Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1017.35040
Droniou, Jérôme; Porretta, Alessio; Prignet, Alain
Parabolic capacity and soft measures for nonlinear equations. (English)
[J] Potential Anal. 19, No.2, 99-161 (2003). ISSN 0926-2601; ISSN 1572-929X/e

Let $\Omega $ be a bounded, open subset of $\bbfR^N$, $T$ a positive number and $Q=[0,T]\times \Omega $. Let $p$ be a real number, with $1<p<\infty $, and let $1/p+1/p'=1$. This paper deals with the parabolic initial-boundary value problem $u_t+A(u)=\mu $ in $Q$, $u=0$ on $[0,T]\times \partial \Omega$, $u(0)=u_0$ in $\Omega $, where $A$ is a nonlinear monotone and coercive operator in divergence form which acts from the space $L^p(0,T;W_0^{1,p}(\Omega))$ into its dual $L^{p'}(0,T; W^{-1,p'}(\Omega))$. The authors first introduced the notion of capacity related to the parabolic $p$-Laplace operator. Then they proved a decomposition theorem for measures that do not charge the sets with null capacity. Using these results they proved the following Theorem: Let $\mu $ be a bounded real measure on $Q$ which does not charge the subsets of $Q$ of null capacity, and let $u_0 \in L^1(\Omega)$. Then there is a unique renormalized solution $u$ of the problem. Moreover, $u$ satisfies the additional regularity: $u\in L^\infty (0,T; L^1(\Omega))$ and $\max (-k,\min (k,u)) \in L^p(0,T;W_0^{1,p}(\Omega))$ for every $k>0$.
[Dagmar Medková (Praha)]

MSC 2000:: *35K55 Nonlinear parabolic equations

Keywords: parabolic equation; capacity

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]