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A regularity theory for parabolic equations with anisotropic non-local operators in $L_{q}(L_{p})$ spaces

Jae-Hwan Choi, Jaehoon Kang, Daehan Park

Abstract

In this paper, we present an $L_q(L_p)$-regularity theory for parabolic equations of the form: $$ \partial_t u(t,x)=\mathcal{L}^{\vec{a},\vec{b}}(t)u(t,x)+f(t,x),\quad u(0,x)=0. $$ Here, $\mathcal{L}^{\vec{a},\vec{b}}(t)$ represents anisotropic non-local operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: $$ \mathcal{L}^{\vec{a},\vec{0}}(t)u(x)=\sum_{i=1}^{d} \int_{\mathbb{R}}\left( u(x^{1},\dots,x^{i-1},x^{i}+y^{i},x^{i+1},\dots,x^{d}) - u(x) \right) \frac{a_{i}(t,y^{i})}{|y^{i}|^{1+α_{i}}} \mathrm{d}y^{i} . $$ To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calderón-Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic non-local operators and parabolic equations with isotropic non-local operators.

A regularity theory for parabolic equations with anisotropic non-local operators in $L_{q}(L_{p})$ spaces

Abstract

In this paper, we present an -regularity theory for parabolic equations of the form: Here, represents anisotropic non-local operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calderón-Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic non-local operators and parabolic equations with isotropic non-local operators.
Paper Structure (8 sections, 13 theorems, 235 equations)

This paper contains 8 sections, 13 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Settings and Main results
  3. Settings
  4. Main results
  5. Proof of Theorem \ref{['23.03.07.13.46']}
  6. Estimations of IASBM's heat kernel
  7. BMO estimation of solution
  8. Applications

Key Result

Lemma 2.6

Let $1<p<\infty$ and $\gamma\in\mathbb{R}$. $(i)$ The space $H_p^{\vec{\phi},\gamma}$ is a Banach space. $(ii)$ For any $\mu\in\mathbb{R}$, the map $(1-\vec{\phi}\cdot\Delta_{\vec{d}})^{\mu/2}$ is an isometry from $H^{\vec{\phi},\gamma}_{p}$ to $H^{\vec{\phi},\gamma-\mu}_{p}$. $(iii)$ If $\mu>0$, th where the constant $C$ is independent of $u$. $(iv)$ For any $u\in H^{\vec{\phi},\gamma+2}_{p}$, we

Theorems & Definitions (33)

  • Remark 2.1
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • Theorem 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 23 more