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. ...

Let $\lambda_1$ denotes the first eigenvalue of the Laplace operator with Dirichlet boundary conditions. The famous Faber-Krahn inequality asserts that the ball minimizes $\lambda_1$ among open sets with a given volume. A natural question is to ask whether the regular $N$-gon also minimizes $\lambda_1$ among all polygons with $N$ sides and given area. Using Steiner symmetrization, G. Polya proves that it is true for triangles and quadrilaterals, but the problem is still completely open for $N\geq 5$. ...

How can we prove Hopf's boundary point lemma, in $C^{1,Dini}$-type domains, for fractional singular/degenerate PDE-s.

This was done for standard singular/degenerate PDE-s in reference [1]. ...