Research outline

Hide
One of the problems that I am interested in is concerned with the minimization of the two phase functional \begin{equation} \inf_{u\in \mathcal A}\int_\Omega \nabla u^p+\lambda_+^p\chi_{\{u>0\}}+\lambda_^p\chi_{\{u\le0\}}, \quad \lambda_+^p\lambda_^p>0 \end{equation} where $\lambda_\pm$ are positive constants, $\chi_E$ is the characteristic function of $E\subset \mathbb R^n$ and the infimum is taken over the class of admissible functions $\mathcal A=\{u \ : \ ug\in W^{1, p}_0(\Omega)\},1<p<\infty$ for a given 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 result is the partial regularity of free boundary, namely \begin{equation}\label{Haus} \mathscr H^{n1}\left(\partial\{u>0\}\setminus \partial_{\text{red}}\{u>0\}\right)=0\tag{1}, \end{equation} where $\mathscr H^{n1}$ is the $(n1)$ 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} for any $1<p<\infty$ using a new method based on stratification of free boundary points of $\partial\{u>0\}.$ Other problems in this field that I have been working on in the past are multiphase segregation problems, singular perturbations $\Delta_p u^\varepsilon=\frac1\varepsilon\beta\left(\frac{u^\varepsilon}\varepsilon\right)$ as $\varepsilon\to 0,$ the two phase Stefan problem and the tangential/transversal behaviour of the free boundary near the fixed boundary.

MongeAmpère type equations and reflector antennae design:
Hide Let $\mathscr U$ be a smooth, bounded domain in $\mathbb R^n.$ For each $x\in \mathscr U$ 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 the we must have the energy balance condition\begin{equation*}\int_{\mathscr U}f=\int_{\mathscr V}g\tag 2.\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 (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 any 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 MongeAmpère type equation of the form $$\det\left[D^2u(x)A(x, u(x), Du(x))\right]=\varphi(x, u(x), Du(x))\tag 3$$ where $\varphi$ depends on $f$ and $g$ as well. The analysis of (3) is very complicated and the smoothness of $u$ also depends on the extrinsic properties of the receiver $\Sigma$. Various problems of this sort are studied in the papers [5], [9], [15] and [16]. 
Hide Let $\mathbf u$ be a vectorfield such that $\mathbf u\in W^{1, n}(\Omega, \mathbb R^n)$ where $\Omega\subset \mathbb R^n$ is a smooth, bounded domain and $\det \nabla \mathbf u=1$ a.e. in $\Omega.$ We want to study the properties of the minimizer $\mathbf u$ of the stored energy functional $$E[u]=\int_{\Omega}\nabla \mathbf u^2,$$ among all vectorfields $\mathbf u\in W^{1, n}(\Omega, \mathbb R^n), \det \nabla \mathbf u=1, \text{a.e. in} \ \Omega.$ It is wellknown 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 PiolaKirchhoff tensor and $(\nabla \mathbf u)^{t}$ is the transpose of the inverse matrix. Observe that $p$ is the hydrostatic pressure manifested as the Lagrange multiplier corresponding to the incompressibility constraint $\det\nabla \mathbf u=1.$
Due to its highly nonlinear structure the system (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 CalderonZygmund 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 MongeAmpere 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. 
Hide 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 socalled 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 $Q'\subset\mathbb R^{d1}$ 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\left\{\int_\Omega\nabla v^pdx+\int_\Omega hvdx: v\in W^ {1,p}_0(\Omega)\text{ and }v\ge \phi\text{ on }\Gamma_\epsilon\right\}, \end{equation} 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 joint papers [17] and [23].