The Fundamental Solution of an Elliptic Equation with Singular Drift

2022/08/31 by Maz'ya, Vladimir, McOwen, Robert · 1 citation
#35A08 (Primary) #35B40 #35J15 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2209.00058

Abstract

For n≥ 3, we study the existence and asymptotic properties of the fundamental solution for elliptic operators in nondivergence form, \mathcal L(x,∂_x)=a_ij(x)∂_i∂_j+b_k(x)∂_k, where the a_ij have modulus of continuity ω(r) satisfying the square-Dini condition and the b_k are allowed mild singularities of order r^-1ω(r). A singular integral is introduced that controls the existence of the fundamental solution. We give examples that show the singular drift b_k∂_k may act as a perturbation that does not dramatically change the fundamental solution of \mathcal L^o=a_ij∂_i∂_j, or it could change an operator \mathcal L^o that does not have a fundamental solution to one that does.

Cited by

• Pointwise bounds on Dirichlet Green's functions for a singular drift term

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