Let E be an open set in RN, and for T>0 let ET denote the cylindrical domain E×[0,T]. We consider quasi-linear, parabolic partial differential equations of the form
ut−divA(x,t,u,Du)=0 weakly in ET,
where the function A(x,t,u,ξ):ET×RN+1→RN is assumed to be measurable with respect to (x,t)∈ET for all (u,ξ)∈R×RN, and continuous with respect to (u,ξ) for a.e.~(x,t)∈ET. Moreover, we assume the structure conditions
{A(x,t,u,ξ)⋅ξ≥C0|ξ|p,A(x,t,u,ξ)|≤C1|ξ|p−1,
for a.e. (x,t)∈ET, ∀u∈R,∀ξ∈RN, where C0 and C1 are given positive constants, and we take p>1. We consider a boundary datum g with
{g∈Lp(0,T;W1,p(E)),g continuous on ¯ET with modulus of continuity ωg(⋅),
and we are interested in the boundary behavior of solutions to the Cauchy-Dirichlet problem
{ut−divA(x,t,u,Du)=0 weakly in ET,u(⋅,t)|∂E=g(⋅,t) a.e. t∈(0,T],u(⋅,0)=g(x,0),
with g as above. We do not impose any {\em a priori} requirements on the boundary of the domain E⊂RN, and we provide an estimate on the modulus of continuity at a boundary point in terms of a Wiener-type integral, defined by a proper elliptic p-capacity. The results depend on the value of p, namely whether 1<p≤2NN+1, 2NN+1<p<2, p≥2.
This is a joint work with Naian Liao (Salzburg University, Austria) and Teemu Lukkari (Aalto University, Finland).
