We study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and compute the first few of them explicitly in terms of the scalar and mean curvatures. This is done by applying the Seeley calculus to the Dirichlet-to-Neumann operator, whose spectrum coincides with ...

We study some direct results for the recently introduced family of modified Baskakov type operators. In particular, we obtain local direct results on ordinary and simultaneous approximation and an estimation of error for linear combinations in terms of higher order modulus of continuity. We have applied the Steklov mean as a tool for the linear approximating method. 2000 Mathematics Subject Cla...

We study elliptic problems at critical growth under Steklov boundary conditions in bounded domains. For a second order problem we prove existence of nontrivial nodal solutions. These are obtained by combining a suitable linking argument with fine estimates on the concentration of Sobolev minimizers on the boundary. When the domain is the unit ball, we obtain a multiplicity result by taking adva...

This is the first in a series of two papers aiming to establish sharp spectral asymptotics for Steklov type problems on planar domains with corners. In the present paper we focus on the two-dimensional sloshing problem, which is a mixed Steklov-Neumann boundary value problem describing small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section. We pr...

We consider the Steklov problem for the linear biharmonic equation. We survey existing results for the positivity preserving property to hold. These are connected with the first Steklov eigenvalue. We address the problem of minimizing this eigenvalue among suitable classes of domains. We prove the existence of an optimal convex domain of fixed measure. Mathematics Subject Classification (2000)....

We prove sharp upper and lower bounds for the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary. The argument involves frequency function methods for harmonic functions in the interior of the surface as well as the construction of exponentially accurate approximations for the Steklov eigenfunctions near the boundary.

New criteria of Lp − Lq boundedness of Hardy-Steklov type operator (1.1) with both increasing on (0, ∞) boundary functions a(x) and b(x) are obtained for 1 < p ≤ q < ∞ and 0 < q < p < ∞, p > 1. This result is applied for two-weighted Lp − Lq characterization of the corresponding geometric Steklov operator (1.3) and other related problems.

In this paper we consider the multiscale computation of a Steklov eigenvalue problem with rapidly oscillating coefficients. The new contribution obtained in this paper is a superapproximation estimate for solving the homogenized Steklov eigenvalue problem and to present a multiscale numerical method. Numerical simulations are then carried out to validate the theoretical results reported in the ...

† Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, Indianapolis, IN 46202-3216, USA z C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11794-3840, USA • St. Petersburg Department of Steklov Mathematical Institute, RAS 27, Fontanka, 191023, St. Petersburg, Russia ‡ Steklov Mathematical Institute, Gubk...