Critical perturbations of elliptic operators by lower order terms

In this work we study issues of existence and uniqueness of solutions of certain boundary value problems for elliptic equations in the upper half-space. More specifically we treat the Dirichlet, Neumann, and Regularity problems for the general second order, linear, elliptic operator under a smallness assumption on the coefficients in certain critical Lebesgue spaces. Our results are perturbative in nature, asserting that if a certain operator L₀ has good properties (as far as boundedness and invertibility of certain associated solution operators), then the same is true for L₁, whenever the coefficients of these two operators are close in certain L[superscript p] spaces. Our approach is through the theory of layer potentials, though the lack of good estimates for solutions of L = 0 force us to use a more abstract construction of these objects, as opposed to the more classical definition through the fundamental solution. On the other hand, these more general objects suggest a wider range of applications for these techniques. The results contained in this thesis were obtained in collaboration with Simon Bortz, Steve Hofmann, Svitlana Mayboroda, and Bruno Poggi. The resulting publications can be found in [BHL+a] and [BHL+b].

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics (MU)

Grantor dc:publisher

University of Missouri--Columbia

Year dc:date.issued

2021

Author dc:creator

• Luna-Garcia, Jose Luis

Advisor dc:contributor.advisor

• Hofmann, Steve

Language dc:language.iso

eng, English

DOI

https://doi.org/10.32469/10355/85798

OAI identifier oai:identifier

oai:mospace.umsystem.edu:10355/85798