Concavity principles for nonautonomous elliptic equations and applications

In the study of concavity properties of positive solutions to nonlinear elliptic partial differential equations the diffusion and the nonlinearity are typically independent of the space variable. In this paper we obtain new results aiming to get almost concavity results for a relevant class of anisotropic semilinear elliptic problems with spatially dependent source and diffusion.Comment: 13 pages, Asymptotic Analysis, to appea

Concavity properties for quasilinear equations and optimality remarks

In this paper we study quasiconcavity properties of solutions of Dirichlet problems related to modified nonlinear Schr\"odinger equations of the type $$-{\rm div}\big(a(u) \nabla u\big) + \frac{a'(u)}{2} |\nabla u|^2 = f(u) \quad \hbox{in $\Omega$},$$ where $\Omega$ is a convex bounded domain of $\mathbb{R}^N$. In particular, we search for a function $\varphi:\mathbb{R} \to \mathbb{R}$, modeled on $f\in C^1$ and $a\in C^1$, which makes $\varphi(u)$ concave. Moreover, we discuss the optimality of the conditions assumed on the source.Comment: To be published on Differential and Integral Equation

Some overdetermined problems related to the anisotropic capacity

We characterize the Wulff shape of an anisotropic norm in terms of solutions to overdetermined problems for the Finsler p-capacity of a convex set Omega subset of R-N, with 1 < p < N. In particular we show that if the Finsler p-capacitary potential u associated to 0 has two homothetic level sets then 0 is Wulff shape. Moreover, we show that the concavity exponent of u is q = -(p - 1)/(N - p) if and only if Omega is Wulff shape

Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian

For domains in $\mathbb{R}^d$, $d\geq 2$, we prove universal upper and lower bounds on the product of the lowest eigenvalue for the Laplacian to the power $p>0$ and the supremum over all starting points of the $p$-moments of the exit time of Brownian motion. It is shown that the lower bound is sharp for integer values of $p$ and that for $p \geq 1$, the upper bound is asymptotically sharp as $d\to\infty$. Our upper bounds improve the known results in the literature even for $p=1$. For all $p>0$, we verify the existence of an extremal domain among the class of domains that are convex and symmetric with respect to all coordinate axes. For these domains we conjecture that the cube is extremal

Optimal Concavity for Newtonian Potentials

In this note I give a short overview about convexity properties of solutions to elliptic equations in convex domains and convex rings and show a result about the optimal concavity of the Newtonian potential of a bounded convex domain in ℝn , n ≥ 3, namely: if the Newtonian potential of a bounded domain is ”sufficiently concave”, then the domain is necessarily a ball. This result can be considered an unconventional overdetermined problem.This paper is based on a talk given by the author in Bologna at the ”Bruno Pini Mathematical Analysis Seminar”, which in turn was based on the paper P. Salani, A characterization of balls through optimal concavity for potential functions, Proc. AMS 143 (1) (2015), 173-183

Concavity properties of solutions to Robin problems

We prove that the Robin ground state and the Robin torsion function are respectively log-concave and $\frac{1}{2}$-concave on an uniformly convex domain $\Omega\subset \mathbb{R}^N$ of class $\mathcal{C}^m$, with $[m -\frac{ N}{2}]\geq 4$, provided the Robin parameter exceeds a critical threshold. Such threshold depends on $N$, $m$, and on the geometry of $\Omega$, precisely on the diameter and on the boundary curvatures up to order $m$.Comment: 24 page

Optimal concavity of the torsion function

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\geq 2$ and $\Omega$ be a bounded open set in $\mathbb{R}^n$ whose torsion function $u$ (i.e. the solution to $\Delta u=-1$ in $\Omega$, vanishing on $\partial\Omega$) satisfies the following property: $\sqrt{M-u(x)}$ is convex, where $M=\max\{u(x)\,:\,x\in\overline\Omega\}$. Then $\Omega$ is an ellipsoid

Optimal concavity of the torsion function

International audienceIn this short note we consider an unconventional overdetermined problem for the torsion function: let n ≥ 2 and Ω be a bounded open set in R n whose torsion function u (i.e. the solution to ∆u = −1 in Ω, vanishing on ∂Ω) satisfies the following property: M − u(x) is convex, where M = max{u(x) : x ∈ Ω}. Then Ω is an ellipsoid