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Completeness theorems on the boundary for a parabolic equation

Alberto Cialdea, Carmine Sebastiano Mare

Abstract

Let $\{v_α\}$ be a system of polynomial solutions of the parabolic equation $a_{hk}\partial_{x_{h}x_{k}}u - \partial_t u =0$ in a bounded $C^1$-cylinder $Ω_{T}$ contained in $\mathbb{R}^{n+1}$. Here $a_{hk}\partial_{x_{h}x_{k}}$ is an elliptic operator with real constant coefficients. We prove that $\{v_α\}$ is complete in $L^{p}(Σ')$, where $Σ'$ is the parabolic boundary of $Ω_{T}$. Similar results are proved for the adjoint equation $a_{hk}\partial_{x_{h}x_{k}} u+ \partial_t u =0$.

Completeness theorems on the boundary for a parabolic equation

Abstract

Let be a system of polynomial solutions of the parabolic equation in a bounded -cylinder contained in . Here is an elliptic operator with real constant coefficients. We prove that is complete in , where is the parabolic boundary of . Similar results are proved for the adjoint equation .
Paper Structure (6 sections, 18 theorems, 105 equations)

This paper contains 6 sections, 18 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The jump formulas for parabolic potentials
  4. The classes $\mathscr{A}^{p}$ and $\mathscr{A}^{p}_{*}$
  5. Parabolic polynomials
  6. Completeness theorems

Key Result

Theorem 1

Let $\varphi\in L^{p}(\Sigma_3)$ ($1\leq p<\infty$). Then for almost every $(x_0,t) \in \Sigma_3$. Here the limit $(x,t)\to (x_0,t)^{+}$ ($(x,t)\to (x_0,t)^{-}$) has to be understood as an internal (external) angular boundary value and the integral on the right hand side exists as a singular integral. Moreover, for any $1<p<\infty$ and for any real $c\neq is invertible on $L^{p}(\Sigma_{3})$.

Theorems & Definitions (36)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 26 more