Related papers: Isoperimetric Inequalities and Sharp Estimate for …

We consider the Dirichlet problem for positive solutions of the equation $-\Delta_p (u) = f(u)$ in a convex, bounded, smooth domain $\Omega \subset\R^N$, with $f$ locally Lipschitz continuous. \par We provide sufficient conditions…

We consider semilinear elliptic equations with double power nonlineaities. The condition to assure the existence of positive solutions is well-known. In the present paper, we remark that the additional condition to assure uniqueness…

We consider the edge- and vertex-isoperimetric probem on finite and infinite hexagonal grids: For a subset W of the hexagonal grid of given cardinality, we give a lower bound for the number of edges between W and its complement, and lower…

The existence of positive weak solutions to a singular quasilinear elliptic system in the whole space is established via suitable a priori estimates and Schauder's fixed point theorem.

In this paper we provide a Bonnesen-style inequality which gives a lower bound for the isoperimetric deficit corresponding to a closed convex curve in terms of some geometrical invariants of this curve. Moreover we give a geometrical…

In this work, we study the existence and nonexistence of solution for strongly coupled elliptic systems to m-parameters.

This article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known $L^{\infty}$ estimates that hold for all nonlinearities. Such estimates are known to…

Using elementary techniques, we prove sharp anisotropic Hardy-Littlewood inequalities for positive multilinear forms. In particular, we recover an inequality proved by F. Bayart in 2018.

We prove radial symmetry of singular solutions to an overdetermined boundary value problem for a class of degenerate quasilinear elliptic equations.

We review recent results on the study of the isoperimetric problem on Riemannian manifolds with Ricci lower bounds. We focus on the validity of sharp second order differential inequalities satisfied by the isoperimetric profile of possibly…

In this article we review a new method for proving the nonexistence of positive solutions of elliptic inequalities in unbounded domains in $\rn$, which was recently introduced by the authors. We expose our method and new results on the two…

This paper deals with the lack of compactness in nonlinear elliptic problems $(P)$. In particular, a domain $\Omega$ is provided where not converging Palais-Smale sequences exist at every energy level. Nevertheless, it is proved that…

The isoperimetric problem asks for the maximum area of a region of given perimeter. It is natural to consider other measurements of a region, such as the diameter and width, and ask for the extreme value of one when another is fixed. The…

We investigate the non-existence and existence of positive solutions to biharmonic elliptic inequalities on manifolds. Using Green function and volume growth conditions, we establish the critical exponent for biharmonic problem.

In this manuscript we review some recent results about approximation of solutions of elliptic problems with high-contrast coefficients. In particular, we detail the derivation of asymptotic expansions for the solution in terms of the…

The classical isoperimetric inequality can be extended to a general normed plane. In the Euclidean plane, the defect in the isoperimetric inequality can be calculated in terms of the signed areas of some singular sets. In this paper we…

We prove the sharp quantitative isoperimetric inequality in the case of the barycentric asymmetry, for bounded sets. This generalizes the $2$-D case recently proved in~\cite{BCH}.

In this paper we deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.

We prove a sharp isoperimetric inequality for measured Finsler manifolds having non-negative Ricci curvature and Euclidean volume growth. We also prove a rigidity result for this inequality, under the additional hypotheses of boundedness of…

We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter.…