Arshak Petrosyan

Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set.Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren’s frequency formula, first established by Caffarelli, Salsa, and Silvestre.

Optimal Regularity and the Free Boundaryin the Parabolic Signorini Problem

We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgrens monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.

Density Estimates for a Degenerate/Singular Phase-Transition Model

We consider a Ginzburg--Landau type phase-transition model driven by a p-Laplacian type equation. We prove density estimates for absolute minimizers and we deduce the uniform convergence of level sets and the existence of plane-like minimizers in periodic media.

Geometric properties of Bernoulli-type minimizers

We consider a Bernoulli-type variational problem and we prove some geometric properties for minimizers, such as: gradient bounds, linear growth from the free boundary, density estimates, uniform convergence of level sets and the existence of plane-like minimizers in periodic media. 2000 Mathematics Subject Classification: Primary 49J10, 35R35; Secondary 35J70.

ON EXISTENCE AND UNIQUENESS IN A FREE BOUNDARY PROBLEM FROM COMBUSTION

ABSTRACT We study a free boundary problem for the heat equation describing the propagation of laminar flames under certain geometric assumptions on the initial data. The problem arises as the limit of a singular perturbation problem, and generally no uniqueness of limit solutions can be expected. However, if the initial data is starshaped, we show that the limit solution is unique and coincides with the minimal classical supersolution. Under certain convexity assumption on the data, we prove first that the limit solution is a classical solution of the free boundary problem for a short time interval, and then that the solution, in fact, stays classical as long as it does not vanish identically.

Convexity and uniqueness in a free boundary problem arising in combustion theory.

We consider solutions to a free boundary problem for the heat equation, describing the propagation of flames. Suppose there is a bounded domain O I QT = Rn x (0,T) for some T > 0 and a function u > 0 in O such that ut = ?u, in O, u = 0 and |Nu| = 1, on G := ?O n QT, u(·,0) = u0, on O0, where O0 is a given domain in Rn and u0 is a positive and continuous function in O0, vanishing on ?O0. If O0 is convex and u0 is concave in O0, then we show that (u,O) is unique and the time sections Ot are convex for every t I (0,T), provided the free boundary G is locally the graph of a Lipschitz function and the fixed gradient condition is understood in the classical sense.

A two-phase problem with a lower-dimensional free boundary

For a bounded domain D ⊂ Rn, we study minimizers of the energy functional ∫ D |∇u| dx+ ∫ D∩(Rn−1×{0}) λχ{u>0} + λ χ{u 0} and Γ− = ∂{u(·, 0) < 0} never touch. Moreover, using Alexandrov-type reflection technique, we can show that in dimension n = 3 the free boundaries are C1 regular on a dense subset.

On the full regularity of the free boundary in a class of variational problems

We consider nonnegative minimizers of the functional Jp(u;Ω)=∫ Ω |∇u| p +λ p p Χ {u>0 }, 1 0} ∩Ω has no singularities and is a real analytic hypersurface if p = 2 and n 0 such that the same result holds if p - 2| < ∈ 0 .

Optimal Regularity in Rooftop-Like Obstacle Problem

We study the solutions of the obstacle problem when the obstacle is smooth on each side of a certain hyperplane but only Lipschitz across. We prove that the optimal regularity of these solutions is on each side of the hyperplane. The proof uses a modification of Almgrens frequency formula.

Some extremal problems for analytic functions

The paper mainly concerns with functions f, analytic in S|Imz|<1 and bounded by a given constant. We state sharp estimates for supR|f′| under the additional condition SupR|f|≤1. Using these estimates we deduce well-known Bernsteins inequality and some of its generalizations for entire functions of a finite type with respect to an arbitrary proximate order. Parallelly we investigate also the next extremal problem, related to the mentioned class of functions: if for some ζ∊S, what is the minimal value for sup R|f|? Also we present the description of extremal functions for these problems.