The variational approach to $s$-fractional heat flows and the limit cases $s\to 0^+$ and $s\to 1^-$

Luca, Lucia De; Crismale, Vito; Kubin, Andrea; Ninno, Angelo; Ponsiglione, Marcello

doi:10.48550/arxiv.2107.13828

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2021

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“…From the probabilistic point of view, we mention that geometric stable stochastic processes with a logarithmic operator as an infinitesimal generator are studied in [21]. Logarithmic-type energies also appear in the recent paper [12], where the asymptotic analysis for the s-fractional heat flow as s → 0 + (and as s → 1 − ) is examined.…”

Section: Introductionmentioning

confidence: 99%

Small order asymptotics for nonlinear fractional problems

Hernández-Santamaría¹,

Saldaña²

2021

Preprint

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We study the limiting behavior of solutions to boundary value nonlinear problems involving the fractional Laplacian of order 2s when the parameter s tends to zero. In particular, we show that least-energy solutions converge (up to a subsequence) to a nontrivial nonnegative least-energy solution of a limiting problem in terms of the logarithmic Laplacian, i.e. the pseudodifferential operator with Fourier symbol ln(|ξ| 2 ). These results are motivated by some applications of nonlocal models where a small value for the parameter s yields the optimal choice. Our approach is based on variational methods, uniform energy-derived estimates, and the use of a new logarithmic-type Sobolev inequality.

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Section: Introductionmentioning

confidence: 99%