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Weyl-type laws for fractional p-eigenvalue problems
Antonio Iannizzotto, Marco Squassina
TL;DR
Higher fractional p-eigenvalues lack the local asymptotic context that motivates this paper's counting-function estimates. The authors construct a min-max sequence using the cohomological index and prove asymptotic estimates for it, while noting that nonlocality makes the bounds asymmetric.
Problem
The paper addresses the asymptotic counting of higher fractional p-eigenvalues, following prior asymptotic estimates for the local p-Laplacian.
Source: §1 · p. 2Method
It defines a nondecreasing min-max sequence of variational eigenvalues using the cohomological index.
Source: §1 · p. 2Results
The problem admits a nondecreasing sequence of positive eigenvalues tending to infinity, for which the paper estimates the counting function at infinity.
Source: §1 · p. 2Takeaways & Limitations
The authors suspect a sharper Weil-type law that implies both stated estimates, at least when sp>N.
Source: §1 · p. 2The nonlocal Gagliardo norm necessitates correction multipliers, producing asymmetric estimates.
Source: §1 · p. 3
Abstract
from arXiv · showhide
We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.
1. Introduction
The paper constructs higher variational eigenvalues for the fractional p-Laplacian and estimates their counting function at infinity. It places these estimates alongside local p-Laplacian asymptotics and attributes their asymmetry to the fractional norm's nonlocal character.
1. Introduction
- For the local p≠2 problem, prior bounds give C1|Ω|λ^(N/p) ≤ N(λ) ≤ C2|Ω|λ^(N/p) for large λ,.Source: §1 · p. 1
- The fractional problem uses W^{s,p}(R^N) and imposes the Dirichlet condition on R^N\Ω, consistent with the operator's nonlocal character.Source: §1 · p. 1
- The higher eigenvalues are defined as a nondecreasing min-max sequence using the cohomological index.Source: §1 · p. 2
- A nondecreasing sequence of positive variational eigenvalues tends to infinity, and the paper estimates its counting function at infinity.Source: §1 · p. 2
- The authors suspect a sharper Weil-type law that would imply both stated estimates, at least when sp>N.Source: §1 · pp. 2–3
- The estimates are asymmetric because the nonlocal Gagliardo norm requires correction multipliers in place of the classical splitting argument.Source: §1 · p. 3
2. Construction of the variational eigenvalues
The paper formulates the fractional p-eigenvalue problem variationally on a fractional Sobolev space and constructs a non-decreasing min-max sequence using a cohomological index. It establishes that the sequence consists of eigenvalues and diverges to infinity.
2. Construction of the variational eigenvalues
- The construction uses cohomological-index monotonicity under odd continuous maps and its value k on the sphere S^(k−1).The paper also compares the index with the Krasnoselskii genus and co-genus.Source: §2 · p. 3
- The fractional Sobolev space X(Ω) can be normed by the Gagliardo seminorm and is uniformly convex.A Poincaré-type inequality justifies this equivalent norm.Source: §2 · p. 4
- The energy functional restricted to the normalized sphere has critical values exactly at the eigenvalues of problem (1.2).The normalization is I(u) = 1; a critical point at level λ yields a λ-eigenfunction.Source: §2 · p. 4
- The non-decreasing sequence is defined by min-max using the cohomological index, rather than the classical genus.Its monotonicity follows from the nested admissible families Fk+1 ⊆ Fk.Source: §2 · p. 5
- Every variational value λk is an eigenvalue, and λk → ∞ as k increases.The proof uses the Palais–Smale condition and the unboundedness of J on S.Source: §2 · p. 5
3. Preparatory results
The preparatory results establish monotonicity and domain-combination bounds for the genus and co-genus of sublevel sets. They also construct continuous odd maps from componentwise data to support those bounds.
Monotonicity
- The genus and co-genus satisfy monotonicity inequalities for the stated domain and level changes.The passage gives inequalities comparing the respective sublevel-set indices across those changes.Source: §3 · pp. 5–6
Lemma 3.2: genus super-additivity up to a correction factor
- For disjoint bounded Lipschitz domains, the genus of their component sublevel sets is bounded above by the genus on the union at the adjusted level.The proof builds a map from the product of component spheres, with total dimension parameter k equal to the sum of the component parameters.Source: §3 · p. 6
Construction of the componentwise map
- The componentwise construction normalizes each nonzero local restriction and sets its local test value to infinity when that restriction has zero Lp norm.The resulting map is odd, and its continuity is checked by considering positive, vanishing, and eventually zero local norms.Source: §3 · p. 6
Lemma 3.3: co-genus sub-additivity up to a correction factor
- For disjoint bounded Lipschitz domains, the co-genus is sub-additive up to a correction factor, using a threshold that at least one component must meet.The proof establishes min_i θ_i(u) ≤ μ′ before constructing the map into the sphere.Source: §3 · p. 7
4. Proof of the main result
The proof of the main theorem combines cube coverings with the preparatory index inequalities to establish the asserted asymptotic estimates. It also proves radial symmetry and monotonicity for first eigenfunctions on a ball, and compares counting functions for alternative variational sequences.
Proof of the asymptotic estimates
- For bounded open Ω with Lipschitz boundary, the proof establishes asymptotic estimates (1.4) and (1.5); it proves (1.5) under sp > N using an exterior cube union of volume at most 2|Ω|.Source: §4 · pp. 8–9
First eigenfunctions on a ball
- On a ball, every positive λ1-eigenfunction is radially symmetric and radially decreasing, while every negative one is radially symmetric and radially increasing.Schwartz symmetrization preserves the Lp norm and does not increase the fractional seminorm; the proof then uses proportionality of the symmetrized function and the original eigenfunction.Source: §4 · p. 9
Alternative variational sequences
- The alternative variational sequences have counting functions satisfying N+(λ) ≤ N(λ) ≤ N−(λ), so (1.4) holds for N− and (1.5) for N+.The paper notes that it is not known whether the sequences themselves coincide.Source: §4 · p. 10