Widths of weighted Sobolev classes with restrictions $f(a)=\dots=f^{(k-1)}(a)=f^{(k)}(b)=\dots=f^{(r-1)}(b)=0$ and spectra of non-linear differential equations

TL;DR

This paper establishes the equivalence between Kolmogorov widths of certain weighted Sobolev classes with boundary restrictions and the spectral numbers of specific non-linear differential equations, under compact embedding conditions.

Contribution

It proves the coincidence of widths and spectral numbers for weighted Sobolev classes with boundary conditions, extending understanding of their spectral properties.

Findings

Kolmogorov widths of weighted Sobolev classes are characterized.

Spectral numbers of non-linear differential equations are identified.

The results hold under compact embedding assumptions.

Abstract

In this paper we prove that Kolmogorov widths of weighted Sobolev classes with restrictions $f(a)=\dots=f^{(k-1)}(a)=f^{(k)}(b)=\dots=f^{(r-1)}(b)=0$ in a weighted Lebesgue space and spectral numbers of some non-linear differential equation coincide. Here we suppose that the embedding is compact.