arXiv subjects
Publications and source records attributed to V. Pchelintsev.
This paper is concerned with spectral estimates for the first Dirichlet eigenvalue of the degenerate $p$-Laplace operator in bounded simply connected domains $Ω\subset \mathbb C$. The proposed approach relies on the conformal analysis of the elliptic operators, which allows us to obtain spectral estimates in domains with non-rectifiable boundaries.
We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains $Ω\subset\mathbb R^2$. With the help of these estimates we obtain asymptotically sharp inequalities of ratios of eigenvalues in the frameworks of the Payne-Pólya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains in terms of conformal (hyperbolic) geometry.
In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate $p$-Laplace operator, $p>2$, in a large class of non-convex domains. This study is based on applications of the geometric theory of composition operators on Sobolev spaces that permits us to estimates constants of Poincaré-Sobolev inequalities and as an application to derive lower estimates of the first non-trivial eigenvalues for the Alhfors domains (i.e. to quasidiscs). This class of domains includes some snowflakes type domains with fractal boundaries.
In this paper we apply estimates of the norms of Sobolev extension operators to the spectral estimates of of the first nontrivial Neumann eigenvalue of the Laplace operator in non-convex extension domains. As a consequence we obtain a connection between resonant frequencies of free membranes and the smallest-circle problem (initially proposed by J.~J.~Sylvester in 1857).
In this paper we study spectral properties of the Neumann-Laplace operator in planar quasiconformal regular domains $Ω\subset\mathbb R^2$. This study is based on the quasiconformal theory of composition operators on Sobolev spaces. Using the composition operators theory we obtain estimates of constants in Poincaré-Sobolev inequalities and as a consequence lower estimates of the first non-trivial eigenvalue of the Neumann-Laplace operator in planar quasiconformal regular domains.