PDEA Seminar: A Relation Between the Dirichlet and Regularity Problem for Parabolic Equations

Abstract: In this talk we consider the relationship between the Dirichlet and Regularity problem for parabolic operators of the form L = -div(A\nabla) + \partial_t. Letting (D_L)_p be the statement that the L^p Dirichlet problem is solvable and similarly (R_L)_p for the regularity problem, we know from Kenig-Pipher that (R_L)_p => (D_L^*)_p’ and the partial converse (D_L^*)_p’ + (R_L)_q => (R_L)_p was proved by Shen. The first of these was already shown in the parabolic setting by Dindos-Dyer, where the non-tangential maximal function of the half-time derivative could be ignored. However, after recent progress by Dindos in working with the half-time derivative we are now able to tackle the second result. This is done by first adapting Shen’s approach for q<p and then interpolating the L^p regularity problem against a suitable endpoint space to get the full generality i.e. that (D_L^*)_p’ => (R_L)_p or (R_L)_q does not hold for any 1<q<\infty. This is joint work with Martin Dindos.

Participate on site or via Zoom link, meeting ID: 67226102216

This is a seminar in the PDEs and Applications seminar series