Correct solvability, embedding theorems and separability for the Sturm-Liouville equation

N. A. Chernyavskaya; L. A. Shuster

doi:10.1007/s40574-015-0024-2

Correct solvability, embedding theorems and separability for the Sturm-Liouville equation

N. A. Chernyavskaya
, L. A. Shuster

Department of Mathematics

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

Abstract

For p [1,∞), f L_p(ℝ) and 0 ≤ q L₁^Loc(ℝ), we show that the weighted function space S⁽²⁾_p (R, q) ={ y AC⁽¹⁾_loc (ℝ) : ∥y" qy∥p + ∥q^1/p y∥p < ∞} is embedded into L p(R) if and only if the equation -y"(x) + q(x)y(x) = f (x), x ℝ, is correctly solvable in L_p(ℝ).

Original language	English
Pages (from-to)	45-52
Number of pages	8
Journal	Bollettino dell'Unione Matematica Italiana
Volume	8
Issue number	1
DOIs	https://doi.org/10.1007/s40574-015-0024-2
State	Published - 8 Aug 2015

Bibliographical note

Publisher Copyright:
© Unione Matematica Italiana 2015.

Keywords

Embedding theorem
Sobolev space
Sturm-Liouville equation

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10.1007/s40574-015-0024-2

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abstract = "For p [1,∞), f Lp(ℝ) and 0 ≤ q L1Loc(ℝ), we show that the weighted function space S(2)p (R, q) =\{ y AC(1)loc (ℝ) : ∥y{"} qy∥p + ∥q1/p y∥p < ∞\} is embedded into L p(R) if and only if the equation -y{"}(x) + q(x)y(x) = f (x), x ℝ, is correctly solvable in Lp(ℝ).",

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AU - Shuster, L. A.

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N2 - For p [1,∞), f Lp(ℝ) and 0 ≤ q L1Loc(ℝ), we show that the weighted function space S(2)p (R, q) ={ y AC(1)loc (ℝ) : ∥y" qy∥p + ∥q1/p y∥p < ∞} is embedded into L p(R) if and only if the equation -y"(x) + q(x)y(x) = f (x), x ℝ, is correctly solvable in Lp(ℝ).

AB - For p [1,∞), f Lp(ℝ) and 0 ≤ q L1Loc(ℝ), we show that the weighted function space S(2)p (R, q) ={ y AC(1)loc (ℝ) : ∥y" qy∥p + ∥q1/p y∥p < ∞} is embedded into L p(R) if and only if the equation -y"(x) + q(x)y(x) = f (x), x ℝ, is correctly solvable in Lp(ℝ).

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KW - Sobolev space

KW - Sturm-Liouville equation

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Correct solvability, embedding theorems and separability for the Sturm-Liouville equation

Abstract

Bibliographical note

Keywords

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