L² Solvability of boundary value problems for divergence form\n parabolic equations with complex coefficients

2016/03/09 by Kaj Nyström, Nyström, Kaj · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1603.02823

openalex publication_date 2016/03/09 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We consider parabolic operators of the form
partial_t+
mathcalL,
\n
mathcalL=-
mboxdiv
, A(X,t)
nabla, in mathbb\nR₊ⁿ⁺²:= (X,t)=(x,x_n+1,t)\∈ mathbb Rⁿ\× mathbb R\× mathbb\nR: x_n+1>0 , n\≥ 1. We assume that A is a (n+1)\×\n(n+1)-dimensional matrix which is bounded, measurable, uniformly elliptic and\ncomplex, and we assume, in addition, that the entries of A are independent of\nthe spatial coordinate x_n+1 as well as of the time coordinate t. For\nsuch operators we prove that the boundedness and invertibility of the\ncorresponding layer potential operators are stable on L²( mathbb\nRⁿ⁺¹, mathbb C)=L²(\∂ mathbb Rⁿ⁺²₊, mathbb C) under complex,\nL^\∞ perturbations of the coefficient matrix. Subsequently, using this\ngeneral result, we establish solvability of the Dirichlet, Neumann and\nRegularity problems for \∂_t+\L, by way of layer potentials\nand with data in L², assuming that the coefficient matrix is a small complex\nperturbation of either a constant matrix or of a real and symmetric matrix.\n

Cited by

• Equivalence between solvability of the Dirichlet and Regularity problem under an L¹ Carleson condition on ∂_t A
• Gaussian estimates for fundamental solutions of higher-order parabolic equations with time-independent coefficients

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