Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 10, pp. 1-10.
Title: Vector-valued Morrey's embedding theorem and Holder continuity
in parabolic problems
Author: Patrick J. Rabier (Univ. of Pittsburgh, PA, USA)
Abstract:
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 $L^q (\Omega)$.
Submitted January 11, 2011. Published January 19, 2011.
Math Subject Classifications: 46E35, 46E40, 35K90, 35K55.
Key Words: Morrey's theorem; embedding; vector-valued
Sobolev space; mixed norm.