| Abstract: |
| In this talk, I will discuss existence, classification, and non-degeneracy results for solutions to singular Liouville-type equations in dimension one. This problem has applications in the mathematical modeling of galvanic corrosion phenomena for ideal electrochemical cells consisting of an electrolyte solution confined in a bounded domain with an electrochemically active portion of the boundary. In higher dimensions, Liouville equations have applications to prescribed curvature problems in conformal geometry, where solutions correspond to constant Q-curvature metrics on Euclidean space, with a singular point at the origin.
After providing a general overview of the existing literature, I will focus on the one-dimensional case and prove that solutions are non-degenerate, under mild assumptions on the singular weight. The proof relies on the use of harmonic extensions and conformal transformations to rewrite the linearized Liouville equation as a Steklov eigenvalue problem on either the intersection or the union of two disks. These results were obtained in collaboration with A. DelaTorre and A. Pistoia. |
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