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Abstract
We introduce a suitable notion of integral operators (comprising the fractional Laplacian as a particular case) acting on functions with minimal requirements at infinity. For these functions, the classical definition would lead to divergent expressions, thus we replace it with an appropriate framework obtained by a cutoff procedure. The notion obtained in this way quotients out the polynomials which produce the divergent pattern once the cutoff is removed. We also present results of stability under the appropriate notion of convergence and compatibility results between polynomials of different orders. Additionally, we address the solvability of the Dirichlet problem. The theory is developed in general in the pointwise sense. A viscosity counterpart is also presented under the additional assumption that the interaction kernel has a sign, in conformity with the maximum principle structure.
Original language  English 

Pages (fromto)  60108 
Number of pages  49 
Journal  Fractional Calculus and Applied Analysis 
Volume  25 
Issue number  1 
DOIs  
Publication status  Published  Feb 2022 
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Minimal surfaces, free boundaries and partial differential equations
ARC Australian Research Council
1/01/19 → 30/06/25
Project: Research

Partial Differential Equations, free boundaries and applications
ARC Australian Research Council
30/11/18 → 30/11/22
Project: Research