L^{p}-Regularity Results for Parabolic Equations with Robin Type Boundary Conditions in Non-rectangular Domains

Abstract

This paper is devoted to the analysis of the following linear parabolic equation $\partial _{t}u-\partial _{x}^{2}u=f,$ subject to
Robin type conditions $\partial _{x}u+\beta u=0$, on the lateral boundary, where coefficient $\beta $ satisfies suitable
nondegeneracy assumptions and possibly depends on the time variable.
The right-hand side $f$ of the
equation is taken in $L^{p}, 1 <p<\infty.$ The problem is set in a domain of the form $
\Omega =\left\{ \left( t,x\right) \in\mathbb{R}^{2}:0<t<1,0<x<t^{\alpha }\right\} $, $\alpha >1/2$. We use
Labbas-Terreni results [23] on the operator's sum method in
the non-commutative case. This work is an extension of the Hilbertian case studied in [15].