On the first eigenvalue
of the normalized p-Laplacian
We prove that, if is an open bounded domain with smooth and connected boundary, for every the first Dirichlet eigenvalue of the normalized -Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (non-optimal) lower bound for the eigenvalue in terms of the measure of , and we address the open problem of proving a Faber-Krahn type inequality with balls as optimal domains.
Key words and phrases:Normalized -Laplacian, viscosity solutions, eigenvalue problem.
2010 Mathematics Subject Classification:49K20, 35J60, 47J10.
1. Introduction and statement of the results
Given an open bounded subset of , we consider the following eigenvalue problem
where denotes the normalized or game-theoretic -Laplacian, defined for any by
where stands for the Hessian of . Equivalently, see [K0], it can be defined as a convex combination of the limit operators as and , since
The normalized -Laplacian has recently received increasing attention, partly because of its application in image processing [K0, Does] and in the description of tug-of-war games (see [PSSW1, PSSW2]). Without claiming to be complete we list [BK18, CFd, CFe, CFf, CF7, EKNT, JK, Juut07, KH, K11, kuhn, MPR1, MPR2] for some related works.
Following Berestycki, Nirenberg, and Varadhan [BNV], in the paper [BiDe2006] (where actually they deal with a wider class of operators), Birindelli and Demengel introduced the first eigenvalue of in as
They proved that calling it first eigenvalue is justified, see [BiDe2006, Theorems 1.3 and 1.4]. In particular they showed that there exists a positive eigenfunction associated with . In other words for problem (1) admits a positive viscosity solution. They also posed the open problem to determine whether is simple. We show that the answer is affirmative. More precisely, we prove:
Let be an open bounded domain in , with smooth and connected. If and are two positive eigenfunctions associated with , then and are proportional, that is there exists such that in .
Here and in the following, smooth means that it is of class . Theorem 1 has the following immediate consequence:
Let be an open bounded domain in , with smooth and connected. If is invariant under elements from a symmetry group such as reflections or rotations, then so are the first eigenfunctions of the normalized -Laplace operator.
In order to obtain Theorem 1 we follow the approach used by Sakaguchi in [Sak]. In particular, it will be clear by inspection of the proof that this method does not work if one drops the assumption that is connected. It is conceivable that the result continues to be true for more general domains, as it is known in the literature for other kinds of operators at least in dimension two (see for instance [BirDem2010, Theorem 4.1]).
As a fundamental preliminary tool, our proof of Theorem 1 exploits a Hopf type lemma (see Lemma 8) and, incidentally, it requires also the strict positivity of the eigenvalue. The latter can be easily established by comparison with the behaviour on balls (see Lemma 6 and Lemma 7). In fact, the observation that for leads to the bounds
where and denote inradius and outer radius of , see the recent papers [blanc, KH]. These bounds are sharp if is a ball, but they are far from optimal if becomes large, e.g. for slender ellipsoids. On the other hand, the problem of finding more accurate bounds for the eigenvalue seems to be an interesting and mostly unexplored question. In this respect (2) is complemented by the following lower estimate for in terms of the Lebesgue measure of .
For every open bounded domain in we have the lower bound
The proof of Theorem 3 will be obtained by the Alexandrov–Bakelman–Pucci method, as addressed by Cabré in [C15] (see also [CDDM]). Unfortunately, it seems to be an intrinsic drawback of this approach to provide a non-optimal estimate. Actually it is natural to conjecture that, as in case of the well-known Faber-Krahn inequality for the -Laplacian, the product should be minimal on balls. In other words, the optimal lower bound expected for the product is the constant . Notice that due to the scaling invariance can be an arbitrary ball here. To prove such an optimal bound seems to be a very interesting and delicate problem. The symmetrization technique usually employed to prove the Faber-Krahn inequality for the -Laplacian does not work here because the normalized -Laplacian operator does not have a variational nature.
To demonstrate that (3) is not optimal for balls let us sketch a quick comparison between the values of and . Clearly, by Theorem 3, the quotient is larger than or equal to . In order to evaluate the presumed accuracy of our estimate, one can evaluate how far it is from . As shown in Lemma 6 below, we have
where denotes the first zero of the Bessel function , with . The plots in Figure 1 left and right, obtained with Mathematica, represent this ratio in two and three dimensions as a function of . Observe that both maps
turn out to be minimal at , with
This shows that the constant in Theorem 3 is not optimal, not even in the linear case .
In the notation of viscosity theory, the equation can be rewritten as
where is defined on and denotes the space of symmetric matrices, with
At the function is discontinuous. In this case, following [CIL] we request from a viscosity solution of (5) that it is a viscosity subsolution of and a viscosity supersolution of . Here is the upper semicontinuous hull and is the lower semicontinuous hull of .
Now since is given by
we have to compute its semicontinuous limits as . Each symmetric matrix has real eigenvalues, and we order them according to magnitude as . Then a simple calculation shows that
In [Bru] these bounds for the normalized -Laplacian are called dominative and submissive -Laplacians and studied in more detail. Anyway, the above considerations serve as a motivation for the following
Given a symmetric matrix , we denote by and its greatest and smallest eigenvalue.
– An upper semicontinuous function is a viscosity subsolution of in if, for every point in and every smooth function which touches from above at (and for which attains a local maximum at ) it holds
– A lower semicontinuous function is a viscosity supersolution of in if, for every point in and every smooth function which touches from below at (and for which attains a local minimum at ) it holds
– A continuous function is a viscosity supersolution to if it is both a viscosity supersolution and a viscosity subsolution.
For and , we denote by the open ball of radius centred at . We also set for brevity .
Lemma 6 (First eigenvalue of the ball).
For any , we have
where denotes the first zero of the Bessel function , for (and the constant is defined in (4)).
Set . We first prove that . By definition, this amounts to show that problem (1) admits a positive viscosity subsolution when . We search for a radial solution and make the ansatz . In terms of the function , problem (1) can be written as (see [KKK])
For the left hand side in the differential equation is just the classical Laplacian, evaluated in polar coordinates for . For other it can be interpreted as a linear Laplacian in a fractional dimension. This was done in [KKK], and a full spectrum and orthonormal system of radial eigenfunctions was derived. The first eigenfunction is a (positive) multiple of . This function is positive in .
Finally, let us show that the equality holds. For this we use an idea from [MPR2], there given for . Assume by contradiction that . Choose such that , and let be a positive solution to problem
Then the function defined on by if and otherwise turns out to satisfy in and on . In view of Remark 5 (i) and (ii), the operator satisfies the assumptions of the comparison result stated in [BiDe2006, Theorem 1.1]. We infer that in , a contradiction. ∎
Lemma 7 (Positivity of the eigenvalue).
For every open bounded domain , we have .
From its definition, it readily follows that is monotone decreasing under domain inclusion, i.e. if . In particular, for every open bounded domain , we have , where . Invoking Lemma 6, we obtain the positivity of . ∎
In the following Lemma we do not assume differentiability of on the boundary. Nevertheless we can bound the difference quotient in interior normal direction from below.
Lemma 8 (Hopf type Lemma).
Assume that satisfies a uniform interior sphere condition, and let be a positive viscosity supersolution of in such that on . Then there exists a constant such that for any
Here denotes the unit outer normal to ,
This follows from realizing that the normalized -Laplacian satisfies the assumptions in [BDL, Theorem 1]. ∎
Proof of Theorem 1. Let and be two positive eigenfunctions associated with . Inspired by the appendix in [Sak] we set
Clearly, we have
We claim that and are strictly positive. Indeed, the functions and are of class up to the boundary (see [BirDem2010, Proposition 3.5] or [APR17, Theorem 1.1]). Then, applying Lemma 8 to and , we see that
Hence, for small enough, is strictly negative on , so that there exists and a neighbourhood of such that in . It follows that
Thus . Arguing in the same way with and interchanged we obtain , and our claim is proved.
Now, to obtain the result, we are going to show that there exists a neighbourhood of such that
This implies in and, in view of the condition in , . The latter equality, combined with (10), implies in as required.
Let us show how to obtain the first equality in (12), the derivation of the second one is completely analogous.
By the regularity of , its unit outer normal can be extended to a smooth unit vector field, still denoted by , defined in an open connected neighbourhood of . Then, by (11) and the regularity of and on , we infer that there exist and an open connected neighbourhood of such that
This implies first of all that the PDE solved by and is nondegenerate in , which in turn, by standard elliptic regularity (see [GT]) yields that and are of class in . Moreover, from the inequality
we infer that
where is the linear operator defined by
In particular, since
and, from (13),
we see that is uniformly elliptic in the connected set . Then, to achieve our proof, it is enough to show that there exists some point where the function vanishes. Indeed, if this is the case, we have:
By the strong maximum principle for uniformly elliptic operators [GT, Theorem 3.5], it will follow that in as required. We point out that, without the connectedness of (and hence of ), the two equalities in (19) might be obtained in two, a priori distinct, connected components of , and this would not be sufficient to infer that and are proportional.
To conclude, let us now show that vanishes at some point in . As an intermediate step we notice that the function must vanish at some point in . Otherwise, we would have:
By applying Hopf’s boundary point lemma for uniformly elliptic operators [GT, Lemma 3.4], we infer that on . By continuity, this inequality, combined with the strict one in that we are assuming by contradiction, implies that in for some . But this contradicts the definition of .
Now, we choose an open bounded set with smooth boundary such that
We assert that there is a point where vanishes (and this point does the job since ). Assume the contrary. Then by continuity we have on for some . Then the two functions and satisfy
In view of Lemma 7, the continuous function is strictly positive in . Now we can apply the comparison principle proved in [LuWang2008, Thm. 2.4], and we infer that
In particular, since contains the point , we have
which gives a contradiction since . ∎
In order to prove Theorem 3, we need some preliminary results.
Let be a positive eigenfunction associated with . The approximations of via supremal convolution are defined for by
Let us start with a preliminary lemma in which we recall some basic well-known properties of the functions . To fix our setting let us define
then for every the supremum in (14) is attained at a point . Thus, setting
so that by definition
In what follows, we shall always assume that is small enough to have . Moreover, let us define
is semiconvex in ;
is a viscosity sub-solution to in ;
as , converge to uniformly in . Hence and converges to in Hausdorff distance;
as , locally uniformly in .
(i) We have , where is the infimal convolution defined by
From [CaSi, Proposition 2.1.5], it readily follows that is semiconcave on , and hence that is semiconvex on .
(ii) The notion of of viscosity subsolution according to Definition 4 can be reformulated by asking that, for every and every in the second order superjet (classically defined as in [CIL]), it holds
Then, in order to prove (ii), it is enough to show that, for every fixed point , any pair belongs to for some other point . In fact, the so-called magic properties of supremal convolution (cf. [CIL, Lemma A.5]) assert precisely that any belongs to , where is a point at which the supremum which defines is attained. Since , it holds .
(iii) For these convergence properties we refer to [CaSi, Thm. 3.5.8], [CFd, Lemma 4].
(iv) Since , this property follows from [CF7, Lemma 10]. ∎
and, for , let be the concave envelope of on the set
being the convex envelope of . Then:
is locally in ;
at any such that , it holds ;
is a viscosity sub-solution to in .
We observe that by [CaSi, Prop.2.1.12] and Lemma 9(i) also is semiconvex. Statements (i) and (ii) follow now from [CDDM, Lemma 5] since, for every fixed , the function satisfies the assumptions of such result on the convex domain .
Statement (iii) follows from part (iii) in Lemma 9 above, combined with the fact that, if a smooth function touches from above at , the smooth function touches from above at . ∎
Proof of Theorem 3. Throughout the proof we write for brevity in place of . Set
By direct computation in polar coordinates, the value of is given by
On the other hand, a natural idea in order to estimate (and hence ) in terms of the measure of , is to apply the change of variables formula to the map , with and being a positive eigenfunction associated with .
This is suggested by the fact that, as one can easily check, is a viscosity solution to
combined with the observation that maps onto , namely
Indeed, for every , the minimum over of the function is necessarily attained a point lying in the interior of (since on