On divergent fractional Laplace equations

Serena Dipierro; Ovidiu Savin; Enrico Valdinoci

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arXiv · 1907.11376

On divergent fractional Laplace equations

Serena Dipierro·Ovidiu Savin·Enrico Valdinoci

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Abstract

We consider the divergent fractional Laplace operator presented in [Dipierro-Savin-Valdinoci, Rev. Mat. Iberoam.] and we prove three types of results. Firstly, we show that any given function can be locally shadowed by a solution of a divergent fractional Laplace equation which is also prescribed in a neighborhood of infinity. Secondly, we take into account the Dirichlet problem for the divergent fractional Laplace equation, proving the existence of a solution and characterizing its multiplicity. Finally, we take into account the case of nonlinear equations, obtaining a new approximation results. These results maintain their interest also in the case of functions for which the fractional Laplacian can be defined in the usual sense.

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Serena Dipierro, Ovidiu Savin, Enrico Valdinoci. 2019-07-26. On divergent fractional Laplace equations. https://arxiv.org/abs/1907.11376

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