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Abstract
We study solutions to Lu=f in Ω⊂R^{n}, being L the generator of any, possibly nonsymmetric, stable Lévy process. On the one hand, we study the regularity of solutions to Lu=f in Ω, u=0 in Ω^{c}, in C^{1,α} domains Ω. We show that solutions u satisfy u/d^{γ}∈C^{ε∘}(Ω‾), where d is the distance to ∂Ω, and γ=γ(L,ν) is an explicit exponent that depends on the Fourier symbol of operator L and on the unit normal ν to the boundary ∂Ω. On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the nonsymmetric setting presents some new interesting features. Finally, we generalize the integration by parts identities in half spaces to the case of bounded C^{1,α} domains. We do it via a new efficient approximation argument, which exploits the Hölder regularity of u/d^{γ}. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.
Original language  English 

Article number  108321 
Journal  Advances in Mathematics 
Volume  401 
DOIs  
Publication status  Published  4 Jun 2022 
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Minimal surfaces, free boundaries and partial differential equations
1/01/19 → 30/06/25
Project: Research

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