| Abstract: |
| The main interest of this talk will be the study of (semi-)linear nonlocal elliptic problems driven by spectral-type operators of the form $\psi(-L_{|D})$ in bounded $C^{1,1}$ domains $D\subset \mathbf R^d$. Here $\psi$ is a complete Bernstein function and $L_{|D}$ is the generator of a killed unimodal L\`evy process. We will present the general framework that covers and extends the theory of the spectral and interpolated fractional Laplacian. The focus will be on the analysis of the nonhomogeneous boundary condition via a weak $L^1$ trace-like boundary operator. This operator is formulated in terms of the Poisson potential with respect to the $d-1$ Hausdorff measure on $\partial D$. The methodology combines stochastic process techniques, potential theory, and spectral analysis. |
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