Semilinear equations for subordinate spectral Laplacian

Biocic Ivan, Università degli Studi di Torino

In this talk we solve semilinear problems in bounded $C^{1,1}$ domains for non-local operators with a non-homogeneous Dirichlet boundary condition, based on the work [1]. The operators cover and extend the case of the spectral fractional Laplacian, and are modelled using the process called subordinate killed Brownian motion. Our focus will be on the potential-probabilistic approach to these problems with an emphasis on methods, intuition, and calculations. This approach is a consequence of recent developments in [2,3]. We present an integral representation of harmonic functions for such non-local operators and give sharp boundary behaviour of Green and Poisson potentials. H\"older regularity of distributional solutions is given as well as a version of Kato's inequality. We explore moderate (i.e. harmonically bounded) solutions and large (i.e. harmonically unbounded) solutions to the semilinear problem. [1] I. Biočić, Semilinear Dirichlet problem for subordinate spectral Laplacian, Communications on Pure and Applied Analysis, \textbf{22} (2023), 851-898. [2] I. Biočić, Z. Vondraček, V. Wagner, Semilinear equations for non-local operators: Beyond the fractional Laplacian, Nonlinear Analysis, \textbf{207} (2021), 112303. [3] P. Kim, R. Song, Z. Vondraček, Potential theory of subordinate killed Brownian motion, Transactions of the American mathematical society, \textbf{371} (2019), 3917-3969.

Area: IS13 - Non-local operators in probability (Giacomo Ascione)

Keywords: Laplacian

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