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Kernel bounds for parabolic operators having first-order degeneracy at the boundary

Luigi Negro, Chiara Spina

Abstract

We study kernel estimates for parabolic problems governed by singular elliptic operators \begin{equation*} \sum_{i,j=1}^{N+1}q_{ij}D_{ij}+c\frac{D_y}{y},\qquad c+1>0, \end{equation*} in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$ under Neumann boundary conditions at $y=0$.

Kernel bounds for parabolic operators having first-order degeneracy at the boundary

Abstract

We study kernel estimates for parabolic problems governed by singular elliptic operators \begin{equation*} \sum_{i,j=1}^{N+1}q_{ij}D_{ij}+c\frac{D_y}{y},\qquad c+1>0, \end{equation*} in the half-space under Neumann boundary conditions at .
Paper Structure (7 sections, 26 theorems, 136 equations)

This paper contains 7 sections, 26 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. The operator $\mathcal{L}=\Delta_{x} +2a\cdot\nabla_xD_y+ B_yu$ in $L^2_{c}$
  3. Weighted Sobolev inequalities
  4. Kernel estimates
  5. Gagliardo-Nirenberg type inequalities for $\mathfrak{b}_{\alpha\psi}$ and $\mathfrak{b}^\ast_{\alpha\psi}$
  6. Derivation of kernel estimates for $e^{z{\mathcal{L}}}$ and $e^{z{\mathcal{L}}^\ast}$
  7. General operators and oblique derivative

Key Result

Proposition 2.1

The forms $\mathfrak{a}$, $\mathfrak{a^*}$ are continuous, accretive and sectorial.

Theorems & Definitions (32)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Lemma 3.6
  • Proposition 3.7
  • ...and 22 more