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

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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 [5], [9], [15] and [16].

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 [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-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 $$\inf_{u\in \mathcal A}\int_\Omega |\nabla u|^p+\lambda_+^p\chi_{\{u>0\}}+\lambda_-^p\chi_{\{u\le0\}}, \quad \lambda_+^p-\lambda_-^p,>0$$ 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 $$\label{Haus} \mathscr H^{n-1}\left(\partial\{u>0\}\setminus \partial_{\text{red}}\{u>0\}\right)=0\tag{1},$$ 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 $$\label{except} \left\{ \begin{array}{lll} \Delta_p u(x)=0 \ \ & \mbox{in}\quad \Omega:=B_1\cap \{u>0\}, \\ u>0\ \ &\mbox{in}\quad \Omega,\\ u(x)=0, \ |\nabla u|=1 \ \ & \mbox{on}\quad \partial\Omega, \\ \end{array} \right.$$ 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 $$\label{g}\{(x',g(x')):x'\in Q'\},$$ 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 $$\label{maineqe} \min\left\{\int_\Omega|\nabla v|^pdx+\int_\Omega hvdx: v\in W^ {1,p}_0(\Omega)\text{ and }v\ge \phi\text{ on }\Gamma_\epsilon\right\},$$ 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].

PAPERS

### PAPERS

[47] 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]

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

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

[44] 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]

[43] Karakhanyan, A.
Lectures on free boundary problems

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

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

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

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

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

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

[36] Karakhanyan, A.
Lecture notes on elliptic equations
ResearchGate

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

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

[33] 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]

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

[31] 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

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

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

[28] 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

[27] 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

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

[25] 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]

[24] 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

[23] 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
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[22] Karakhanyan, A.
Regularity for a quasilinear continuous casting problem
Journal de Mathématiques Pures et Appliquées 109 (2018), 182–201
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[21] Karakhanyan, A., Bucur, C.
Potential theoretic approach to Schauder estimates for the fractional Laplacian
Proceedings of the AMS 145 (2017), no. 2, 637--651
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[20] 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
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[19] 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
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[18] Karakhanyan, A.
Regularity for energy-minimizing area-preserving deformations
Journal of Elasticity 114 (2014) no. 2, 213--223
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[17] 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
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[16] 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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[15] 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
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[14] Karakhanyan, A.
Optimal Regularity for phase transition problems with convection
l'Institut Henri Poincare, Analyse non lineaire 32 (2015), no. 4, 715--740
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[13] Karakhanyan, A., Rodrigues, J.
The Stefan problem with constant convection
Full text pdf

[12] 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
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[11] Karakhanyan, A.
Sufficient conditions for regularity of area-preserving deformations
Manuscripta Mathematica 138 (2012), 463--476
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[10] 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
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[9] Karakhanyan, A., Wang, Xu-Jia
On the reflector shape design
Journal of Differential Geometry 84 (2010), no. 3, 561--610
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[8] 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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[7] 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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[6] 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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[5] 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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[4] 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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[3] 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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[2] 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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[1] 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.