Asymptotic analysis of the Dirichlet fractional Laplacian in domains becoming unbounded

Abstract

In this paper we analyze the asymptotic behavior of the Dirichlet fractional Laplacian (−∆ R n+k ) s , with s ∈ (0, 1), on bounded domains in R n+k that become unbounded in the last k-directions. A dimension reduction phenomenon is observed and described via Γ-convergence.Partially supported by PRID project VAPROGE , ℓ ∈ (0, ∞) .

Study of fractional Poincaré inequalities on unbounded domains

“…The other inequality is obtained by constructing suitable test functions on truncated domains Ω = B m (0, ) × ω and then finally letting tend to infinity. Very recently, and after our result has appeared, an independent proof of the above theorem was also given by [2]. The proof of [2] is completely different from ours and is based on Fourier space methods.…”

“…Very recently, and after our result has appeared, an independent proof of the above theorem was also given by [2]. The proof of [2] is completely different from ours and is based on Fourier space methods. Moreover they require ω to have Lipschitz boundary.…”

“…Assume by contradiction that there is equality in (35). Then u * ∈ H s U (R) ⊂ H s (−1,1) (R) is also a minimizer for P 2 1,s ((−1, 1)) and has to be strictly positive in (−1, 1) by Proposition 6.1 (b). This is a contradiction to u * (0) = 0.…”

“…For the range s ∈ (0, 1 2 ] this is trivial for P 1 n,s by Theorem 1.1 (1). Regarding P 2 n,s , examples of domains have been constructed in [11] to establish that the finite ball condition is not sufficient for P 2 n,s > 0 in the range s ∈ (0, 1 2 ]. The case s > 1/2 is more subtle and a satisfactory characterization of the domains for which P 1 n,s or P 2 n,s > 0 is still unknown.…”

“…Whereas if P 2 n,s (Ω) > 0 we say fractional Poincaré inequality holds true. P 1 n,s and P 2 n,s differ significantly in many properties, see for instance Proposition 2.3, where we have summarized the known results on bounded domains. One of the relevant differences concerns the domain monotonicity: P 2 n,s (Ω 1 ) ≥ P 2 n,s (Ω 2 ) if Ω 1 ⊂ Ω 2 .…”