Related papers: An asymptotic shape optimization problem for Riesz…
We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via…
We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a…
We consider regularization methods based on the coupling of Tikhonov regularization and projection strategies. From the resulting constraint regularization method we obtain level set methods in a straight forward way. Moreover, we show that…
In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace,…
We use moment techniques to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate (from below) the infinite dimensional optimization problems in this…
This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem.…
Given an infeasible, unbounded, or pathological convex optimization problem, a natural question to ask is: what is the smallest change we can make to the problem's parameters such that the problem becomes solvable? In this paper, we address…
In this paper, we study some optimization problems in uniformly convex and uniformly smooth Bochner spaces. We consider four cases of the underlying subsets: closed and convex subsets, closed and convex cones, closed subspaces and closed…
We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means…
In mathematical modelling, the data and solutions are represented as measurable functions and their quality is oftentimes captured by the membership to a certain function space. One of the core questions for an analysis of a model is the…
The aim of this paper is to shed more light on some recent ideas about Lyapunov exponents and clarify the formal structures behind these ideas. In particular, we show that the vector of (averaged) Lyapunov exponents of a smooth…
It is known that the torsional rigidity for a punctured ball, with the puncture having the shape of a ball, is minimum when the balls are concentric and the first eigenvalue for the Dirichlet Laplacian for such domains is also a maximum in…
We consider the volume-constrained minimization of the sum of the perimeter and the Riesz potential. We add an external potential of the form $\|x\|^{\beta}$ that provides the existence of a minimizer for any volume constraint, and we study…
We consider regularization of non-convex optimization problems involving a non-linear least-squares objective. By adding an auxiliary set of variables, we introduce a novel regularization framework whose corresponding objective function is…
The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class. A significant progress on this problem has been recently achieved by Nadirashvili-Sire and…
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 $\lambda_2(\Om)$ denotes the second eigenvalue of the…
We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact planar set, subject to attractive Robin boundary conditions. Under either a constraint of fixed perimeter or area, we…
In this work we study the asymptotic distribution of eigenvalues in one-dimensional open sets. The method of proof is rather elementary, based on the Dirichlet lattice points problem, which enable us to consider sets with infinite measure.…
The paper is devoted to the study of a quantitative Weinstock inequality in higher dimension for the first non trivial Steklov eigenvalue of Laplace operator for convex sets. The key rule is played by a quantitative isoperimetric inequality…
An eigenvalue problem arising in optimal insulation related to the minimization of the heat decay rate of an insulated body is adapted to enforce a positive lower bound imposed on the distribution of insulating material. We prove the…