Vector-valued Morrey's embedding theorem and Hölder continuity in parabolic problems

Rabier, Patrick J.

Electronic Journal of Differential Equations, Tome 2011 (2011).

Résumé

Summary: If $I\subset \mathbb{R}$ is an open interval and $\Omega \subset \mathbb{R}^N$ an open subset with $\partial \Omega $ Lipschitz continuous, we show that the space $W^{1,p}(I,L^q (\Omega ))\cap L^p(I,W^{1,q}(\Omega ))$ is continuously embedded in $C^{0,\frac{1}{p'}-\frac{N}{q}}(\overline{\Omega \times I})\cap L^{\infty }(\Omega \times I)$ if $p,q\in (1,\infty )$ and $q>Np'$. When $p=q$, this coincides with Morrey's embedding theorem for $W^{1,p}(\Omega \times I)$. While weaker results have been obtained by various methods, including very technical ones, the proof given here follows that of Morrey's theorem in the scalar case and relies only on widely known properties of the classical Sobolev spaces and of the Bochner integral. This embedding is useful to formulate nonlinear evolution problems as functional equations, but it has other applications. As an example, we derive apparently new space-time Holder continuity properties for $u_t=Au+f,u(\cdot ,0)=u_0$ when $A$ generates a holomorphic semigroup on $L^q (\Omega)$.

Détail

Classification : 46E35, 46E40, 35K90, 35K55
Keywords: Morrey's theorem, embedding, vector-valued Sobolev space, mixed norm