| Abstract: |
| I will present a recent work on the analysis of a parameterized family of energies associated with transmission problems that effectively couple two distinct models across an interface. Specifically, we examine the coupling between a model based on the regional fractional Laplacian and another model employing a nonlocal operator with a position-dependent interaction kernel.
Both operators are inherently nonlocal and act on functions defined within their respective domains. The coupling occurs via a transmission condition across a hypersurface interface. The heterogeneous interaction kernel of the nonlocal operator leads to an energy space endowed with a well-defined trace operator. This, combined with well-established trace results of fractional Sobolev spaces, facilitates the imposition of a transmission condition across an interface. The family of problems will be parametrized by two key parameters that measure nonlocality and differentiability. For each pair of parameters, we demonstrate existence of a solution to the resulting variational problems. Furthermore, we investigate the limiting behavior of these solutions as a function of the parameters. |
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