Hayk Aleksanyan

A well-known question in classical differential geometry and geometric analysis asks for a description of possible boundaries of K-surfaces, which are smooth, compact hypersurfaces in R^d having constant Gauss curvature equal to K ≥ 0. This question generated a considerable amount of remarkable results in the last few decades. Motivated by these de...

We introduce a new lattice growth model, which we call boundary sandpile. The model amounts to potential-theoretic redistribution of a given initial mass on Z^d (d≥2) onto the boundary of an (a priori) unknown domain. The latter evolves through sandpile dynamics, and has the property that the mass on the boundary is forced to stay below a prescribe...

We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence-type elliptic operators. The construction is applied in two settings. First, we show that solutions to boundary layer problems for divergence-type elliptic equations set in halfspaces and with infi...

In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of $L^2 \mapsto L^2$ type for the operator, as well as for the corresponding maximal function. If the hypersurface is flat, we consider a particular class of a nonlinear phase functions, and apply our analysis...

The paper investigates uniform and almost everywhere convergence of the greedy algorithm by the Haar system. Necessary and sufficient conditions for renorming the functions of Haar system are obtained, which guarantee uniform convergence for functions from C[0, 1] and almost everywhere convergence for functions from L_1[0, 1].

The main purpose of the present paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). For this aim, we introduce a new model of Laplacian growth on the lattice \mathbb{Z}^d (d\geq 2) which continuously deforms occupied regions of the divisible sandpile model of Levine...

In this note we study periodic homogenization of Dirichlet problem for divergence type elliptic systems when both the coefficients and the boundary data are oscillating. One of the key difficulties here is the determination of the fixed boundary data corresponding to the limiting (homogenized) problem. This issue has been addressed in recent papers...

Let $u_\e$ be a solution to the system $$ \mathrm{div}(A_\e(x) \nabla u_{\e}(x))=0 \text{ \ in } D, \qquad u_{\e}(x)=g(x,x/\e) \text{ \ on }\partial D, $$ where $D \subset \R^d $ ($d \geq 2$), is a smooth uniformly convex domain, and $g$ is $1$-periodic in its second variable, and both $A_\e$ and $g$ are sufficiently smooth. Our results in this pa...

In this paper we prove convergence results for the homogenization of the Dirichlet problem with rapidly oscillating boundary data in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as $L^p$...

In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex and smooth domain, and smooth operator and boundary data, we prove pointwise convergence results, namely|uε(x)−u...

We study the convergence of greedy algorithmwith regard to renormalized trigonometric system. Necessary and sufficient conditions are found for system’s normalization to guarantee almost everywhere convergence, and convergence in L p (T) for 1 < p < ∞ of the greedy algorithm, where T is the unit torus. Also the non existence is proved for normaliza...

We characterize the all weighted greedy algorithms with respect to Franklin system which converge uniformly for continuous functions and almost everywhere for integrable functions. In case, when the algorithm fails to satisfy our classification criteria, we construct a continuous function for which the corresponding approximation diverges unbounded...