Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle

In this paper we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue $\lambda_{F}(p,\Omega)$ of the anisotropic $p$-Laplacian, $1<p<+\infty$. Our aim is to enhance how, by means of the $\mathcal P$-function method, it is possible to get several sharp estimates for $\lambda_{F}(p,\Omega)$ in terms of several geometric quantities associated to the domain. The $\mathcal P$-function method is based on a maximum principle for a suitable function involving the eigenfunction and its gradient

Droplet condensation and isoperimetric towers

We consider a variational problem in a planar convex domain, motivated by statistical mechanics of crystal growth in a saturated solution. The minimizers are constructed explicitly and are completely characterized

Total variation denoising in l1l^1 anisotropy

We aim at constructing solutions to the minimizing problem for the variant of Rudin-Osher-Fatemi denoising model with rectilinear anisotropy and to the gradient flow of its underlying anisotropic total variation functional. We consider a naturally defined class of functions piecewise constant on rectangles (PCR). This class forms a strictly dense subset of the space of functions of bounded variation with an anisotropic norm. The main result shows that if the given noisy image is a PCR function, then solutions to both considered problems also have this property. For PCR data the problem of finding the solution is reduced to a finite algorithm. We discuss some implications of this result, for instance we use it to prove that continuity is preserved by both considered problems.Comment: 34 pages, 9 figure

Phase field approach to optimal packing problems and related Cheeger clusters

In a fixed domain of $\Bbb{R}^N$ we study the asymptotic behaviour of optimal clusters associated to $\alpha$-Cheeger constants and natural energies like the sum or maximum: we prove that, as the parameter $\alpha$ converges to the "critical" value $\Big (\frac{N-1}{N}\Big ) _+$, optimal Cheeger clusters converge to solutions of different packing problems for balls, depending on the energy under consideration. As well, we propose an efficient phase field approach based on a multiphase Gamma convergence result of Modica-Mortola type, in order to compute $\alpha$-Cheeger constants, optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions

Anisotropic total variation flow of non-divergence type on a higher dimensional torus

We extend the theory of viscosity solutions to a class of very singular nonlinear parabolic problems of non-divergence form in a periodic domain of an arbitrary dimension with diffusion given by an anisotropic total variation energy. We give a proof of a comparison principle, an outline of a proof of the stability under approximation by regularized parabolic problems, and an existence theorem for general continuous initial data, which extend the results recently obtained by the authors.Comment: 27 page