The exponential transform of a domain $\Omega$ in $\mathbb{R}^n$ ($n\geq 2$) is defined by $$ E(x)= \exp [-\frac{2}{|S^{n-1}|}\int_\Omega \frac{dy}{|x-y|^{n}}], \quad x\in\mathbb{R}^n\setminus\overline{\Omega}, $$ where $|S^{n-1}|$ denotes the area of the unit sphere. The exponential transform is a positive function in the exterior of $\Omega$, with zero boundary values at all reasonably regular points of $\partial\Omega$. The problem is to study in more detail the relationship between the regularity of $\partial\Omega$ and the boundary behavior of $E(x)$. More specifically we conjecture that if $\partial\Omega$ is smooth real analytic then $E(x)$ has a real analytic continuation across $\partial\Omega$. ...

Let $\mu$ be a positive measure with compact support in $\mathbb{R}^n$, $n\geq 3$, and let $K$ be the convex hull of the support of $\mu$. Let $u\geq 0$ be the smallest lower semicontinuous function satisfying $$ u\geq 0, \quad \mu+\Delta u\leq 1, $$ and let $\Omega=\{u>0\}$ be the non-coincidence set for this obstacle problem. The problem is to show that $\Omega$ can be written as a union of balls with centers in $K$. This amounts to saying that $\partial\Omega$ is subject to a natural, and sharp, upper bound for the curvature. ...

Conjecture: Any (weak/viscosity) solution $u$ to the semilinear elliptic equation $$ \hbox{div} (f(x,u)\nabla u) = 0, \qquad \hbox{in } B_1 $$ with $f$ being Dini continuous in $x$-variable, and having only finite number of discontinuities in $u$-variable, and uniform $C^\alpha $ regularity to the right and left of the discontinuities, should have universally bounded derivatives in $B_{1/2}$, with norms depending on $\|u\|_\infty$, ellipticity constants and the space dimension. ...

The following, and several similar problems, was discussed in paper [1]. ...

We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain $\Omega$ that varies over all subdomains of a given bounded domain $D$ of $R^d$. We show in a rather elementary way the existence of a solution that is in general a quasi open set. Under very mild conditions we show that the optimal domain is actually open and with finite perimeter. Some counterexamples show that in general this does not occur. Consider now the shape optimization problem $$\min\Big\{F(\Omega)\ :\ \Omega\subset D\Big\}$$ with $$F(\Omega)=\min\left\{\int_\Omega\Big[\frac1p|\nabla u|^p-f(x)u\Big]\,dx\ :\ u\in W^{1,p}_0(\Omega)\right\}+|\Omega|$$ and with $f\in W^{-1,p'}$. We know that an optimal domain, in the class of $p$ quasi-open sets, always exists and, when $f\in L^q$ with $q>d/p$, it is an open set. Two questions arise: ...

Consider a $p$-harmonic function in the unit ball $\mathbb{B}(0,1)\subset\mathbb{R}^n$, where $1< p< \infty$, $p\not=2$, and $n\ge3$. Suppose that $u(x)=0$ for all $x\in \mathbb{B}(0,1/2)$. Does it follow that $u$ is identically zero in $\mathbb{B}(0,1)$? ...

Let $D$ be a bounded open domain and assume for some $u\in W_0^{1,2}(D)$ and $\alpha>0$ the following is true

(i) $-\Delta u=\text{sgn} (u-\alpha)$ in $D$,

(ii) $\int_D \text{sgn} (u-\alpha)dx=0$,

(iii) $u>0$ in $D$.

Show that $D$ is a ball. ...