FORTIS
ADIUVAT

**Mittag-Leffler Institute, 2015**

RESEARCH INTERESTS

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**The parallel reflector problem with near-field data:**

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.

\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
\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:**

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 [17] and [23].

PAPERS

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[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]

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]

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

Proceedings of the London Mathematical Society In press

arXiv:1910.06997 [math.AP]

Regularity for the two phase singular perturbation problems

Proceedings of the London Mathematical Society In press

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 Vol. 80 (2019), 679–713

arXiv:1902.06675 [math.AP]

Limit behaviour of a singular perturbation problem for the biharmonic operator

Applied Mathematics and Optimization Vol. 80 (2019), 679–713

arXiv:1902.06675 [math.AP]

[43] Karakhanyan, A.

Lectures on free boundary problems

Lectures on free boundary problems

[42] Karakhanyan, A.

A nonlocal free boundary problem with Wasserstein distance

arXiv:1904.06270 [math.AP]

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

l'Institut Henri Poincare, Analyse non lineaire In press

arXiv:1811.07620 [math.AP]

Full and partial regularity for a class of nonlinear free boundary problems

l'Institut Henri Poincare, Analyse non lineaire In press

arXiv:1811.07620 [math.AP]

[40]
Karakhanyan, A.

Structure of singularities in the nonlinear nerve conduction problem

arXiv:1906.05383 [math.AP]

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

Pure and Applied Analysis Vol. 2 (2020), No. 4, 875–942

arXiv:1808.07696 [math.AP]

A free boundary problem driven by the biharmonic operator

Pure and Applied Analysis Vol. 2 (2020), No. 4, 875–942

arXiv:1808.07696 [math.AP]

[38]
Karakhanyan, A., Sabra A.

Refractor surfaces determined by near-field data

Journal of Differential Equations Vol. 269 (2020), Issue 2, 1278–1318

arxiv:1810.07094 [math.AP]

Refractor surfaces determined by near-field data

Journal of Differential Equations Vol. 269 (2020), Issue 2, 1278–1318

arxiv:1810.07094 [math.AP]

[37]
Karakhanyan, A.

Singular Yamabe problem for scalar flat metrics on the sphere

arXiv:1711.01669 [math.AP]

Singular Yamabe problem for scalar flat metrics on the sphere

arXiv:1711.01669 [math.AP]

[35]
Dipierro, S., Karakhanyan, A., Valdinoci, E.

New trends in free boundary problems

Advanced Nonlinear Studies 17 (2017), no. 2, 319--332

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New trends in free boundary problems

Advanced Nonlinear Studies 17 (2017), no. 2, 319--332

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[34]
Karakhanyan, A.

Remarks on the thin obstacle problem and constrained Ginibre ensembles

Communications in PDEs In press

arXiv:1702.00466 [math.AP]

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]

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]

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

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A nonlinear free boundary problem with a self-driven Bernoulli condition

Journal of Functional Analysis 273 (2017), no. 11, 3549–3615

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[29]
Karakhanyan, A.

Capillary surfaces arising in singular perturbation problems

Analysis and PDE in press

arXiv:1701.08232 [math.AP]

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

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A class of unstable free boundary problems

Analysis and PDE 10 (2017), no. 6, 1317–1359

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[27]
Karakhanyan, A.

Blaschke's rolling ball theorem and the Trudinger-Wang monotone bending

Journal of Differential Equations 260 (2016), no. 7, 6322–6332

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Blaschke's rolling ball theorem and the Trudinger-Wang monotone bending

Journal of Differential Equations 260 (2016), no. 7, 6322–6332

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[26]
Karakhanyan, A., Shahgholian, H.

Boundary behavior for a singular perturbation problem

Nonlinear Analysis Series A, TMA 138 (2016), 176–188

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Boundary behavior for a singular perturbation problem

Nonlinear Analysis Series A, TMA 138 (2016), 176–188

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[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]

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

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Stratification of free boundary points for a two-phase variational problem

Advances in Mathematics 328 (2018), 40-81

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[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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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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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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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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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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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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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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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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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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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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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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[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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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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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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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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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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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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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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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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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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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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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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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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Nonlinear free boundary problems with singular source terms

Monatsh. Math. 142 (2004), no. 1-2, 7--16

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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.