Electron. J. Diff. Equ.,
Vol. 2011 (2011), No. 10, pp. 110.
Vectorvalued Morrey's embedding theorem and Holder continuity
in parabolic problems
Patrick J. Rabier
Abstract:
If
is an open interval and
an open subset with
Lipschitz continuous, we show that the space
is continuously
embedded in
if
and
.
When
,
this coincides with Morrey's
embedding theorem for
. 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 spacetime Holder continuity
properties for
when
generates a
holomorphic semigroup on
.
Submitted January 11, 2011. Published January 19, 2011.
Math Subject Classifications: 46E35, 46E40, 35K90, 35K55.
Key Words: Morrey's theorem; embedding; vectorvalued
Sobolev space; mixed norm.
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Patrick J. Rabier
Department of Mathematics,
University of Pittsburgh
Pittsburgh, PA 15260, USA
email: rabier@imap.pitt.edu 
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