Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian

We prove that that the 1-Riesz capacity satisfi es a Brunn-Minkowski inequality, and that the capacitary function of the 1/2-Laplacian is level set convex.Comment: 9 page

Equilibrium shapes of charged droplets and related problems: (mostly) a review

We review some recent results on the equilibrium shapes of charged liquid drops. We show that the natural variational model is ill-posed and how this can be overcome by either restricting the class of competitors or by adding penalizations in the functional. The original contribution of this note is twofold. First, we prove existence of an optimal distribution of charge for a conducting drop subject to an external electric field. Second, we prove that there exists no optimal conducting drop in this setting

Shape Optimization Problems for Metric Graphs

We consider the shape optimization problem $$\min\big\{{\mathcal E}(\Gamma)\ :\ \Gamma\in{\mathcal A},\ {\mathcal H}^1(\Gamma)=l\ \big\},$$ where ${\mathcal H}^1$ is the one-dimensional Hausdorff measure and ${\mathcal A}$ is an admissible class of one-dimensional sets connecting some prescribed set of points ${\mathcal D}=\{D_1,\dots,D_k\}\subset{\mathbb R}^d$. The cost functional ${\mathcal E}(\Gamma)$ is the Dirichlet energy of $\Gamma$ defined through the Sobolev functions on $\Gamma$ vanishing on the points $D_i$. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.Comment: 23 pages, 11 figures, ESAIM Control Optim. Calc. Var., (to appear

A note on the hausdorff dimension of the singular set for minimizers of the mumford-shah energy

We give a more elementary proof of a result by Ambrosio, Fusco and Hutchinson to estimate the Hausdorff dimension of the singular set of minimizers of the Mumford-Shah energy (see [2, Theorem 5.6]). On the one hand, we follow the strategy of the above mentioned paper; but on the other hand our analysis greatly simplifies the argument since it relies on the compactness result proved by the first two Authors in [4, Theorem 13] for sequences of local minimizers with vanishing gradient energy, and the regularity theory of minimal Caccioppoli partitions, rather than on the corresponding results for Almgren's area minimizing sets

On a class of weighted Gauss-type isoperimetric inequalities and applications to symmetrization

We solve a class of weighted isoperimetric problems of the form $$\min\left\{\int\_{\partial E}w e^V\,dx:\int\_E e^V\,dx={\rm constant}\right\}$$ where $w$ and $V$ are suitable functions on $\R^d$. As a consequence, we prove a comparison result for the solutions of degenerate elliptic equations

A spectral shape optimization problem with a nonlocal competing term

We study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. We also show that if the Riesz repulsion is large, then minimizers do not exist.Comment: 32 page

Optimization problems for solutions of elliptic equations and stability issues

Introduction: In this thesis we address some problems related to two topics in the Calculus of Variations which have attracted a growing interest in recent decades. In a general simpli cation we can refer to those two elds as optimization problems for shapes and for solutions of elliptic equations and quantitative stability problems for geometric and functional inequalities. It is worth bearing in mind that the historical and mathematical development of these two classes of problems have merged and looking at them as separate elds may not be the best approach to adopt. On the other hand a division of the works presented may simplify the reading, for this reason the thesis is divided into two main parts. In the rst, Part I, containing three chapters, we deal with optimization problems related to the shape optimization eld. In Part II, we address the quantitative stability of three problems, the rst one regarding a class of isoperimetric inequalities, the second one about a spectral optimization problem and the third one concerning a class of functional inequalities

Reifenberg flatness for almost-minimizers of the perimeter under minimal assumptions

The aim of this note is to prove that almost-minimizers of the perimeter are Reifenberg flat, for a very weak notion of minimality. The main observation is that smallness of the excess at some scale implies smallness of the excess at all smaller scales

A note on the Hausdorff dimension of the singular setfor minimizers of the Mumford-Shah energy

We give a more elementary proof of a result by Ambrosio, Fusco and Hutchinson to estimate the Hausdorff dimension of the singular set of minimizers of the Mumford-Shah energy (see [Calc. Var. Partial Differential Equations 16 (2003), no. 2, 187-215, Theorem 5.6]). On the one hand, we follow the strategy of the above mentioned paper; but on the other hand our analysis greatly simplifies the argument since it relies on the compactness result proved by the first two authors in [J. Math. Pures Appl. 100 (2013), 391-409, Theorem 13] for sequences of local minimizers with vanishing gradient energy, and the regularity theory of minimal Caccioppoli partitions, rather than on the corresponding results for Almgren's area minimizing set