Related papers: Some overdetermined problems related to the anisot…

In this paper, we consider the anisotropic $\alpha$-Gauss curvature flow for complete noncompact convex hypersurfaces in the Euclidean space with the anisotropy determined by a smooth closed uniformly convex Wulff shape. We show that for…

We study the Steklov eigenvalue problem for the $\infty-$orthotropic Laplace operator defined on convex sets of $\mathbb{R}^N$, with $N\geq2$, considering the limit for $p\to+\infty$ of the Steklov problem for the $p-$orthotropic Laplacian.…

Given a bounded smooth domain $\Omega$ in $\mathbb{R}^2$, we study the following anisotropic elliptic problem $$ \begin{cases} -\nabla\big(a(x)\nabla \upsilon\big)= a(x)\big[e^{\upsilon}-s\phi_1-4\pi\alpha\delta_q-h(x)\big]\,\,\,\,…

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…

In this paper, a class of anisotropic weights having the form of $|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ in dimensions $n\geq2$ is considered, where $x=(x',x_{n})$ and $x'=(x_{1},...,x_{n-1})$. We first find the optimal…

We consider nonlocal equations of the type \[ (-\Delta_{p})^{s}u = \mu \quad \text{in }\Omega, \] where $\Omega \subset \mathbb{R}^{n}$ is either a bounded domain or the whole $\mathbb{R}^{n}$, $\mu$ is a Radon measure on $\Omega$, $0<s<1$…

Local minimizers for the anisotropic isoperimetric problem in the small-volume regime on closed Riemannian manifolds are shown to be geodesically convex and small smooth perturbations of tangent Wulff shapes, quantitatively in terms of the…

We study positive solutions $u_p$ of the nonlinear Neumann elliptic problem $\Delta u =u$ in $\Omega $, $\partial u/\partial\nu = |u|^{p-1}u$ on $\partial\Omega$, where $\Omega $ is a bounded open smooth domain in $\mathbb{R}^2$. We…

We use the upper and lower potential functions and Bowen's formula estimating the Hausdorff dimension of the limit set of a regular semigroup generated by finitely many $C^{1+\alpha}$-contracting mappings. This result is an application of…

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also…

In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P)\left\{\begin{split} -\Delta_{H,p}u&=\frac{\la f(x)}{u^{q(x)}}+g(u)\text{ in }\Omega,\\ u&>0\text{ in…

Given an elliptic integrand of class $ \mathscr{C}^{3} $, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with…

We investigate the structure of the nodal set of solutions to an unstable Alt-Phillips type problem \[ -\Delta u = \lambda_+(u^+)^{p-1}-\lambda_-(u^-)^{q-1} \] where $1 \le p<q<2$, $\lambda_+ >0$, $\lambda_- \ge 0$. The equation is…

We study the condenser capacity $\mathrm{cap}_p(E,\Omega)$ on \emph{unbounded} open sets $\Omega$ in a proper connected metric space $X$ equipped with a locally doubling measure supporting a local $p$-Poincar\'e inequality, where…

Given a smooth bounded domain $\Omega$ in $\mathbb{R}^2$, we study the following anisotropic Neumann problem $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=\lambda a(x) u^{p-1}e^{u^p},\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\,…

We investigate extremality properties of shape functionals which are products of Newtonian capacity $\cp(\overline{\Om})$, and powers of the torsional rigidity $T(\Om)$, for an open set $\Om\subset \R^d$ with compact closure…

We derive various sharp upper bounds for the $p$-capacity of a smooth compact set $K$ in the hyperbolic space $\mathbb{H}^n$ and the Euclidean space $\mathbb{R}^n$. Firstly, using the inverse mean curvature flow, for the mean convex and…

We examine the two elliptic systems given by [(G)_{\lambda,\gamma} \quad -\Delta u = \lambda f'(u) g(v), \quad -\Delta v = \gamma f(u) g'(v) \quad in $ \Omega$,] and [(H)_{\lambda,\gamma} \quad -\Delta u = \lambda f(u) g'(v), \quad -\Delta…

Let $\Omega $ be a bounded domain in $\mathbb{R} ^N $, and let $u\in C^1 (\overline{\Omega }) $ be a weak solution of the following overdetermined BVP: $-\nabla (g(|\nabla u|)|\nabla u|^{-1} \nabla u )=f(|x|,u)$, $ u>0 $ in $\Omega $ and…

Let $\phi$ be an $L^2$-normalized Hecke--Maa{\ss} cusp form for $\mathrm{PGL}_n(\mathbb{Z}[i])$ on the locally symmetric space $X:=\mathrm{PGL}_n(\mathbb{Z}[i])\backslash \mathrm{PGL}_n(\mathbb{C}) / \mathrm{PU}_n$. If $\Omega$ is a compact…