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Extrapolation of solvability of the parabolic $L^p$ Neumann problem on bounded Lipschitz cylinders

Martin Dindoš, YingYi Liu

Abstract

A recent result of the first author with Li and Pipher has established the extrapolation of solvability of the $L^p$ parabolic Neumann problem on unbounded graph domains of the form $Ω=\{(x',x_n):\,x_n>\varphi(x')\}\times\mathbb R$, where $\varphi:\mathbb R^{n-1}\to\mathbb R$ is a Lipschitz function. The result shows that under the assumptions that the $L^p$ parabolic Neumann problem for the equation $Lu=-\partial_t u+\mbox{div}(A\nabla u)=0$ in $Ω$ and also the $L^{p'}$ parabolic Dirichlet problem for the adjoint equation $L^*u=\partial_t u+\mbox{div}(A\nabla u)=0$ in $Ω$ are solvable, then also the $L^q$ parabolic Neumann problem for the equation $Lu=0$ in $Ω$ 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.

Extrapolation of solvability of the parabolic $L^p$ Neumann problem on bounded Lipschitz cylinders

Abstract

A recent result of the first author with Li and Pipher has established the extrapolation of solvability of the parabolic Neumann problem on unbounded graph domains of the form , where is a Lipschitz function. The result shows that under the assumptions that the parabolic Neumann problem for the equation in and also the parabolic Dirichlet problem for the adjoint equation in are solvable, then also the parabolic Neumann problem for the equation in is solvable for all . However the mentioned paper does not answer the question whether the same claim is also true for domains of the form , where 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 is a bounded Lipschitz domain.
Paper Structure (10 sections, 7 theorems, 105 equations, 1 figure)

This paper contains 10 sections, 7 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions
  3. Energy solutions
  4. Basic estimates and known results
  5. Proof of Theorem \ref{['MT']}
  6. Strategy of the proof
  7. Large atoms
  8. Small atoms, part 1
  9. Unbounded domain related to $\Omega$
  10. Small atoms, part 2

Key Result

Theorem 1.3

Let $\Omega=\mathcal{O}\times\mathbb R$, where $\mathcal{O}\subset\mathbb R^n$ is a bounded Lipschitz domain (as defined by Definition DefLipDomain). Consider the PDE such that E:elliptic holds for a.e. $(X,t)\in \Omega$. Assume that for some $p\in(1,\infty)$ the $L^p$ Neumann problem Np$^L$ and the $L^{p'}$ Dirichlet problem Dq$^{L^*}$ for the adjoint PDE $L^*u=0$ are both solvable (as defined i

Figures (1)

  • Figure 1: Visualization of the domains $\mathcal{O}$ and $\tilde{\Omega}$.

Theorems & Definitions (15)

  • Theorem 1.3
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.7
  • Definition 2.12: \ref{['Dq']}
  • Definition 2.13: \ref{['Np']}
  • Lemma 3.1: Caccioppoli's inequality
  • Lemma 3.2
  • ...and 5 more