Critical perturbations of elliptic operators by lower order terms

Date: 
Wednesday, April 28, 2021 - 4:00pm
Location: 
Zoom presentation
Speaker: 
Jose Luis Luna Garcia

Abstract:

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 L_0 has good properties (as far as boundedness and invertibility of certain associated solution operators), then the same is true for L_1, whenever the coefficients of these two operators are close in certain L^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.

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