A regularity theory for parabolic equations with anisotropic non-local operators in L_q(L_p) spaces

2023/08/01 by Jae-Hwan Choi, Choi, Jae-Hwan, Jae-Hoon Kang +3 · 1 citation
Computer Science · Mathematics · #35B65 #45K05 #47G20 #60H30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2308.00347

openalex publication_date 2023/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we present an L_q(L_p)-regularity theory for parabolic equations of the form: ∂_t u(t,x)=L^a,b(t)u(t,x)+f(t,x), u(0,x)=0. Here, L^a,b(t) represents anisotropic non-local operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: L^a,0(t)u(x)=∑_i=1^d ∫_ℝ( u(x¹,…,x^i-1,xⁱ+yⁱ,xⁱ⁺¹,…,x^d) - u(x) ) \fraca_i(t,yⁱ)|yⁱ|^1+α_i dyⁱ . 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.

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• Some remarks on Campanato Theorem and the Anisotropic Bessel Spaces

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