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In this paper, we present several sharp upper bounds for the numerical radii of the diagonal and off-diagonal parts of the $2\times2$ block operator matrix $\begin{bmatrix}A&B\\ C&D\end{bmatrix}$. Among extensions of some results of…

We study radial viscosity solutions to the equation \[ -\ |Du\ |^{q-2}\Delta_{p}^{N}u=f(\ |x\ |)\quad\text{in }B_{R}\subset\mathbb{R}^{N}, \] where $f\in C[0,R)$, $p,q\in(1,\infty)$ and $N\geq2$. Our main result is that $u(x)=v(\ |x\ |)$ is…

In this paper, we establish the existence of prescribed mean curvature (PMC) hypersurfaces in conformal product manifolds with (possibly empty) $C^{1,\alpha}$ fixed graphical boundaries under a barrier condition. This result generalizes…

This paper is devoted to the investigation of the boundary regularity for the Poisson equation $${{cc} -\Delta u = f & \text{in} \Omega u= 0 & \text{on} \partial \Omega$$ where $f$ belongs to some $L^p(\Omega)$ and $\Omega$ is a…

In this article we consider the following boundary value problem \begin{equation*}\label{abs} \left\{ \begin{aligned} F(x,u,Du,D^{2}u)+c(x)u+ p(x)u^{-\alpha}&=0~\text{in}~\Omega\\ u&=0~~\text{on}~~\partial\Omega, \end{aligned} \right.…

Let $(M, g)$ be a compact Riemannian manifold with boundary $\partial M$. Given a function $f$ on $\partial M$, we consider the problem of finding a conformal metric of $g$ with zero scalar curvature in $M$ and prescribed mean curvature $f$…

We consider the class of semi-stable positive solutions to semilinear equations $-\Delta u=f(u)$ in a bounded domain $\Omega\subset\mathbb R^n$ of double revolution, that is, a domain invariant under rotations of the first $m$ variables and…

Let $n\ge 2$ and $s\in (n-2,n)$. Assume that $\Omega\subset \mathbb{R}^n$ is a one-sided bounded non-tangentially accessible domain with $s$-Ahlfors regular boundary and $\sigma$ is the surface measure on the boundary of $\Omega$, denoted…

Let $f$ be a continuous and non-decreasing function such that $f>0$ on $(0,\infty)$, $f(0)=0$, $\sup \_{s\geq 1}f(s)/s< \infty$ and let $p$ be a non-negative continuous function. We study the existence and nonexistence of explosive…

This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain $\Omega\subset\mathbb{R}^n$, with or without a source term in the right hand side of the equation…

In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial_t u - \mbox{div} (A \nabla u)+B\cdot \nabla u=0$ on cylindrical domains of the form $\Omega=\mathcal O\times\mathbb R$…

In this paper we consider positive supersolutions of the nonlinear elliptic equation \[- \Delta u = \rho(x) f(u)|\nabla u|^p, \qquad \hfill \mbox{ in } \Omega,\] where $0\le p<1$, $ \Omega$ is an arbitrary domain (bounded or unbounded) in $…

In this article, we deal with the fine boundary regularity, a weighted H\"{o}lder regularity of weak solutions to the problem involving the fractional $(p,q)$ Laplacian denoted by $(-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u = f(x)$ in…

Given a mean-convex domain $\Omega\subset \R^n$ with boundary of class $C^{2,1}$, we provide a representation formula for a boundary integral of the type \[ \int_{\partial \Omega} f(k(x)) \, d\mathcal{H}^{n-1} \] where $k\geq 0$ is the mean…

In this paper, we mainly discuss the local regularity of the solution to the following problem \begin{align*} \begin{cases} -\dive({\bf{A}}(x)\nabla u(x))=f(x),&~x\in\Omega,\\ u(x)=0,&~x\in\partial\Omega, \end{cases} \end{align*} where…

In this work we address the question of the existence of nonradial domains inside a nonconvex cone for which a mixed boundary overdetermined problem admits a solution. Our approach is variational, and consists in proving the existence of…

We give upper and lower bounds for the spectral radius of a nonnegative matrix by using its average 2-row sums, and characterize the equality cases if the matrix is irreducible. We also apply these bounds to various nonnegative matrices…

Suppose that $\lambda - T$ is left-invertible in $L(H)$ for all $\lambda \in \Omega$, where $\Omega$ is an open subset of the complex plane. Then an operator-valued function $L(\lambda)$ is a left resolvent of $T$ in $\Omega$ if and only if…

We develop a new technique for studying the boundary limiting behavior of a holomorphic function on a domain $\Omega$ -- both in one and several complex variables. The approach involves two new localized maximal functions. As a result of…

We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where…