Mittag-Leffler Institute, 2015 • Research Interests: Nonlinear PDEs, Geometric Analysis
• Capillary surfaces, K-surfaces
• Monge-Ampere equation, reflector surfaces
• Phase transitions, Free boundary problems
• Grants:
• EP/K024566/1, PI, 2013-15, 126,258 GBP
• Polonez grant, 2015/19/P/ST1/02618, CoI, 2016-18, 825,664 zł ( 169,803 GBP)
• EP/S03157X/1, PI, 2019-24, 606,851 GBP
RESEARCH INTERESTS

### A BRIEF OUTLINE OF MY RECENT WORK

Click on the button to toggle between showing and hiding content. The parallel 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 $$\det\left[D^2u(x)-A(x, u(x), Du(x))\right]=\varphi(x, u(x), Du(x))\tag 3$$ where $\varphi$ depends on $f$ and $g$ as well. The analysis of (3) is very complicated and the smoothness of $u$ depends, among other things, on the extrinsic properties of the receiver surface $\Sigma$. Various problems of this sort are studied in the papers , ,  and .

Let $\mathbf u\in W^{1, 2}(\Omega,\mathbb R^n)$ be a vector field such that $\text{cof}(\nabla {\bf u})\in W^{1,2}(\Omega,\mathbb R^n)$ 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   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  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-Ampere 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.

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 \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 $\mathcal A=\{u \ : \ u-g\in W^{1, p}_0(\Omega)\},1<p<\infty$ 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:=B_1\cap \{u>0\}, \\ u>0\ \ &\mbox{in}\quad \Omega,\\ u(x)=0, \ |\nabla u|=1 \ \ & \mbox{on}\quad \partial\Omega, \\ \end{array} \right. \end{equation} 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 carachteristic 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.

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\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  and .

PAPERS

### PAPERS

 Espin, T., Karakhanyan, A.
Boundary estimates for solutions of the Monge-Ampere equation satisfying Dirichlet-Neumann type
conditions in annular domains
Methods and Applications of Analysis In press
arxiv:2006.09814 [math.AP]

 Karakhanyan, A.
A remark on an obstacle problem with lower regularity
arXiv:1910.06992 [math.AP]

 Karakhanyan, A.
Regularity for the two phase singular perturbation problems
arXiv:1910.06997 [math.AP]

 Dipierro, S., Karakhanyan, A., Valdinoci, E.
Limit behaviour of a singular perturbation problem for the biharmonic operator
Applied Mathematics and Optimization In press
arXiv:1902.06675 [math.AP]

 Karakhanyan, A.
Lectures on free boundary problems

 Karakhanyan, A.
A nonlocal free boundary problem with Wasserstein distance
arXiv:1904.06270 [math.AP]

 Karakhanyan, A.
Full and partial regularity for a class of nonlinear free boundary problems
arXiv:1811.07620 [math.AP]

 Karakhanyan, A.
Structure of singularities in the nonlinear nerve conduction problem
arXiv:1906.05383 [math.AP]

 Dipierro, S., Karakhanyan, A., Valdinoci, E.
A free boundary problem driven by the biharmonic operator
arXiv:1808.07696 [math.AP]

 Karakhanyan, A., Sabra A.
Refractor surfaces determined by near-field data
Journal of Differential Equations In press
arxiv:1810.07094 [math.AP]

 Karakhanyan, A.
Singular Yamabe problem for scalar flat metrics on the sphere
arXiv:1711.01669 [math.AP]

 Karakhanyan, A.
Lecture notes on elliptic equations
ResearchGate

 Dipierro, S., Karakhanyan, A., Valdinoci, E.
New trends in free boundary problems
Advanced Nonlinear Studies 17 (2017), no. 2, 319--332
Full text pdf

 Karakhanyan, A.
Remarks on the thin obstacle problem and constrained Ginibre ensembles
Communications in PDEs In press
arXiv:1702.00466 [math.AP]

 Dipierro, S., Karakhanyan, A., Valdinoci, E.
Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients
Discrete and Cont. Dynamical Systems In press
arXiv:1701.03131 [math.AP]

 Karakhanyan, A.
A geometric approach to regularity for nonlinear free boundary problems
arXiv:1702.00465 [math.AP]

 Dipierro, S., Karakhanyan, A., Valdinoci, E.
A nonlinear free boundary problem with a self-driven Bernoulli condition
Journal of Functional Analysis 273 (2017), no. 11, 3549–3615
Full text pdf

 Aleksanyan, H., Karakhanyan, A.
$K$-surfaces with free boundaries
arXiv:1705.04842 [math.AP]

 Karakhanyan, A.
Capillary surfaces arising in singular perturbation problems
Analysis and PDE in press
arXiv:1701.08232 [math.AP]

 Dipierro, S., Karakhanyan, A., Valdinoci, E.
A class of unstable free boundary problems
Analysis and PDE 10 (2017), no. 6, 1317–1359
Full text pdf

 Karakhanyan, A.
Blaschke's rolling ball theorem and the Trudinger-Wang monotone bending
Journal of Differential Equations 260 (2016), no. 7, 6322–6332
Full text pdf

 Karakhanyan, A., Shahgholian, H.
Boundary behavior for a singular perturbation problem
Nonlinear Analysis Series A, TMA 138 (2016), 176–188
Full text pdf

 Dipierro, S., Karakhanyan, A.
A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two
Communications in PDEs In press
arXiv:1509.00277 [math.AP]

 Dipierro, S., Karakhanyan, A.
Stratification of free boundary points for a two-phase variational problem
Advances in Mathematics 328 (2018), 40-81
Full text pdf

 Karakhanyan, A., Stromqvist, M.
Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization
Calc. Var. and Partial Differential Equations 55 (2016), no. 6, Art. 138
Full text pdf

 Karakhanyan, A.
Regularity for a quasilinear continuous casting problem
Journal de Mathématiques Pures et Appliquées 109 (2018), 182–201
Full text pdf

 Karakhanyan, A., Bucur, C.
Potential theoretic approach to Schauder estimates for the fractional Laplacian
Proceedings of the AMS 145 (2017), no. 2, 637--651
Full text pdf

 Karakhanyan, A., Shahgholian, H.
On a conjecture of De Giorgi related to homogenization
Annali di Matematica Pura ed Applicata 196 (2017), no. 6, 2167--2183
Full text pdf

 Karakhanyan, A.
Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation
Discrete and Cont. Dynamical Systems 36 (2016), no. 1, 261--277
Full text pdf

 Karakhanyan, A.
Regularity for energy-minimizing area-preserving deformations
Journal of Elasticity 114 (2014) no. 2, 213--223
Full text pdf

 Karakhanyan, A., Stromqvist, M.
Application of uniform distribution to homogenization of a thin obstacle problem with $p$-Laplacian
Communications in PDEs 39 (2014), no. 10 1870--1897
Full text pdf

 Karakhanyan, A.
An inverse problem for the refractive surfaces with parallel lighting
SIAM Journal of Mathematical Analysis 48 (2016), no. 1, 740--784
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 Karakhanyan, A.
Existence and regularity of the reflector surfaces in $\mathbb R^{n+1}$
Archive for Rational Mechanics and Analysis 213 (2014), no. 3, 833--885
Full text pdf

 Karakhanyan, A.
Optimal Regularity for phase transition problems with convection
l'Institut Henri Poincare, Analyse non lineaire 32 (2015), no. 4, 715--740
Full text pdf

 Karakhanyan, A., Rodrigues, J.
The Stefan problem with constant convection
Full text pdf

 Karakhanyan, A., Shahgholian, H.
Analysis of a free boundary at contact point with Lipschitz data
Transactions of the AMS 367 (2015), no. 7, 5141--5175
Full text pdf

 Karakhanyan, A.
Sufficient conditions for regularity of area-preserving deformations
Manuscripta Mathematica 138 (2012), 463--476
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 Caffarelli L., Karakhanyan, A., and Lin Fang-Hua
The geometry of solutions to a segregation problem for non-divergence systems
J. of Fixed Point Theory and Appl. 5 (2009), no. 2, 319--351
Full text pdf

 Karakhanyan, A., Wang, Xu-Jia
On the reflector shape design
Journal of Differential Geometry 84 (2010), no. 3, 561--610
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 Karakhanyan, A.
On the regularity of weak solutions to refractor problem
Arm. J. Math. 2 (2009), no. 1, pp. 28--37
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 Chaudhuri, N., Karakhanyan, A.
On Derivation of Euler-Lagrange Equations for incompressible energy-minimizers
Cal. of Var. and Partial Differential Equations 36 (2009), no. 4, 627--645
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 Caffarelli, L., Karakhanyan, A.
Lectures on gas flow in porous media
Applied and Numerical Harmonic Analysis Birkhauser Boston, Inc., Boston, MA, 2010, 133--157
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 Karakhanyan, A., Wang, Xu-Jia
The reflector design problem
Proceedings of Inter. Congress of Chinese Mathematicians 2 (2007), no 1-4, 1--24
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 Karakhanyan, A.
On the Lipschitz regularity of solutions of minimum problem with free boundary
Interfaces and Free Boundaries 10 (2008), no. 1, 79--86
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 Karakhanyan, A., Kenig, C. and Shahgholian, H.
The behavior of the free boundary near the fixed boundary for a minimization problem
Calc. Var. and Partial Differential Equations 28 (2007), no. 1, 15--31
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 Karakhanyan, A.
Up-to boundary regularity for a singular perturbation problem of $p$-Laplacian type
J. Differential Equations 226 (2006), no. 2, 558--571
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 Hakobyan, A., Karakhanyan, A.
Nonlinear free boundary problems with singular source terms
Monatsh. Math. 142 (2004), no. 1-2, 7--16
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### SYMPOSIA

The Maxwell minisymposia in PDEs are one day events usually with 2-3 international speakers taking place at the International Centre for Mathematical Sciences in Edinburgh. They aim to bring the applied and pure analysts from the Heriot-Watt and Edinburgh universities together and stimulate the interest in PDEs in Scotland and the UK.