Related papers: Power concavity for elliptic and parabolic boundar…

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem \begin{equation} \tag{$P$} \left\{\begin{array}{ll} \partial_t u=\Delta u +f(x,t,u,\nabla u) &…

In this article we consider a special type of degenerate elliptic partial differential equations of second order in convex domains that satisfy the interior sphere condition. We show that any positive viscosity solution $u$ of $-|\nabla…

We study the Dirichlet problem for the weighted Schr\"odinger operator \[-\Delta u +Vu = \lambda \rho u,\] where $\rho$ is a positive weighting function and $V$ is a potential. Such equations appear naturally in conformal geometry and in…

In this paper, we study properties of solutions of the Dominative $p$-Laplace equation with homogeneous Dirichlet boundary conditions in a bounded convex domain $\Omega$. For the equation $-\mathcal{D}_p u= 1$, we show that $\sqrt{u}$ is…

In a domain of the Euclidean space, we estimate from below the distance to the boundary of global maximum points of solutions of elliptic and parabolic equations with homogeneous Dirichlet boundary values. As reference cases, we first…

In this paper, we prove that the first (positive) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator \[ L(u)=\Delta u-(\nabla u,x), \] is strongly log-concave if the domain is bounded and convex, which improves the conclusion in [6].…

The unique existence of a weak solution to the homogeneous closed Dirichlet problem on certain D-star-shaped domains is proven for a mixed elliptic-hyperbolic equation. Equations of this kind arise in models for electromagnetic wave…

The paper addresses the doubly elliptic eigenvalue problem $$\begin{cases} -\Delta u=\lambda u \qquad &\text{in $\Omega$,}\\ u=0 &\text{on $\Gamma_0$,}\\ -\Delta_\Gamma u +\partial_\nu u =\lambda u\qquad &\text{on $\Gamma_1$,} \end{cases}…

We consider an inverse boundary value problem for the doubly nonlinear parabolic equation \[ \epsilon(x)\partial_t u^m-\nabla\cdot\bigl(\gamma(x)|\nabla u|^{p-2}\nabla u\bigr)=0 \quad\text{in }(0,T)\times\Omega, \] where…

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic…

We study the log-concavity of the first Dirichlet eigenfunction of the Laplacian for convex domains. For positively curved surfaces satisfying a condition involving the curvature and its second derivatives, we show that the first…

In this article, we study certain type of boundary behaviour of positive solutions of the heat equation on the upper half-space of $\R^{n+1}$. We prove that the existence of the parabolic limit of a positive solution of the heat equation at…

In this paper we prove that solutions to several shape optimization problems in the plane, with a convexity constraint on the admissible domains, are polygons. The main terms of the shape functionals we consider are either E f ($\Omega$),…

This article studies the Dirichlet problem for a class of degenerate fully nonlinear elliptic equations on Riemannian manifolds with \textit{mean concave} boundary in the sense that the mean curvature of the boundary is…

We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue sigma_1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Sigma with genus gamma and k boundary…

We consider the asymptotic behavior of nonlinear nonlocal flows $u_t+(-\La)^{1/2}u=0$ to find the geometric property of the solutions in nonlinear eigenvalue problem: (-\La)^{1/2}\vp=\lambda\vp posed in a strictly convex domain…

In this paper, we prove the concavity of $p$-entropy power of probability densities solving the $p$-heat equation on closed Riemannian manifold with nonnegative Ricci curvature. As applications, we give new proofs of $L^p$-Euclidean Nash…

We present explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable…

We describe a general method for constructing Heisenberg uniqueness pairs $(\Gamma,\Lambda)$ in the euclidean space $\mathbb{R}^{n}$ based on the study of boundary value problems for partial differential equations. As a result, we show, for…

We consider second order elliptic systems of partial differential equations subject to Dirichlet and Neumann boundary conditions. We prove analyticity of the elementary symmetric functions of the eigenvalues, and compute Hadamard-type…