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In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace…

In a smoothly bounded convex domain $\Omega\subset R^n$ with $n\ge 1$, a no-flux initial-boundary value problem for \[ \left\{ \begin{array}{l} u_t=\Delta \big(u\phi(v)\big), v_t=\Delta v-uv, \end{array} \right. \] is considered under the…

We give a localized regularity condition for energy conservation of weak solutions of the Euler equations on a domain $\Omega\subset \mathbb{R}^d$, $d\ge 2$, with boundary. In the bulk of fluid, we assume Besov regularity of the velocity…

We propose a derivative-free trust-region method based on finite-difference gradient approximations for smooth optimization problems with convex constraints. The proposed method does not require computing an approximate stationarity…

Let $\Omega \subset \mathbb{R}^2$ be a bounded, convex domain and let $-\Delta \phi_1 = \mu_1 \phi_1$ be the first nontrivial Laplacian eigenfunction with Neumann boundary conditions. The Hot Spots conjecture claims that the maximum and…

We consider the torsional rigidity and the principal eigenvalue related to the $p$-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. The limit…

We provide upper and lower bounds for the mean ${\mathscr M}(H)$ of $\sup_{t\geqslant 0} \{B_H(t) - t\}$, with $B_H(\cdot)$ a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter $H\in(0,1)$. We find…

The aim of this paper is twofold. First, we cut off a part of a convex surface by a plane near a ridge point and characterize the limiting behavior of the surface measure in $S^2$ induced by this part of surface when the plane approaches…

We prove well-posedness in reflexive Sobolev spaces of weak solutions to the stationary Stokes problem with Navier slip boundary condition over bounded domains $\Omega$ of $\mathbb{R}^n$ of class $W^{2-1/s}_s$, $s>n$. Since such domains are…

We give a H\"ormander type $L^2-$estimate for the $\bar{\partial}-$equation with respect to the measure $\delta_\Omega^{-\alpha}dV$, $\alpha<1$, on any bounded pseudoconvex domain with $C^2-$boundary. Several applications to the function…

We proceed with the investigation of the problem $(P_\lambda): $ $-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega$, where $\Omega$ is a…

We consider the following singularly perturbed Neumann problem \begin{eqnarray*} \ve^2 \Delta u -u +u^p = 0 \, \quad u>0 \quad {\mbox {in}} \quad \Omega, \quad {\partial u \over \partial \nu}=0 \quad {\mbox {on}} \quad \partial \Omega,…

In this note we construct smooth bounded domains $\Omega \subset \mathbb R^2$, other than disks, for which the overdetermined problem $$ \left\{ \begin{alignedat}{2} \Delta u + \lambda u &= 0 &\qquad& \text{ in } \Omega, \newline u &= b…

Let $\Omega\subset\mathbb R^2$ be a bounded domain of class $C^{2+\alpha}$, $0<\alpha<1$. We show that if $u$ is the maximal solution of $\Delta u = 4\exp(2u)$, which tends to $+\infty$ as $(x,y)\to\partial\Omega$, then the hyperbolic…

An accurate functional inequality for Div-BV positive symmetric tensors $A$ in a bounded domain $U\subset\mathbb{R}^n$ arises whenever the tangential part of the normal trace $\gamma_\nu A\sim A\vec\nu$ is a finite measure over $\partial…

This paper is concerned with the Neumann initial-boundary value problem for the two-species chemotaxis system with consumption of chemoattractant \begin{equation*} u_t=\Delta u-\chi_1\nabla\cdot(u\nabla w), \end{equation*} \begin{equation*}…

We consider a scalar valued elliptic partial differential equation on a sufficiently smooth domain $\Omega$, subject to a regularized Tresca friction-type boundary condition on a subset $\Gamma$ of $\partial \Omega$. The friction threshold,…

For $0<\alpha<1$ let $V(\alpha)$ denote the supremum of the numbers $v$ such that every $\alpha$-H\"older continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2…

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider $T(\Omega)/(M(\Omega)|\Omega|)$ and $M(\Omega)\lambda_1(\Omega) $, where $\Omega$ is a…

We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let $\Omega\subset {\bf R}^n$ ($n\geq 2$) be a smooth bounded domain and let $\Phi:{\bf R}^2\to {\bf R}$ be a $C^1$ function, with $\Phi(0,0)=0$,…