School of Mathematics
The University of Edinburgh and
Maxwell Institute of Mathematical Sciences
Edinburgh, UK
email: M.Dindos@ed.ac.uk
School of Informatics
The University of Edinburgh
Edinburgh, UK
email: s2029366@ed.ac.uk
Martin Dindos, Yingyi Liu
Abstract:
A recent result of the first author with Li and Pipher established the
extrapolation of solvability of the \(L^p\) parabolic Neumann problem on unbounded
graph domains of the form \(\Omega=\{(x',x_n):x_n>\phi(x')\}\times\mathbb{R}\),
where \(\phi:\mathbb{R}^{n-1}\to\mathbb{R}\) is a Lipschitz function.
The result shows that under the assumptions that the \(L^p\) parabolic Neumann problem
for the equation \(Lu=-\partial_t u+\operatorname{div}(A\nabla u)=0\) in
\(\Omega\) and the \(L^{p'}\) parabolic Dirichlet problem for the adjoint equation
\(L^*u=\partial_t u+\operatorname{div}(A^T\nabla u)=0\) in \(\Omega\) are solvable,
then also the \(L^q\) parabolic Neumann problem for the equation \(Lu=0\) in \(\Omega\)
is solvable for all \(1< q< p\).
However, the mentioned paper does not answer the question whether the same claim
is also true for domains of the form \(\mathcal O\times\mathbb{R}\), where \(\mathcal O\)
is a bounded Lipschitz domain (in spatial variables) since this case does not follow
from our argument for the unbounded case. Indeed, the bounded Lipschitz cylinder case
requires a significantly different approach which we present in this article and
establish an analogous result when \(\mathcal O\) is a bounded Lipschitz domain.
Submitted May 4, 2026. Published July 13, 2026.
Math Subject Classifications: 35K20, 35K10.
Key Words: Parabolic Neumann problem; bounded Lipschitz cylinder; extrapolation of solvability;
non-tangential maximal function; atomic Hardy space; parabolic Dirichlet problem.
DOI: 10.58997/ejde.2026.53
Show me the PDF file (500 KB), TEX file for this article.
|
Martin Dindos School of Mathematics The University of Edinburgh and Maxwell Institute of Mathematical Sciences Edinburgh, UK email: M.Dindos@ed.ac.uk |
|---|---|
|
Yingyi Liu School of Informatics The University of Edinburgh Edinburgh, UK email: s2029366@ed.ac.uk |
Return to the EJDE web page