Torsion Functions and the Cheeger Problem: A Fractional Approach

1 Introduction

The Cheeger constant h⁢(Ω) of a bounded domain Ω⊂ℝN (N>1) is defined by

where E is a smooth subset of Ω and the nonnegative values P⁢(E) and |E| denote, respectively, the distributional perimeter and the N-dimensional Lebesgue measure of E. A subset E that minimizes the quotient is a Cheeger set of Ω.

In [7] Kawohl and Fridman proved that

where λ1,p⁢(Ω) is the first eigenvalue of the Dirichlet p-Laplacian operator, that is, the least real number λ such that the Dirichlet problem

has a nontrivial solution. (Recall that the p-Laplacian operator is defined by Δp⁢u:=div⁢(|∇⁡u|p-2⁢∇⁡u), p>1.)

The first eigenvalue λ1,p⁢(Ω) is also variationally characterized by

with |⋅|r standing for the usual norm of Lr⁢(Ω), 1≤r≤∞ (this notation will be adopted from now on).

In [2] a different characterization of the Cheeger constant of Ω was obtained:

where ϕp denotes the p-torsion function of Ω, that is, the solution of the Dirichlet problem

which is known as the p-torsional creep problem (see [6]).

We remark that

since the minimum is attained at ϕp/|ϕp|1.

The fractional version of problem (1.1) consists in minimizing that quotient when P⁢(E) is substituted by Ps⁢(E), given by

where χE stands for the characteristic function of the smooth subdomain E. The value Ps⁢(E) is called the (nonlocal) s-perimeter of E. So, the fractional Cheeger problem is the minimization problem

and hs⁢(Ω) is called the s-Cheeger constant of Ω.

For 1<p<∞ and s∈(0,1), the fractional (s,p)-Laplacian (-Δ)ps is the nonlinear nonlocal operator defined by

This definition is consistent, up to a normalization constant depending on p, s and N, with the usual p-Laplacian operator.

The first (fractional) eigenvalue λ1,ps⁢(Ω) of (-Δ)ps is the least number λ such that the problem

has a nontrivial weak solution (see [8, 4]). Its variational characterization is given by (see [1])

where

is the (s,p)-seminorm of Gagliardo in ℝN of a measurable function u and W0s,p⁢(Ω) is a suitable fractional Sobolev space defined in the sequel (see Definition 2.1).

In [1] Brasco, Lindgren and Parini proved the s-Cheeger version of the result originally obtained by Kawohl and Friedman [7] for the Cheeger problem

In this paper, by assuming that Ω is a Lipschitz bounded domain, we show, in the spirit of the paper [2], that the fractional version of the torsional creep problem

is intrinsically connected to both the s-Cheeger problem and the first eigenproblem for the fractional Dirichlet p-Laplacian, as p goes to 1.

This connection will be developed in Section 2, where we introduce the (s,p)-torsion function of Ω, that is, the weak solution ϕps of (1.5). We will derive the estimates

and

where B1 denotes the unit ball of ℝN.

Then, taking (1.4) into account, we will combine (1.6) with (1.7) in order to conclude the main result of this paper:

Moreover, in Section 2 we prove that ϕps minimizes, in W0s,p⁢(Ω)∖{0}, the Rayleigh quotient [u]s,pp/|u|1p. As an immediate consequence of this fact, we show that ϕps is a radial function when Ω is a ball.

2 The Main Results

From now on Ω denotes a Lipschitz bounded domain of ℝN, N≥2, and 0<s<1<p<Ns.

The functions in W0s,p⁢(Ω) have a natural extension to ℝN and, although u=0 in ℝN∖Ω, the identity

shows dependence on values in ℝN∖Ω.

It is worth mentioning that W0s,p⁢(Ω) is a reflexive Banach space and that this space coincides with the closure of C0∞⁢(Ω) relative to the norm

if ∂⁡Ω is Lipschitz (see [1, Proposition B.1]).

Moreover, thanks to the fractional Poincaré inequality (see [1, Lemma 2.4])

we have that [⋅]s,p is also a norm in W0s,p⁢(Ω), equivalent to the norm defined in (2.1).

We refer the reader to [3] for fractional Sobolev spaces.

The existence of a weak solution of (2.2) when f∈L1⁢(Ω) follows from direct minimization in W0s,p⁢(Ω) of the functional

whereas the uniqueness comes from, for instance, the comparison principle for the fractional p-Laplacian (see [8, Lemma 9]). The same principle shows that if f is nonnegative, then the weak solution u is nonnegative as well. When f∈L∞⁢(Ω) and Ω is sufficiently smooth, say with boundary at least of class C1,1, the weak solutions are α-Hölder continuous up to the boundary for some α∈(0,1), see [5].

When f≡1 the Dirichlet problem (2.2) will be referred to as the (s,p)-fractional torsional creep problem and its unique weak solution will be called (s,p)-torsion function. Let us denote this function by ϕps. We have ϕps≥0 and

In particular, by taking φ=ϕps we obtain

The next result recovers [5, Lemma 4.1].

It is also well known that the first eigenfunctions of the fractional p-Laplacian belong to L∞⁢(Ω) and are either positive or negative almost everywhere in Ω. Moreover, they are scalar multiple of each other. So, let us denote by eps the positive and L∞-normalized first eigenfunction. It follows that |eps|∞=1 and

meaning that

Of course, by taking φ=eps in (2.7) we obtain

As mentioned in the introduction, λ1,ps⁢(Ω) is variationally characterized by (1.2).

We would like to emphasize the following consequence of (2.10): ϕps>0 almost everywhere in Ω.

A Faber–Krahn inequality also holds true for the first fractional eigenvalue.

The next estimate is obtained by applying standard set-level techniques; however, the bounds obtained are adequate to study the asymptotic behavior as p→1+.