New arXiv:2007.13195v2 [math.AP] 3 Aug 2020 · 2020. 8. 4. · Dipartimento di Matematica e...
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Symmetrization for fractional elliptic problems: a direct approach Vincenzo Ferone * and Bruno Volzone † Abstract We provide new direct methods to establish symmetrization results in the form of mass concentration (i.e. integral) comparison for fractional elliptic equations of the type (-Δ) s u = f (0 <s< 1) in a bounded domain Ω, equipped with homoge- neous 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. 1 Introduction The aim of this note is to develop some new techniques regarding the application of symmetrization methods to Dirichlet fractional elliptic problems of the type (1.1) (-Δ) s u = f in Ω, u =0 on R N \ Ω, where Ω ⊂ R N (N ≥ 1) is a smooth bounded open set, the source term f = f (x) is assumed to belong to L p (Ω) for suitable p ≥ 1 and s ∈ (0, 1). In the case of local equations, it is well known that under Schwarz symmetrization the solution to the homogeneous Dirichlet problem for an elliptic equation increases in terms of rearrangements. Namely, consider the Dirichlet problem (1.2) - ( a ij z x i ) x j = f in Ω, z =0 on ∂ Ω, * Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, 80143 Napoli, Italy. E-mail: [email protected]† Dipartimento di Scienze e Tecnologie, Università degli Studi di Napoli “Parthenope”, 80133 Napoli, Italy. E-mail: [email protected]1 arXiv:2007.13195v2 [math.AP] 3 Aug 2020
New arXiv:2007.13195v2 [math.AP] 3 Aug 2020 · 2020. 8. 4. · Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli FedericoII,80143Napoli,Italy
