Symmetrization for Fractional Elliptic Problems: A Direct Approach

We provide new direct methods to establish symmetrization results in the form of a mass concentration (that is, integral) comparison for fractional elliptic equations of the type ( - Δ ) s u = f ( 0 < s < 1 ) in a bounded domain Ω , equipped with homogeneous Dirichlet boundary conditions. The...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2021-03, Vol.239 (3), p.1733-1770
Main Authors: Ferone, Vincenzo, Volzone, Bruno
Format: Article
Language:English
Subjects:
Boundary conditions
Classical Mechanics
Complex Systems
Dirichlet problem
Elliptic functions
Fluid- and Aerodynamics
Mathematical and Computational Physics
Physics
Physics and Astronomy
Theoretical
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Summary:We provide new direct methods to establish symmetrization results in the form of a mass concentration (that is, integral) comparison for fractional elliptic equations of the type ( - Δ ) s u = f ( 0 < s < 1 ) in a bounded domain Ω , equipped with homogeneous Dirichlet boundary conditions. The classical pointwise Talenti rearrangement inequality in [ 47 ] is recovered in the limit s → 1 . Finally, explicit counterexamples constructed for all s ∈ ( 0 , 1 ) highlight that the same pointwise estimate cannot hold in a nonlocal setting, thus showing the optimality of our results.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-020-01601-8