Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A class of non-cylindrical domains for parabolic equations

Alberto Domínguez Corella, Jorge Rivera-Noriega

TL;DR

This work defines and analyzes a class of non-cylindrical domains in $\\mathbb{R}^{n+1}$ on which Dirichlet-type problems for parabolic equations, such as the heat equation, can be posed. It introduces parabolic homogeneity and a Lip$(1,1/2)$ framework, culminating in an implicit-function theorem tailored to this regularity to describe boundary graphs. The main contribution is the construction of non-cylindrical star-like domains $\\Omega=\\{(s,r\\omega):\\omega\in S^{n-1}, 0\le r<\\varphi(s,\\omega)\\}$ with $\\varphi$ Lip$(1,1/2)$ and bounds, proving these domains are compatible with the existing $L^p$-Dirichlet theory for parabolic equations. This broadens the class of domains where parabolic boundary-value problems can be analyzed using caloric measures and related parabolic techniques, linking the new non-cylindrical domains to earlier star-like Lipschitz results.

Abstract

We present a class of non-cylindrical domains where Dirichlet-type problems for parabolic equations, such as the heat equation, can be posed and solved. The regularity for the boundary of this class of domains is a mixed Lipschitz condition, as described in the bulk of the paper. The main tool is an adequate version of the implicit function theorem for functions with this kind of regularity. It is proved that the class introduced herein is of the same type as domains previously considered by several authors.

A class of non-cylindrical domains for parabolic equations

TL;DR

This work defines and analyzes a class of non-cylindrical domains in on which Dirichlet-type problems for parabolic equations, such as the heat equation, can be posed. It introduces parabolic homogeneity and a Lip framework, culminating in an implicit-function theorem tailored to this regularity to describe boundary graphs. The main contribution is the construction of non-cylindrical star-like domains with Lip and bounds, proving these domains are compatible with the existing -Dirichlet theory for parabolic equations. This broadens the class of domains where parabolic boundary-value problems can be analyzed using caloric measures and related parabolic techniques, linking the new non-cylindrical domains to earlier star-like Lipschitz results.

Abstract

We present a class of non-cylindrical domains where Dirichlet-type problems for parabolic equations, such as the heat equation, can be posed and solved. The regularity for the boundary of this class of domains is a mixed Lipschitz condition, as described in the bulk of the paper. The main tool is an adequate version of the implicit function theorem for functions with this kind of regularity. It is proved that the class introduced herein is of the same type as domains previously considered by several authors.
Paper Structure (5 sections, 7 theorems, 93 equations)

This paper contains 5 sections, 7 theorems, 93 equations.

Table of Contents

  1. Motivation
  2. Basic definitions and main results
  3. Parabolic homogeneity in $\mathbb{R}^{n+1}$
  4. Lip$(1,1/2)$–type functions and their version of the implicit function theorem
  5. Non–cylindrical domains in $\mathbb{R}^{n+1}$

Key Result

Theorem 1

If $\Omega$ is a star-like Lipschitz domain, then it is a Lipschitz domain.

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2: Implicit function theorem for Lipschitz functions
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm:parabolic-metric']}
  • Definition 3
  • Theorem 4
  • proof
  • Theorem 5: Implicit function theorem
  • ...and 7 more