Critical Perturbations for Linear Elliptic Equations

Abstract: In this talk we develop a perturbation theory for the L^2 solvability of certain Boundary Value Problems for linear elliptic equations with complex coefficients in the upper half space. While we expect the methods to apply to general systems and higher order equations, we will focus here on the general scalar second order equation, for which most of the main difficulties are already present: For instance a lack of boundedness and continuity of solutions, precluding the use of a pointwise-defined fundamental solution.

Our theory is based on solvability via the method of layer potentials. As such the main points to consider are boundedness and invertibility, in the appropriate functional spaces, of the corresponding operators and their boundary traces. For the boundedness issue we employ the theory of local Tb theorems, to obtain control on certain square functions that allow us to conclude the desired bounds on the layer potentials. The invertibility will be treated through the analyticity of the boundary traces as a function of the coefficients of the equation.

Of technical interest is that our methods allow us to obtain nontangential maximal function estimates for the layer potential solutions so constructed.

This is joint work with Simon Bortz, Steve Hofmann, Svitlana Mayboroda, and Bruno Poggi.

analysis of PDEsclassical analysis and ODEs

Audience: researchers in the topic

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HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

Meetings will be weekly, and usually will occur on Monday, with some exceptions. For each talk, the Zoom link is made available in the website too. Contact Bruno Poggi at poggi008@umn.edu if you would like to subscribe to the seminar mailing list.

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