arXiv subjects

Aram Karakhanyan

Publications and source records attributed to Aram Karakhanyan.

30 records · Page 2Linked to original sources

arXiv2019

A nonlocal free boundary problem with Wasserstein distance

We study the probability measures $ρ\in \mathcal M(\mathbb R^2)$ minimizing the functional \[ J[ρ]=\iint \log\frac1{|x-y|}dρ(x)dρ(y)+d^2(ρ, ρ_0), \] where $ρ_0$ is a given probability measure and $d(ρ, ρ_0)$ is the 2-Wasserstein distance of $ρ$ and $ρ_0$. % We prove the existence of minimizers $ρ$ and show that the potential $U^ρ=-\log|x|\ast ρ$ solves a degenerate obstacle problem, the obstacle being the transport potential. Every minimizer $ρ$ is absolutely continuous with respect to the Lebesgue measure. The singular set of the free boundary of the obstacle problem is contained in a rectifiable set, and its Hausdorff dimension is $< n-1$. Moreover, $U^ρ$ solves a nonlocal Monge-Ampére equation, which after linearization leads to the equation $ρ_t={\hbox{div}}(ρ\nabla U^ρ)$. The methods we develop use Fourier transform techniques. They work equally well in high dimensions $n\ge2$ for the energy \[ J[ρ]=\iint |x-y|^{2-n}dρ(x)dρ(y)+d^2(ρ, ρ_0). \]

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arXiv2018

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

In this paper we classify the nonnegative global minimizers of the functional \[ J_F(u)=\int_ΩF(|\nabla u|^2)+λ^2χ_{\{u>0\}}, \] where $F$ satisfies some structural conditions and $χ_D$ is the characteristic function of a set $D\subset \mathbb R^n$. We compute the second variation of the energy and study the properties of the stability operator. The free boundary $\partial\{u>0\}$ can be seen as a rectifiable $n-1$ varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded and use Allard's monotonicity formula to prove the existence of tangent cones modulo a set of small Hausdorff dimension. In particular we prove that if $n=3$ and the ellipticity constants of the quasilinear elliptic operator generated by $F$ are close to 1 then the conical free boundary must be flat.

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arXiv2018

Refractor surfaces determined by near-field data

In this paper we study the near-field refractor problem with point source at the origin and prescribed target on the given receiver surface $Σ$. This nonvariational problem can be studied in the framework of prescribed Jacobian equations. We construct the corresponding generating function and show that the Aleksandrov and the Brenier type solutions are equivalent. Our main result establishes local smoothness of Aleksandrov's solutions when the data is smooth and when the medium containing the source has smaller refractive index than the medium containing the target. This is done by deriving the Monge-Ampere type equation that smooth solutions satisfy and establishing the validity of the MTW condition for a large class of receiver surfaces, which in turn implies the local $C^2 $ regularity of the refactor.

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arXiv2017

Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients

We provide an integral estimate for a non-divergence (non-variational) form second order elliptic equation $a_{ij}u_{ij}=u^p$, $u\ge 0$, $p\in[0, 1)$, with bounded discontinuous coefficients $a_{ij}$ having small BMO norm. We consider the simplest discontinuity of the form~$x\otimes x|x|^{-2}$ at the origin. As an application we show that the free boundary corresponding to the obstacle problem (i.e. when~$p=0$) cannot be smooth at the points of discontinuity of~$a_{ij}(x)$. To implement our construction, an integral estimate and a scale invariance will provide the homogeneity of the blow-up sequences, which then can be classified using ODE arguments.

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arXiv2017

A class of unstable free boundary problems

We consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter. The main difference with the existing literature is that the total energy is here a nonlinear superposition of the either local or nonlocal surface tension effect with the elastic energy. In sharp contrast with the linear case, the problem considered in this paper is unstable, namely a minimizer in a given domain is not necessarily a minimizer in a smaller domain. We provide an explicit example for this instability. We also give a free boundary condition, which emphasizes the role played by the domain in the geometry of the free boundary. In addition, we provide density estimates for the free boundary and regularity results for the minimal solution. As far as we know, this is the first case in which a nonlinear function of the perimeter is studied in this type of problems. Also, the results obtained in this nonlinear setting are new even in the case of the local perimeter, and indeed the instability feature is not a consequence of the possibly nonlocality of the problem, but it is due to the nonlinear character of the energy functional.

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arXiv2017

On a conjecture of De Giorgi related to homogenization

For a periodic vector field $\bf F$, let ${\bf X}^ε$ solve the dynamical system \begin{equation*} \frac{d{\bf X}^ε}{dt} = {\bf F}\left(\frac {{\bf X}^ε}ε\right) . \end{equation*} In \cite{DeGiorgi} Ennio De Giorgi enquiers whether from the existence of the limit ${\bf X}^0(t):=\lim\limits_{ε\to 0}{\bf X}^ε(t)$ one can conclude that $ \frac{d{\bf X}^0}{dt}= constant$. Our main result settles this conjecture under fairly general assumptions on $\bf F$, which may also depend on $t$-variable. Once the above problem is solved, one can apply the result to the transport equation, in a standard way. This is also touched upon in the text to follow.

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arXiv2017

Regularity for a quasilinear continuous casting problem

In this paper study the regularity of continuous casting problem \begin{equation} \hbox{div}(|\nabla u|^{p-2}\nabla u-{\bf v} β(u))=0\tag{$\sharp$} \end{equation} for prescribed constant velocity $\bf v$ and enthalpy $β(u)$ with jump discontinuity at $u=0$. We establish the following estimates: local log-Lipschitz $p>2$ for $u$ (and BMO for $\nabla u$) for two phase, Lipschitz $p>1$ for one phase and linear growth up-to boundary near the contact points. We also prove that the free boundary is continuous curve in the direction of $\bf v$ in two spatial dimensions. The proof is based on a delicate argument exploiting Sard's theorem for $W^{2, 2+η}, η>0$ functions and circumventing the lack of comparison principle for the solutions of ($\sharp$).

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arXiv2017

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

We study a Bernoulli type free boundary problem with two phases $$J[u]=\int_Ω|\nabla u(x)|^2\,dx+Φ\big({\mathcal M}_-(u), {\mathcal{M}}_+(u)\big), \quad u-\bar u\in W^{1,2}_0(Ω), $$ where $\bar u\in W^{1,2}(Ω)$ is a given boundary datum. Here, ${\mathcal M}_1$ and ${\mathcal M}_2$ are weighted volumes of $\{u\le 0\}\cap Ω$ and $\{u>0\}\cap Ω$, respectively, and $Φ$ is a nonnegative function of two real variables. We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases. In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem). Another property of this type of problems is that the minimizer in $Ω$ is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem. Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits of $u$ are minimizers of the Alt-Caffarelli-Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit. As a special case, we can deal with the energy levels generated by the volume term $Φ(0, r_2)=r_2^{\frac{n-1}n}$, which interpolates the Athanasopoulos-Caffarelli-Kenig-Salsa energy, thanks to the isoperimetric inequality. In particular, we develop a detailed optimal regularity theory for the minimizers and for their free boundaries.

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arXiv2017

New trends in free boundary problems

We present a series of recent results on some new classes of free boundary problems. Differently from the classical literature, the problems considered have either a "nonlocal" feature (e.g., the interaction or/and the interfacial energy may depend on global quantities) or a "nonlinear" flavor (namely, the total energy is the nonlinear superposition of energy components, thus losing the standard additivity and scale invariances of the problem).

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arXiv2015

An inverse problem for the refractive surfaces with parallel lighting

In this article we examine the regularity of two types of weak solutions to a Monge-Ampère type equation which emerges in a problem of finding surfaces that refract coaxial light rays emitted from source domain and striking a given target after refraction. Historically, ellipsoids and hyperboloids of revolution were the first surfaces to be considered in this context. The mathematical formulation commences with deriving the energy conservation equation for sufficiently smooth surfaces, regarded as graphs of functions to be sought, and then studying the existence and regularity of two classes of suitable weak solutions constructed from envelopes of hyperboloids or ellipsoids of revolution. Our main result in this article states that under suitable conditions on source and target domains and respective intensities these weak solutions are locally smooth.

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arXiv2015

A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two

We continue the analysis of the two-phase free boundary problems initiated in \cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free boundary. There, we also defined the functional $$ϕ_p(r,u,x_0)=\frac1{r^4}\int_{B_r(x_0)}\frac{|\nabla u^+(x)|^p}{|x-x_0|^{N-2}}dx\int_{B_r(x_0)}\frac{|\nabla u^-(x)|^p}{|x-x_0|^{N-2}}dx$$ where $x_0$ is a free boundary point, i.e. $x_0\in\partial\{u>0\}$ and $u$ is a minimizer of the functional $$J(u):=\int_Ω|\nabla u|^p +λ_+^p\,χ_{\{u>0\}} +λ_-^p\,χ_{\{u\le 0\}}, $$ for some bounded smooth domain $Ω\subset {\mathbb R}^N$ and positive constants $λ_\pm$ with $Λ:=λ_+^p-λ^p_->0$. Here we show the discrete monotonicity of $ϕ_p(r,u,x_0)$ in two spatial dimensions at non-flat points, when $p$ is sufficiently close to 2, and then establish the linear growth. A new feature of our approach is the anisotropic scaling argument discussed in Section 4.

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arXiv2013

Analysis of a free boundary at contact points with Lipschitz data

In this paper we consider a minimization problem for the functional $$ J(u)=\int_{B_1^+}|\nabla u|\sp 2+λ_{+}^2χ_{\{u>0\}}+λ_{-}^2χ_{\{u\leq0\}}, $$ in the upper half ball $B_1^+\subset\R^n, n\geq 2$ subject to a Lipschitz continuous Dirichlet data on $\partial B_1^+$. More precisely we assume that $0\in \partial \{u>0\}$ and the derivative of the boundary data has a jump discontinuity. If $0\in \bar{\partial(\{u>0\} \cap B_1^+)}$ then (for $n=2$ or $n>3$ and one-phase case) we prove, among other things, that the free boundary $\partial \{u>0\}$ approaches the origin along one of the two possible planes given by $$ γx_1 = \pm x_2, $$ where $γ$ is an explicit constant given by the boundary data and $λ_\pm$ the constants seen in the definition of $J(u)$. Moreover the speed of the approach to $γx_1=x_2$ is uniform.

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