An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

Title:

An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

Creator:

Chernyavskaya, Nina A. and Shuster, Leonid A.

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:f0a1eec6-7ccd-47cf-8356-e9433086e7c1
uuid:f0a1eec6-7ccd-47cf-8356-e9433086e7c1

Subject:

Sobolev space, embedding theorem, and Sturm-Liouville equation

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

We consider the weighted space $W_1^{(2)}(\mathbb R,q)$ of Sobolev type $$ W_1^{(2)}(\mathbb R,q)=\left \{y\in A_{\rm loc}^{(1)}(\mathbb R)\colon \|y''\|_{L_1(\mathbb R)}+\|qy\|_{L_1(\mathbb R)}<\infty \right \} $$ and the equation $$ - y''(x)+q(x)y(x)=f(x),\quad x\in \mathbb R. \leqno (1) $$ Here $f\in L_1(\mathbb R)$ and $0\le q\in L_1^{\rm loc}(\mathbb R).$ \endgraf We prove the following: \item {1)} The problems of embedding $W_1^{(2)}(\mathbb R,q)\hookrightarrow L_1(\mathbb R)$ and of correct solvability of (1) in $L_1(\mathbb R) $ are equivalent; \item {2)} an embedding $W_1^{(2)}(\mathbb R,q)\hookrightarrow L_1(\mathbb R) $ exists if and only if $$\exists a>0\colon \inf _{x\in \mathbb R}\int _{x-a}^{x+a} q(t) {\rm d} t>0.$$.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

709-716

Source:

Czechoslovak Mathematical Journal | 2012 Volume:62 | Number:3

Harvested from:

CDK

Metadata only:

false