Dorin Bucur

Optimal Partitions for Eigenvalues

We introduce a new numerical method for approximating partitions of a domain minimizing the sum of Dirichlet-Laplacian eigenvalues of any order. First we prove the equivalence of the original problem and a relaxed formulation based on measures. Using this result, we build a numerical algorithm to approximate optimal configurations. We describe numerical experiments aimed at studying the asymptotic behavior of optimal partitions with large numbers of cells.

How to Choose Interpolation Data in Images

We introduce and discuss shape-based models for finding the best interpolation data when reconstructing missing regions in images by means of solving the Laplace equation. The shape analysis is done in the framework of

A Duality Approach for the Boundary Variation of Neumann Problems

\Gamma

On the attainable eigenvalues of the Laplace operator

-convergence, from two different points of view. First, we propose a continuous PDE model and get pointwise information on the “importance” of each pixel by a topological asymptotic method. Second, we introduce a finite dimensional setting into the continuous model based on fat pixels (balls with positive radius) and study by

Multiphase shape optimization problems

\Gamma

Influence of wall roughness on the slip behaviour of viscous fluids

-convergence the asymptotics when the radius vanishes. In this way, we obtain relevant information about the optimal distribution of the best interpolation pixels. We show that the resulting optimal data sets are identical to sets that can also be motivated using level set ideas and approximation theoretic considerations. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.

Shape optimisation problems governed by nonlinear state equations

In two dimensions, we study the stability of the solution of an elliptic equation with Neumann boundary conditions for nonsmooth perturbations of the geometric domain. Using harmonic conjugates, we relate this problem to the shape stability of the solution of an elliptic equation with Dirichlet boundary conditions. As a particular case, we prove the stability of the solution under a topological constraint (uniform number of holes), which is analogous to Sveraks result for Dirichlet boundary conditions.

Anatomy of the shape Hessian via lie brackets

We consider the subset E of

Asymptotic behaviour of optimal spectral planar domains with fixed perimeter

\RR^2

Lipschitz Regularity of the Eigenfunctions on Optimal Domains

of all points whose first and second components, respectively, coincide with the first and second eigenvalues of the Laplace operator