Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solve the PDE $$ \begin{cases} -\Delta_p u=f\;\text{in $D$}\\ u\in W_0^{1,p}(D) \end{cases} $$ ...

Suppose that $1< p< \infty$, $p \not= 2$ , and that the space dimension $n\ge 3$. Let $u$ and $v$ be $p$-harmonic functions in the unit ball $\mathbb{B}^n$ of $\mathbb{R}^n$. Suppose that for all $x\in\mathbb{B}^n$ we have $u(x)\le v(x)$ and that $u(0)=v(0)$. Prove that $u\equiv v$. ...

Consider the homogenization problem of the elliptic system \begin{equation} - \nabla \cdot A \left( \frac{x}{\varepsilon} \right) \nabla u (x) = 0, \ \ x \in D, \tag{1} \end{equation} in a domain $D\subset \mathbb{R}^d$, ($d\geq 2$), and with oscillating Dirichlet boundary data \begin{equation} u(x) = g \left(x , \frac{x}{\varepsilon} \right), \ \ x \in \partial D. \tag{2} \end{equation} ...

Let $U$ be a bounded open set in $\mathbb{R}^2 $. Let $u$ be the minimizer of $$\int_U |\nabla u|^2 -u\,dx \tag{$\ast$ }$$ over the set $\{u\in H^1_0(U) \mid |\nabla u|\le 1 \}$. This is the classical elastic-plastic torsion problem. Let $E:=\{|\nabla u|< 1\}$ be the elastic region, and let $P:=\{ |\nabla u|=1\}$ be the plastic region. It can be shown that there is a function $f:\partial U \to [0,\infty )$ such that the plastic region is consisted of connected components of the form $$\{x+t\nu(x) \mid x\in\partial U, f(x)\ge t>0\},$$ where $\nu$ is the inward unit normal to $\partial U$. (Note that $f$ actually parametrizes the free boundary.) In [2] it is shown that when a part of $\partial U$ is a line segment, or a segment of a circle, then the number of plastic components attached to it are finite. This is also repeated in the book [1], where also the question of finiteness of the number of all plastic components is raised. ...

The free boundary regularity for singular control problems is important in order to be able to construct an optimal control (given by a reflection of the stochastic process). For so called optimal dividend problems (a class of problems on optimal corporate capital structure), this is characterized by the interface between an elliptic condition and a gradient constraint in one direction. ...

Let $u \in C^\infty({\mathbb{R}}^n,[-1,1])$ be a solution of $(-\Delta)^{s/2}u=u-u^3$ in ${\mathbb{R}}^n$, with $\partial_{x_n}u>0$ and $$ \lim_{x_n\to\pm\infty} u(x',x_n)=\pm1,$$ for any $x'\in {\mathbb{R}}^{n-1}$. For which dimension $n$ and fractional exponent $s\in(0,2)$ can one conclude that $u$ is necessarily one-dimensional? ...

The fractional Allen-Cahn equation in a domain $\Omega\subset{\mathbb{R}}^n$ may be seen as the Euler-Lagrange equation of the energy functional $$ {\mathcal{E}}_\Omega(u):= c\,\iint_{Q_\Omega} \frac{|u(x)-u(y)|^2}{|x-y|^{n+s}}\,dx\,dy+\int_\Omega (1-u^2(x))^2\,dx,$$ where $c>0$ is an appropriate normalizing constant and $$ Q_\Omega:=\big(\Omega\times\Omega\big)\cup\big(\Omega\times\Omega^c\big)\cup\big(\Omega^c\times\Omega\big).$$ One says that $u:{\mathbb{R}}^n\to{\mathbb{R}}$ is a minimizer in $\Omega$ if ${\mathcal{E}}_\Omega(u)\leq {\mathcal{E}}_\Omega(u+\varphi)$ for any $\varphi\in C^\infty_0(\Omega)$. One also says that $u$ is a local minimizer when it is a minimizer in any domain $\Omega\subset{\mathbb{R}}^n$. ...