Related papers: On existence and nonexistence of isoperimetric ine…

We prove an isoperimetric inequality of the Rayleigh-Faber-Krahn type for a nonlinear generalization of the first twisted Dirichlet eigenvalue. More precisely, we show that the minimizer among sets of given volume is the union of two equal…

We provide a sharp lower bound for the perimeter of the inner parallel sets of a convex body $\Omega$. The bound depends only on the perimeter and inradius $r$ of the original body and states that \[|\partial\Omega_t| \geq…

In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem under the assumption of small volumes. Applications of the main theorem to asymptotic expansions of the isoperimetric problem are…

We study isoperimetric inequalities on "slabs", namely weighted Riemannian manifolds obtained as the product of the uniform measure on a finite length interval with a codimension-one base. As our two main applications, we consider the case…

In a Riemannian manifold with a smooth positive function that weights the associated Hausdorff measures we study stable sets, i.e., second order minima of the weighted perimeter under variations preserving the weighted volume. By assuming…

Let $p_i\geq 2$ and consider the following anisotropic $p$-Laplace equation $$ -\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\Big(\Big|\frac{\partial u}{\partial x_i}\Big|^{p_i-2}\frac{\partial u}{\partial x_i}\Big)=g(x)f(u),\,\,u>0\text{ in…

The normalised volume measure on the $\ell_p^n$ unit ball ($1\leq p\leq 2$) satisfies the following isoperimetric inequality: the boundary measure of a set of measure $a$ is at least $cn^{1/p}\tilde{a}\log^{1-1/p}(1/\tilde{a})$, where…

Let $(M^n,g,e^{-\phi}dV_g,e^{-\phi}dA_g,m)$ be a compact smooth metric measure space with boundary with $n\geqslant 3$. In this article, we consider several Yamabe-type problems on a compact smooth metric measure space with or without…

In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calder\'on's and Zygmund's theory of first order…

We consider the isoperimetric inequality involving the $s$-perimeter and the $t$-perimeter with $0<s<t<1$, and show that the ball is a local minimizer of the (scale-invariant) isoperimetric ratio $\mathcal{F}(E):=P_t(E)^{\frac{1}{n-t}}/…

The discrete isoperimetric inequality states that among all n -gons with a fixed area, the regular n -gon has the least perimeter. We prove analogues of the discrete isoperimetric inequality (involving circumradius or inradius) for cyclic…

We obtain a uniform $L^{\infty}(\Omega)$ a priori bound, for any positive weak solutions to elliptic problem with a nonlinearity $f$ slightly subcritical, slightly superlinear, and regularly varying. To achieve our result, we first obtain a…

We prove the existence of minimizers in the class of negative definite measures on compact subsets of momentum space in the homogeneous setting under several side conditions (constraints). The method is to employ Prohorov's theorem. Given a…

In the present manuscript we address and solve for the first time a nonlocal discrete isoperimetric problem. We consider indeed a generalization of the classical perimeter, what we call a nonlocal bi-axial discrete perimeter, where, not…

Nonconvex functionals with spherical symmetry are studied. Existence of one and radial symmetry of all global minimizers is shown with an approach based on convex relaxation.

It is shown that $2\beta_1(\G)\leq h(\G)$ for any countable group $\G$, where $\beta_1(\G)$ is the first $\ell^2$-Betti number and $h(\G)$ the uniform isoperimetric constant. In particular, a countable group with non-vanishing first…

Let $\Omega$ be a bounded open interval, let $p>1$ and $\gamma>0$, and let $m:\Omega\rightarrow\mathbb{R}$ be a function that may change sign in $\Omega $. In this article we study the existence and nonexistence of positive solutions for…

We prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\in\re$, $\Omega\subset\re^2$ then the inequality $$…

We prove a quantitative version of the isoperimetric inequality for a non local perimeter of Minkowski type. We also apply this result to study isoperimetric problems with repulsive interaction terms, under convexity constraints. We show…

We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem…