| Title:
|
The regularity of the positive part of functions in $L^2(I; H^1(\Omega)) \cap H^1(I; H^1(\Omega)^*)$ with applications to parabolic equations (English) |
| Author:
|
Wachsmuth, Daniel |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
57 |
| Issue:
|
3 |
| Year:
|
2016 |
| Pages:
|
327-332 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $u\in L^2(I; H^1(\Omega))$ with $\partial_t u\in L^2(I; H^1(\Omega)^*)$ be given. Then we show by means of a counter-example that the positive part $u^+$ of $u$ has less regularity, in particular it holds $\partial_t u^+ \notin L^1(I; H^1(\Omega)^*)$ in general. Nevertheless, $u^+$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations. (English) |
| Keyword:
|
Bochner integrable function |
| Keyword:
|
projection onto non-negative functions |
| Keyword:
|
parabolic equation |
| MSC:
|
35K10 |
| MSC:
|
46E35 |
| idZBL:
|
Zbl 06674883 |
| idMR:
|
MR3554513 |
| DOI:
|
10.14712/1213-7243.2015.168 |
| . |
| Date available:
|
2016-09-22T15:24:15Z |
| Last updated:
|
2018-10-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145837 |
| . |
| Reference:
|
[1] Gajewski H., Gröger K., Zacharias K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen.Akademie-Verlag, Berlin, 1974. MR 0636412 |
| Reference:
|
[2] Grün G.: Degenerate parabolic differential equations of fourth order and a plasticity model with non-local hardening.Z. Anal. Anwendungen 14 (1995), no. 3, 541–574. MR 1362530, 10.4171/ZAA/639 |
| Reference:
|
[3] Roubíček T.: Nonlinear Partial Differential Equations with Applications.International Series of Numerical Mathematics, 153, Birkhäuser, Basel, 2013. Zbl 1270.35005, MR 3014456 |
| Reference:
|
[4] J. Wloka J.: Partielle Differentialgleichungen.Teubner, Stuttgart, 1982. Zbl 0482.35001, MR 0652934 |
| . |
