Transmission problems originally arose in elasticity theory, and they are nowadays of great interest due to their many applications in different areas in science. Typically, in these problems, there is a fixed interface where solutions may change abruptly, and the primary focus is to study their behavior across this surface.
In this talk, we will discuss the existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with C^{1,\alpha} interfaces. For this, we develop a novel geometric stability argument based on the mean value property. These results are part of my Ph.D. dissertation, and they are joint work with Luis A. Caffarelli and Pablo R. Stinga.
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