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)....

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...

In the paper, a two-grid discretization scheme is discussed for the Steklov eigenvalue problem. With the scheme, the solution of the Steklov eigenvalue problem on a fine grid is reduced to the solution of the Steklov eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. Using spectral approximation theory, it is shown theoretically that the tw...

The conjecture by Steklov was solved negatively by Rakhmanov in 1979. His original proof was based on the formula for orthogonal polynomial obtained by adding point masses to the measure of orthogonality. In this note, we show how this polynomial can be obtained by applying the method developed recently for proving the sharp lower bounds for the problem by Steklov.

The positivity-preserving property for the inverse of the biharmonic operator under Steklov boundary conditions is studied. It is shown that this property is quite sensitive to the parameter involved in the boundary condition. Moreover, positivity of the Steklov boundary value problem is linked with positivity under boundary conditions of Navier and Dirichlet type.

This paper explains how to obtain the existence and uniqueness of weak solutions for weighted p-Laplacian boundary problem Steklov Neumann, relying on Browder’s theorem under conditions monotonous function f. The also discusses behavior problems through numerical results. Additionally, illustrate effectiveness our approach, examples both linear nonlinear equations are presented.