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**Mittag-Leffler Institute, 2015**

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

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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
$\mathbf u\in W^{1, 2}(\Omega,\mathbb R^n)$ 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.

\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].

This problem is studied in the papers [17] and [23].

PAPERS

*
*

[51]
Huang, Y., Karakhanyan, A.

The well-posedness of cylindrical jets with surface tension

arxiv:2311.13748 [math.AP]

The well-posedness of cylindrical jets with surface tension

arxiv:2311.13748 [math.AP]

[50]
Huang, Y., Karakhanyan, A.

On explicit geometric solutions of some inviscid flows with free boundary

arxiv:2311.13756 [math.AP]

On explicit geometric solutions of some inviscid flows with free boundary

arxiv:2311.13756 [math.AP]

[48]
Dipierro, S., Karakhanyan, A., Valdinoci, E.

Classification of global solutions of a free boundary problem in the plane

Interfaces and Free Boundaries vol. 25, no. 3, 455–490

arxiv:2203.11663 [math.AP]

Classification of global solutions of a free boundary problem in the plane

Interfaces and Free Boundaries vol. 25, no. 3, 455–490

arxiv:2203.11663 [math.AP]

[47] Espin, T., Karakhanyan, A.

Boundary estimates for solutions of the Monge-Ampere equation satisfying Dirichlet-Neumann type

conditions in annular domains

Nonlinear Analysis 238C (2024) 113399

arxiv:2006.09814 [math.AP]

Boundary estimates for solutions of the Monge-Ampere equation satisfying Dirichlet-Neumann type

conditions in annular domains

Nonlinear Analysis 238C (2024) 113399

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

Calculus of Variations and PDEs In press

arXiv:1904.06270 [math.AP]

A nonlocal free boundary problem with Wasserstein distance

Calculus of Variations and PDEs In press

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

Interfaces and Free Boundaries In press

arXiv:1906.05383 [math.AP]

Structure of singularities in the nonlinear nerve conduction problem

Interfaces and Free Boundaries In press

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

Full text pdf

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]

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

Full text pdf

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]

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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- This course is a rigorous introduction to the wave, heat, and Laplace equations
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