Related papers: Regularity of the free boundary for a semilinear v…

We are motivated by studying a boundary-value problem for a class of semilinear degenerate elliptic equations \begin{align}\tag{P}\label{P} \begin{cases} - \Delta_x u - |x|^{2\alpha} \dfrac{\partial^2 u}{\partial y^2} = f(x,y,u) &…

Given bounded vector field $b : \mathbb R^d \to \mathbb R^d$, scalar field $u : \mathbb R^d \to \mathbb R$ and a smooth function $\beta : \mathbb R \to \mathbb R$ we study the characterization of the distribution $\mathrm{div}(\beta(u)b)$…

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…

The classical result of A. Ambrosetti and G. Prodi [1], in the form of M.S. Berger and E. Podolak [4], gives the exact number of solutions for the problem \[ \Delta u+g(u)= \mu \phi _1(x)+e(x) \;\; \mbox{in $D$} , \;\; u=0 \;\; \mbox{on…

We continue the analysis of the two-phase free boundary problems initiated in \cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free…

In this paper we study the initial boundary value problem for the system $\mbox{div}(\sigma(u)\nabla\varphi)=0,\ \ u_t-\Delta u=\sigma(u)|\nabla\varphi|^2$. This problem is known as the thermistor problem which models the electrical heating…

We study local regularity properties of local minimizer of scalar integral functionals of the form $$\mathcal F[u]:=\int_\Omega F(\nabla u)-f u\,dx$$ where the convex integrand $F$ satisfies controlled $(p,q)$-growth conditions. We…

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $\Delta^2 u = \lambda f(u)$ on a general bounded domain $\Omega$ in $ \IR^N$, with the Navier boundary condition $ u=\Delta u =0 $ on $ \pOm$. Here $…

We consider the class of semi-stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain $\Omega$ of $R^n$ (with $\Omega$ convex in some results). This class includes all local minimizers, minimal, and extremal…

For $\nu,\nu_i,\mu_j\in(0,1)$, we analyze the semilinear integro-differential equation on the one-dimensional domain $\Omega=(a,b)$ in the unknown $u=u(x,t)$ \[…

We are concerned with the inverse boundary problem of determining anomalies associated with a semilinear elliptic equation of the form $-\Delta u+a(\mathbf x, u)=0$, where $a(\mathbf x, u)$ is a general nonlinear term that belongs to a…

We consider the optimization problem of minimizing $\int_{\Omega}G(|\nabla u|) dx$ in the class of functions $W^{1,G}(\Omega)$, with a constrain on the volume of $\{u>0\}$. The conditions on the function $G$ allow for a different behavior…

We discuss the Dirichlet problem of the quasi-linear elliptic system \begin{eqnarray*} -e^{-f(U)}div(e^{f(U)}\bigtriangledown U)+&{1/2}f'(U)|\bigtriangledown U|^2&=0, {in $\Omega$}, & U|_{\partial\Omega}&=\phi. \end{eqnarray*} Here $\Omega$…

In this paper, using a discrete Schwarz rearrangement on lattice graphs developed in \cite{DSR}, we study the existence of global minimizers for the following functional $I:H^1\left(\mathbb{Z}^N\right)\to \R$, $$I(u)=\frac{1}{2}…

We investigate regularity properties of minimizers for non-autonomous convex variational integrands $F(x, \mathrm{D} u)$ with linear growth, defined on bounded Lipschitz domains $\Omega \subset \mathbb{R}^n$. Assuming appropriate…

We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus \Omega)^2} = f(x,u) & \quad \mbox{in } \Omega \\ u = 0 &…

We provide a priori error estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form $-{div} (A\nabla u) + Vu + f(u^2) u = \lambda u$, $\|u\|_{L^2}=1$. We…

We prove {the first} regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain $\Omega$ is obtained as the integral of a cost function $j(u,x)$…

In this paper, we study the boundary value problem of the classical semilinear parabolic equations $$ u_t-\Delta u=|u|^{p-1}u, \ \ in \ \ \Omega\times (0,T) $$ and $u=0$ on the boundary $\partial\Omega\times [0,T)$ and $u=\phi$ at $t=0$,…

In this article, we study the existence of non-trivial weak solutions for the following boundary-value problem \begin{gather*} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial y^2}=f(x,y,u) \quad\text{ in…