## Parabolic boundary value problems with rough coefficients

##### Abstract

This thesis is motivated by some of the recent results of the solvability of elliptic PDE in
Lipschitz domains and the relationships between the solvability of different boundary value
problems. The parabolic setting has received less attention, in part due to the time irreversibility
of the equation and difficulties in defining the appropriate analogous time-varying domain. Here
we study the solvability of boundary value problems for second order linear parabolic PDE in
time-varying domains, prove two main results and clarify the literature on time-varying domains.
The first result shows a relationship between the regularity and Dirichlet boundary value
problems for parabolic equations of the form Lu = div(A∇u)−ut = 0 in Lip(1, 1/2) time-varying
cylinders, where the coefficient matrix A = [aij(X, t)] is uniformly elliptic and bounded. We
show that if the Regularity problem (R)p for the equation Lu = 0 is solvable for some 1 < p < then the Dirichlet problem (D*) 1 p, for the adjoint equation L*v = 0 is also solvable, where
p' = p/(p − 1). This result is analogous to the one established in the elliptic case.
In the second result we prove the solvability of the parabolic Lp Dirichlet boundary value
problem for 1 < p ≤ ∞ for a PDE of the form ut = div(A∇u)+B ·∇u on time-varying domains
where the coefficients A = [aij(X, t)] and B = [bi(X, t)] satisfy a small Carleson condition. This
result brings the state of affairs in the parabolic setting up to the current elliptic standard.
Furthermore, we establish that if the coefficients of the operator A and B satisfy a vanishing
Carleson condition, and the time-varying domain is of VMO-type then the parabolic Lp Dirichlet
boundary value problem is solvable for all 1 < p ≤ ∞. This is related to elliptic results where
the normal of the boundary of the domain is in VMO or near VMO implies the invertibility
of certain boundary operators in Lp for all 1 < p < ∞. This then (using the method of layer
potentials) implies solvability of the Lp boundary value problem in the same range for certain
elliptic PDE. We do not use the method of layer potentials, since the coefficients we consider
are too rough to use this technique but remarkably we recover Lp solvability in the full range of
p’s as the elliptic case. Moreover, to achieve this result we give new equivalent and localisable
definitions of the appropriate time-varying domains.