity is below a certain threshold. Some numerical computations are also provided, together with a list of open questions.Comment: 11 pages, 7 figures, published article on Notices Amer. Math. Soc. is available at http://www.ams.org/publications/journals/notices/201708/rnoti-p830.pd

Minimization of λ2(Ω)\lambda_2(\Omega) with a perimeter constraint

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In $N$ dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to set inclusion and $\gamma$ lower semicontinuous.Comment: Indiana University Mathematics Journal (2009) to appea

Symmetry breaking for a problem in optimal insulation

We consider the problem of optimally insulating a given domain $\Omega$ of ${\mathbb{R}}^d$; this amounts to solve a nonlinear variational problem, where the optimal thickness of the insulator is obtained as the boundary trace of the solution. We deal with two different criteria of optimization: the first one consists in the minimization of the total energy of the system, while the second one involves the first eigenvalue of the related differential operator. Surprisingly, the second optimization problem presents a symmetry breaking in the sense that for a ball the optimal thickness is nonsymmetric when the total amount of insulator is small enough. In the last section we discuss the shape optimization problem which is obtained letting $\Omega$ to vary too.Comment: 12 pages, 0 figure

Shape Optimization Problems with Internal Constraint

We consider shape optimization problems with internal inclusion constraints, of the form \min\big\{J(\Omega)\ :\ \Dr\subset\Omega\subset\R^d,\ |\Omega|=m\big\}, where the set \Dr 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.Comment: 18 pages, 0 figure

Optimal partitions for Robin Laplacian eigenvalues

We prove the existence of an optimal partition for the multiphase shape optimization problem which consists in minimizing the sum of the first Robin Laplacian eigenvalue of $k$ mutually disjoint {\it open} sets which have a $\mathcal H ^ {d-1}$-countably rectifiable boundary and are contained into a given box $D$ in $R^d

Sign changing solutions of Poisson's equation

Let $\Omega$ be an open, possibly unbounded, set in Euclidean space $\R^m$ with boundary $\partial\Omega,$ let $A$ be a measurable subset of $\Omega$ with measure $|A|$, and let $\gamma \in (0,1)$. We investigate whether the solution v_{\Om,A,\gamma} of $-\Delta v=\gamma{\bf 1}_{\Omega \setminus A}-(1-\gamma){\bf 1}_{A}$ with $v=0$ on $\partial \Omega$ changes sign. Bounds are obtained for $|A|$ in terms of geometric characteristics of \Om (bottom of the spectrum of the Dirichlet Laplacian, torsion, measure, or $R$-smoothness of the boundary) such that {\rm essinf} v_{\Om,A,\gamma}\ge 0. We show that {\rm essinf} v_{\Om,A,\gamma}<0 for any measurable set $A$, provided |A| >\gamma |\Om|. This value is sharp. We also study the shape optimisation problem of the optimal location of $A$ (with prescribed measure) which minimises the essential infimum of v_{\Om,A,\gamma}. Surprisingly, if \Om is a ball, a symmetry breaking phenomenon occurs.Comment: 27 pages, 2 figures, various minor typos have been correcte