The reflector problem with near-field data:

Let $\mathscr U$ be a smooth, bounded domain in $\mathbb R^n.$ For every $x\in \mathscr U$ we issue a ray parallel to the $x_{n+1}$ axis in $\mathbb R^{n+1}$ which after reflection from an unknown reflector surface $\Gamma_u$ strikes the given surface $\Sigma$ (called receiver) at a point $Z$.
   
Clearly, the emitted rays from $\mathscr U$ and their reflections generate intensity rates on $\mathscr U$ and $\mathscr V$ (the gain domain after reflection), say $f$ and $g$, respectively. If there is no loss of energy then we must have the energy balance condition \begin{equation*} \int_{\mathscr U}f=\int_{\mathscr V}g\tag2. \end{equation*} Now the problem, we are interested in, can be formulated as follows: Let the output $(f, \mathscr U)$ and gain $(g, \mathscr V)$ be given such that the energy balance condition (2) holds. Does there exist a surface $\Gamma_u$ such that the corresponding reflector mapping $Z=\mathscr R_u(x)$ maps $\mathscr R_u:\mathscr U\to \mathscr V$ and $$\int_{E}f=\int_{\mathscr R_u(E)}g$$ for every Borel set $E\subset \mathscr U?$

There is a weak formulation of this problem allowing to construct a generalised solution for fairly wide class of data. As for the regularity of $\Gamma_u$, regarded as the graph of function $u$, then mathematically one has to study the Monge-Ampère type equation of the form $$ \det\left[D^2u-A\right]=\varphi,\tag 3 $$ where $A$ is a matrix valued function of $x$, $u(x)$,$Du(x)$, and $\varphi$ is a function of $x$, $u(x)$,$Du(x)$ depending on $f$ and $g$ as well. The analysis of (3) is very complicated and the smoothness of $u$ depends, among other things,  on the extrinsic properties of the receiver surface $\Sigma$.

The model with a point source of light was a a famous open problem mentioned among the 100 open problems of Shing-Tung Yau, as number 24. Its full solution was given in the joint paper [9] with Xu-Jia Wang.

Various problems of this sort are studied in the papers [5], [15] and [16].

The Alt-Caffarelli-Friedman problem: Consider the variational problem with two phases \begin{equation} \inf_{u\in \mathcal A}\int_\Omega |\nabla u|^p+\lambda_+^p\chi_{\{u>0\}}+\lambda_-^p\chi_{\{u\le0\}}, \quad \end{equation} where $\lambda_\pm$ are positive constants, $\lambda_+^p-\lambda_-^p,>0$, $\chi_E$ is the characteristic function of $E\subset \mathbb R^n$, and the infimum is taken over the class of admissible functions $\mathcal A=\{u \ : \ u-g\in W^{1, p}_0(\Omega)\},1<p<\infty$ for a given Dirichlet data $g\in W^{1,p}(\Omega), \Omega\subset \mathbb R^n.$ The classical case $p=2$ has been studied by Wilhelm Alt, Luis Caffarelli and Avner Friedman in 1984 where the central results are the local Lipschitz regularity of $u$ and partial regularity  of free boundary, namely \begin{equation}\label{Haus} \mathscr H^{n-1}\left(\partial\{u>0\}\setminus \partial_{\text{red}}\{u>0\}\right)=0\tag{1}, \end{equation} where $\mathscr H^{n-1}$ is the $(n-1)$ dimensional Hausdorff measure and $\partial_{\text{red}}\{u>0\}$ is the reduced boundary of the positivity set $\{u>0\}.$ There are numerous generalizations of this problem for the one phase case, (i.e. when the minimizer $u$ is nonnegative) however the general problem without sign restriction had remained widely open due to the lack of monotonicity formulae.  Recently, in the joint paper with  S.Dipierro, we proved \eqref{Haus} and established the local Lipschitz regularity for any $1<p<\infty$ using a new method based on   a stratification argument for the free boundary points.

$\sigma_{n-1}$ minimal surfaces and the singular perturbation problem: Weak solutions of the ACF problem can be obtained as limits of the singular perturbation problems $ \Delta_p u^\varepsilon=\frac1\varepsilon\beta\left(\frac{u^\varepsilon}\varepsilon\right) $ as $\varepsilon\to 0,$ where $0\le \beta\in C^\infty_0(0,1)$ with connected support. For the one phase problem, i.e. $u^\varepsilon\ge 0$ the limit function solves the following Bernoulli type free boundary problem \begin{equation}\label{except} \left\{ \begin{array}{lll} \Delta_p u(x)=0 \ \ & \mbox{in}\quad \Omega, \\ u>0\ \ &\mbox{in}\quad \Omega,\\ u(x)=0, \ |\nabla u|=1 \ \ & \mbox{on}\quad \partial\Omega, \\ \end{array} \right. \end{equation} where $\Omega:=B_1\cap \{u>0\}$. If $p=2$ then it is known that the blow-ups of $u$ at the free boundary points must be homogenous functions of degree one, i.e. $u(x)=rg(\theta)$, $\sigma\in D:=\mathbb S^{n-1}\cap \{g>0\}$. It turns out that the surface $\mathcal M$ parametrized by $X(\theta)=\theta g+\nabla_D g$ has non vanishing Gauss curvature and $\sigma_{n-1}(\kappa)=0$, where $\kappa=(\kappa_1, \dots, \kappa_{n-1})$ is the principal curvature vector and $$ \sigma_m(\kappa)=\sum_{1\le i_1<\dots< i_{m}\le m}\kappa_{i_1}\dots\kappa_{i_{m}} $$ is the $m$th elementary symmetric function. In particular, one can see that the Alt-Caffarelli double-cone corresponds to a piece of catenoid trapped in $\mathbb S^{n-1}$. Moreover, the surface $\mathcal M$ is perpendicular to $\mathbb S^{n-1}$. It is known that if $u\ge 0$ is a global minimizer of the ACF functional with $p=2$ then $\mathcal M$ is a flat disc provided that $n\le 4$. An interesting open problem is to show that this resut is true for $n=5,6$. For $p\not =2$ the classification of the global minimizers is known only for the one phase problem with $n=2$.  

The obstacle problem with fully nonlinear elliptic operators:

Let  $F:\mathcal S\to \mathbb R$ be a uniformly elliptic operator defined on the space of symmetric $n\times n$ matrices $\mathcal S$. We wish to study the viscosity solutions of the problem $$ F(D^2 u(x))=\beta(x) \chi_{\{u(x)>0\}}, \quad \mbox{in} \ B_1 $$ where $\beta$ is some given function and $\chi_{\{u(x)>0\}}$ the characteristic function of the positivity set of $u$. If $\beta =1$, $u\ge 0$,  and $F(D^2 u)=\text{Trace}(D^2 u)$ then this is the classical obstacle problem. If $\beta =-1$ and $u$ is allowed to change sign then we get the unstable obstacle problem. The problem of optimal regularity of $u$ (under suitable structural hypotheses on $F$) for the problem $F(D^2 u)=\chi_{\{u(x)>0\}}$ with sign changing solution (i.e. two phase obstacle problem) is still open.

Let $\mathbf u\in W^{1, 2}(\Omega,\mathbb R^n)$ be a vector field such that the cofactor matrix $\text{cof}(\nabla {\bf u})\in W^{1,2}(\Omega,\mathbb R^n)$, where $\Omega\subset \mathbb R^n$ is a smooth, bounded domain. We want to establish optimal regularity for the minimizers $\mathbf u$ of the stored energy functional  $$ E[u]=\int_{\Omega}|\nabla \mathbf u|^2, $$ among all vectorfields subject to the hard Jacobian constraint $\det \nabla \mathbf u=1, \text{a.e. in} \ \Omega.$  It is well-known that the sufficiently regular local minimizers solve the system
\begin{equation*}
\left\{\begin{array}{lll}\tag 4
  \operatorname{div} \mathbf  T=0 & \textrm{in}\ \Omega,\\
\det \nabla \mathbf u=1 & \textrm{a.e. in } \Omega,
\end{array}
\right.
\end{equation*}
where $\mathbf  T=\nabla \mathbf u+p(\nabla \mathbf u)^{-t}$ is the first Piola-Kirchhoff tensor, $(\nabla \mathbf u)^{-t}$ is the transpose of the inverse matrix, and  $p$ is the hydrostatic pressure manifested as the Lagrange multiplier corresponding to the incompressibility constraint $\det\nabla \mathbf u=1.$

Thus $({\bf u}, p)$ are the unknowns to be determined from the system (4). Due to the  highly nonlinear nature of (4)  only few regularity results are  known.  In the joint paper with N.Chaudhuri [7]  we found an explicit representation for $p$ as a sum of Calderon-Zygmund type singular operators provided that $\mathbf u\in L\log(2+L).$ Later in [18] I  showed that if $n=2$ and $p\ge 0$ then there is a convex function $\psi$ such that $$D_{ij}^2\psi(y) =p(\mathbf u^{-1}(y))\delta_{ij}+\sigma_{ij}(y),$$ where $\sigma_{ij}(y)=\sum_mu^i_m(\mathbf u^{-1}(y))u_m^j(\mathbf u^{-1}(y)).$ Using some basic concepts from the classical Monge-Ampère equations it follows that $p$ is locally square integrable, and consequently $\mathbf u^{-1}\in C^{\frac12}_{loc}$. The open and challenging problem is to show that the local minimizers are locally Lipschitz continuous. 

In his 1994 paper Ennio De Giorgi addressed the question of homogenizing the system of ODEs $$\frac{dx_i}{dt}=F_i\left(\frac{x_1}\epsilon, \dots, \frac{x_d}\epsilon\right)\quad i=1, \dots, d$$ and conjectured that the limit, as $\epsilon\to 0,$ is a linear function of $t,$ i.e. $x_i^0(t)=a_i^0+b_i^0t.$ In the joint paper with H.Shahgholian we have proven that for $d=1, d=2$ this is indeed the case under fairly general assumptions on $\mathbf F=(F_1, \dots, F_d)$ and for $d\ge 3$ with the so-called shear flow. Other work on homogenization is related with the thin obstacle problem. Let \[ T_\epsilon = \bigcup_{k\in \mathbb Z^d}\{\epsilon k+a_\epsilon T\}, \] and let \[\Gamma_\epsilon = \Gamma\cap T_\epsilon.\] We assume that $\Gamma$ is a strictly convex surface in $\mathbb R^d$ that locally admits the representation \begin{equation}\label{g}\{(x',g(x')):x'\in Q'\},\end{equation} where $Q'\subset\mathbb R^{d-1}$ is a cube. For example, $\Gamma$ may be a compact convex surface, or may be defined globally as a graph of a convex function. Without loss of generality we assume that $x_d = g(x').$ We will also study homogenization of the thin obstacle problem for the $p-$Laplacian with an obstacle defined on $\Gamma_\epsilon$. Our goal is to determine the asymptotic behaviour, as $\epsilon\to0,$ of the problem \begin{equation}\label{maineqe} \min \int_\Omega|\nabla v|^pdx+hvdx, \end{equation} where the minimum is taken over the class of functions $ \left\{ v\in W^{1,p}_0(\Omega) \text{ and }v\ge \phi\text{ on }\Gamma_\epsilon \right\}, $ for given $h\in L^q(\Omega),$ $1/p+1/q=1$ and $\phi\in W^{1,p}_0(\Omega)\cap L^\infty(\Omega)$ is the (thin) obstacle.
This problem is studied in the papers [17] and [23].