IJPAM: Volume 36, No. 1 (2007)

PARABOLIC HÖLDER NORMS AND
MEASURES IN NON-SMOOTH DOMAINS

Caroline Sweezy
Department of Mathematical Sciences
College of Arts and Sciences
New Mexico State University, 3MB
P.O. Box 30001, Las Cruces, New Mexico, 88003-8801, USA
e-mail: csweezy@nmsu.edu


Abstract.Let $u(x,t)$ be a solution to an initial/boundary value problem for a second order parabolic equation on a bounded domain $% \Omega _{T}$ in $R^{n+1}$. Let $\omega =\omega ^{(X_{0},T_{0})}$ denote the boundary measure of $\Omega _{T}$, generated by the operator $\partial /\partial t-L$. Given some technical restrictions on the parabolic operator and on the domain $\Omega _{T}$, conditions for $\mu $, a Borel measure on $% \Omega _{T}$, and $\nu d\omega $, a measure on the parabolic boundary of $% \Omega _{T}$, are found and shown to be sufficient to control the $% L^{q}(\Omega _{T},\mu )$ norm of a local Hölder norm of $u(x,t)$ by the $% L^{p}(\partial _{p}\Omega _{T},\nu d\omega )$ norm of $u$'s boundary values.

Received: January 29, 2007

AMS Subject Classification: 35K20, 42B25

Key Words and Phrases: second order differential parabolic equations, strictly parabolic divergence form operators, Hölder norms, the Dirichlet problem, bounded domains with rough lateral boundary, measures and weights, semi-discrete Littlewood-Paley functions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 36
Issue: 1