| Title:
|
An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation (English) |
| Author:
|
Chernyavskaya, Nina A. |
| Author:
|
Shuster, Leonid A. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
3 |
| Year:
|
2012 |
| Pages:
|
709-716 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $$\exists a>0\colon \inf _{x\in \mathbb R}\int _{x-a}^{x+a} q(t) {\rm d} t>0.$$ (English) |
| Keyword:
|
Sobolev space |
| Keyword:
|
embedding theorem |
| Keyword:
|
Sturm-Liouville equation |
| MSC:
|
34B24 |
| MSC:
|
34B40 |
| MSC:
|
46E35 |
| idZBL:
|
Zbl 1265.34106 |
| idMR:
|
MR2984630 |
| DOI:
|
10.1007/s10587-012-0041-6 |
| . |
| Date available:
|
2012-11-10T21:11:20Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143021 |
| . |
| Reference:
|
[1] Chernyavskaya, N., Shuster, L.: Estimates for the Green function of a general Sturm-Liouville operator and their applications.Proc. Am. Math. Soc. 127 (1999), 1413-1426. Zbl 0918.34032, MR 1625725, 10.1090/S0002-9939-99-05049-2 |
| Reference:
|
[2] Chernyavskaya, N., Shuster, L.: A criterion for correct solvability of the Sturm-Liouville equation in the space $L_p(R)$.Proc. Am. Math. Soc. 130 (2002), 1043-1054. Zbl 0994.34014, MR 1873778, 10.1090/S0002-9939-01-06145-7 |
| Reference:
|
[3] Chernyavskaya, N., Shuster, L.: A criterion for correct solvability in $L_p(\Bbb R)$ of a general Sturm-Liouville equation.J. Lond. Math. Soc., II. Ser. 80 (2009), 99-120. Zbl 1188.34036, MR 2520380, 10.1112/jlms/jdp012 |
| Reference:
|
[4] Grinshpun, E., Otelbaev, M.: On smoothness of solutions of nonlinear Sturm-Liouville equation in $L_1(-\infty,\infty)$.Izv. Akad. Nauk Kaz. SSR, Ser. Fiz.-Mat. 5 (1984), 26-29 Russian. MR 0774312 |
| Reference:
|
[5] Mynbaev, K., Otelbaev, M. O.: Weighted Functional Spaces and the Spectrum of Differential Operators.Moskva: Nauka 286 (1988), Russian. English summary. Zbl 0651.46037, MR 0950172 |
| Reference:
|
[6] Ojnarov, R.: Separability of the Schrödinger operator in the space of summable functions.Dokl. Akad. Nauk SSSR 285 (1985), 1062-1064. MR 0820597 |
| Reference:
|
[7] Ojnarov, R.: Some properties of the Sturm-Liouville operator in $L_p$.Izv. Akad. Nauk Kaz. SSR, Ser. Fiz.-Mat. 152 (1990), 43-47. MR 1089974 |
| Reference:
|
[8] Otelbaev, M. O.: On coercive estimates of solutions of difference equations.Tr. Mat. Inst. Steklova 181 (1988), Russian 241-249. Zbl 0661.39003, MR 0945435 |
| Reference:
|
[9] Otelbaev, M.: On smoothness of a solution of a nonlinear parabolic equation.In 10th Czechoslovak-Soviet Meeting ``Application of Fundamental Methods and Methods of Theory of Functions to Problems of Mathematical Physics'', Stara Gura, 26.09.--01.10. 1988 37. |
| . |
