Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds
-
,
and
Abstract
Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds,
are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed.
Sharp assumptions ensuring
1 Introduction and main results
Let Ω be a connected open set in an n-dimensional Riemannian manifold
where
is also admissible, provided that
We are concerned with eigenfunctions of the p-Laplace operator in Ω, subject to homogeneous Neumann boundary conditions on
for some
if
A unified definition of an eigenfunction u of the problems under consideration amounts to requiring that
for some
Classical variational methods ensure that the eigenvalue problems in question do admit non-trivial (i.e. non-constant)
eigenfunctions under suitable assumptions on Ω; see, e.g., [37]. This is the case
if Ω has a compact closure and a regular boundary – a Lipschitz domain, for instance.
The same conclusion holds if
The present paper is aimed at offering minimal assumptions on the geometry of Ω for any eigenfunction of problem (1.2) to belong to
for some constant c depending on Ω, γ and p.
Estimates in
for some constant c depending on Ω, γ, p and q, are also established under slightly weaker conditions on Ω.
The description of the geometry of Ω adopted in our results calls into play certain functions defined in terms of inequalities of geometric-functional nature for subsets of Ω. They are the isoperimetric function and the p-isocapacitary function of Ω.
The isoperimetric function
for every measurable set
Let us emphasize that, in view of our applications, only the asymptotic behavior of the isoperimetric function
Here, the relation “O near zero” between two functions means that they are bounded by each other, up to positive multiplicative constants, for small values of their argument.
The behavior near 0 is also the only piece of information about
the p-isocapacitary function which is needed for our purposes. This function is denoted by
true for every measurable set
If Ω is sufficiently regular, as specified in connection with equation (1.6) for instance, then
near 0.
More details can be found in the next section. Let us just mention here that the regularity of a domain Ω affects the decay of the functions
The isoperimetric function and the p-isocapacitary function were introduced in the papers [40, 41] to characterize the domains Ω in
In particular, the special choice
All criteria that will be proposed are invariant under the replacement of
means that
The conditions in terms of the isoperimetric function
Theorem 1.1 (Bounds for eigenfunctions via
λ
Ω
).
Assume that
Assume that
(1.9)Then
(1.10)for every
Assume that
(1.11)where
(1.12)and inequality ( 1.3 ) holds for some constant
Assumptions (1.9) and (1.11) are optimal, in a sense specified in the next theorem, for (1.10) and (1.12), respectively, to hold
in classes of sets Ω with a prescribed decay of
Such an assumption is
consistent with the fact that, as noticed above,
A manifold of revolution.
Theorem 1.2 (Sharpness of bounds via
λ
Ω
).
Let
Given any
(1.14)and the p -Laplacian on
Let
(1.15)but
(1.16)Assume in addition that condition ( 1.13 ) is in force. Then there exists an n -dimensional Riemannian manifold
and such that the p -Laplacian on
Although the isocapacitary function has a less transparent geometric meaning than the isoperimetric function, and its behavior can be more difficult to detect, it is in a sense
more appropriate in the framework at hand. Its
use provides yet finer conditions for
A manifold with a family of clustering submanifolds.
Theorem 1.3 (Bounds for eigenfunctions via
ν
Ω
,
p
).
Assume that
Assume that
(1.17)Then
for every
Assume that
(1.18)Then
and inequality ( 1.3 ) holds for some constant
The sharpness of condition (1.18) will be exhibited via manifolds
or
These requirements reflect the fact that the behavior of
Theorem 1.4 (Sharpness of bounds via
ν
Ω
,
p
).
Let
Given any
(1.21)and the p -Laplacian on
Let
(1.22)but
(1.23)Assume in addition that ν satisfies the
(1.24)and such that the p -Laplacian on
Although the existence of eigenfunctions is not a main focus of this paper, we conclude this section by pointing out that it is ensured under the assumptions of Theorem 1.1 (i), and even under those of Theorem 1.3 (i).
Theorem 1.5 (Existence of eigenfunctions).
Assume that
Theorem 1.5 follows from an application of the Ljusternik–Schnirelman variational principle as in [37], thanks to the compactness of the embedding
2 Background
Let Ω be an open set in an n-dimensional Riemannian manifold
where
Assume that
of Ω is defined by
Obviously, the function
In particular, if Ω is connected, then
as shown via an analogous argument as in [46, Lemma 3.2.4].
The Sobolev space
and is endowed with the norm
Here,
The homogeneous Sobolev space
If
the set Ω is connected, and ω is an open set
such that
Note that replacing ω by another set with the same properties results in an equivalent norm.
The isocapacitary function
Each function
In view of a classical result in potential theory (see, e.g., [39, Corollary 2.25]), we adopt
the following definition of capacity of a condenser. Let
Also, we define
for every measurable set
The p-isocapacitary function
of Ω is then given by
The function
If
Hence, in particular,
provided that Ω is connected.
When
as shown by an analogous argument to the one in [46, Lemma 2.2.5].
For any measurable function u on Ω, we define the distribution function
Note that here
The median of u is defined by
Since u and
for every
Given
where the representative of u appearing on the right-hand side is the one which renders the coarea formula true. Via (a version on manifolds of) [46, Lemma 2.2.2/1], one can deduce that, if
then
3 Proofs of Theorems 1.1 and 1.3
Our main task in the present section is to establish Theorem (1.3). Theorem 1.1 will then easily follow, thanks to inequality (2.3).
The proof of Theorem (1.3) (ii) relies upon an analysis of an integral equation fulfilled by the signed rearrangement of any eigenfunction of problem (1.2). This integral equation is derived after obtaining an equation involving integrals of the eigenfunction over its level sets. Assumption (1.18) is the piece of information which, through inequality (2.6), enables us to deduce the existence and uniqueness of solutions to the integral equation in suitable spaces.
The proof of Theorem (1.3) (i) makes use of an iteration argument, which in turn rests on the Sobolev-type inequality contained in the following lemma. The inequality in question is standard in regular domains. The objective of the lemma is to show that it is also supported by domains which merely satisfy assumption (1.17).
Lemma 3.1.
Let
for every
Proof.
Fix any
Let us set
Clearly,
for every
Owing to the monotonicity of capacity,
Let
for
From inequalities (3.3)–(3.5), one can infer that there exists a constant c such that
Consequently,
and, trivially,
Inequalities (3.2), (3.6) and (3.7) imply that
for some constant c.
Now, let
for some constant c and for every
We are now in a position to prove our criteria for bounds of eigenfunctions to problem (1.2).
Proof of Theorem 1.3.
(i) Let u be an eigenfunction of problem (1.2) and let
Trivially,
On the other hand,
Thus,
for some constant
In order to estimate the right-hand side of equation (3.10), we observe that
Multiplying through inequality (3.10) by
Hence, owing to equation (3.13),
Choose ε in a such a way that
for some constant
for some constant
This shows that
for some constant
Thus,
(ii) Assume that u is an eigenfunction of problem (1.2), and choose its representative which supports the coarea formula for Sobolev functions.
Given
for
Consider the function
As recalled in equation (2.4), the function
is locally absolutely continuous, since, thanks to the coarea formula,
Being the composition of monotone locally absolutely continuous functions, the function V is also locally absolutely continuous, and
Here, and in what follows, the superscript “
for a.e.
From equations (3.17) and (3.18), one infers that
where
Let
Set
and observe that
and
By setting, for simplicity,
Let us choose
for
Our aim is now to prove that the equation
has a solution
We preliminarily observe that, by Fubini’s theorem and inequality (2.6),
Hence, owing to assumption (1.18), given
We claim that
for
Assume now that inequality (3.24) holds for some
namely inequality (3.24) with k replaced by
We next
distinguish the cases when
Assume first that
for
Inequality (3.26) is thus verified for
that is, inequality (3.26) with k replaced by
Assume now that
for some constants
In this case, one has that
for
where the second inequality holds by (3.28), the third by (3.25), the fourth by (3.29), the fifth by the fact that
Inequality (3.26) when
Now, we already know that equation (3.22) admits a solution
If
Hence, owing to Minkowski’s integral inequality and to inequality (3.23),
which is a contradiction since
Suppose next that
We begin by showing that
Actually, given any
On the other hand, Minkowski’s integral inequality tells us that
Moreover,
for
Coupling inequality (3.34) with (3.35) yields
From inequalities (3.33) and (3.36), one infers that
whence inequality (3.32) follows.
Next, observe that
The following chain holds:
where the first inequality is due to (3.28), the second to Hölder’s inequality, the third to (3.29) and the fourth to (3.32). Combining inequalities (3.37) and (3.38), and an application of Minkowski’s integral inequality as in (3.31), enable one to deduce that
Inequality (3.39) yields a contradiction, provided that δ is chosen small enough.
We have therefore shown that, for sufficiently small ε, the function
It remains to prove inequality (1.3). To this end, from equation (3.21)
one can deduce that, for every
Choose
Therefore,
Owing to the monotonicity of
whereas if
Since
The same argument, applied to
Altogether, we conclude that
whence inequality (1.3) follows. ∎
Proof of Theorem 1.1.
(i) By assumption (1.9), for every
if
Owing to the arbitrariness of ε, passing to the limit as
(ii) Inequality (2.3) and Fubini’s theorem ensure that
Thereby, assumption (1.11) implies that equation (1.18) is fulfilled as well. The conclusion hence follows via Theorem 1.3 (ii). ∎
4 Sharpness
The sharpness of the results from Theorems 1.1 and 1.3 will be demonstrated in our proofs of Theorems 1.2 and 1.4 via model “manifolds of revolution”, patterned as in Figure 1 of Section 1, and defined as follows.
Let
Given
Here,
for any integrable function
We shall make use of functions
Moreover, the p-Laplace operator takes the form
Thus, equation (1.1) on
The membership of u in
Thus, u is an eigenfunction of problem (1.2) if it satisfies condition (4.5) and
for every locally absolutely continuous function
It will be convenient to perform a change of variables, in order to get rid of the coefficient
where
where we have set
equation (4.4) turns into
where we have set
A locally absolutely continuous function
for every locally absolutely continuous function
We introduce now a few notations to be employed in what follows. Let I be an interval of the form
endowed with the norm
Moreover, we define the Sobolev space
equipped with the norm
Conditions on the weight function η for the embedding
to be compact are given in the following proposition.
Proposition 4.1.
Let
If
Then embedding (4.11) is compact.
Proof.
Assume that
Fix
and
Pick a compactly supported continuously differentiable function
By setting
Since
Thus, as a consequence of standard weighted Hardy-type inequalities
(see, e.g., [41, Theorems 1.3.2/2 and 1.3.2/3]), inequalities (4.14) and (4.15)
ensure that there exists a constant
and
Hence,
Now, consider any bounded sequence
for some constants
On the other hand, owing to the compactness of the embedding
The following theorem extends the results of [18, Theorem 4.1 and Corollary 4.2] to the case when
Theorem 4.2.
Let
Set
and let
be the function defined by
and such that
The metric of the n -dimensional manifold of revolution
(4.22)and
The following conditions are equivalent:
The following conditions are equivalent:
The construction of the manifolds of revolution provided by the following proposition relies on Theorem 4.2.
Proposition 4.3.
Let
Proof.
To begin with, recall that the
Let
Hence,
Define the function
Owing to the monotonicity of the function ϑ, we have that
Moreover,
Hence, via equations (4.23) and (4.24) and the
Define now the function
From equations (4.24) and (4.25), we obtain that
When
and infer that
From either equations (4.27) and (1.19), or equations (4.29) and (1.20), one deduces that,
for every
Furthermore, since
property (4.24) ensures that
for any given
Let us next notice that
Indeed, if
and the latter holds, owing to assumptions (1.22) and (1.23).
If
for some non-decreasing function
which holds thanks to assumptions (1.22) and (1.23), and to the monotonicity of the function ς. Consequently, equation (4.32) holds for every
By equations (4.30)–(4.32), the conclusion follows from [18, Proposition 4.3 ] and Theorem 4.2. ∎
Proof of Theorem 1.4.
(i) Given
Consider first the case when
The function η is defined as follows. Let
Let
solves equation (4.8) in
Thus, the function v, defined by
solves equation (4.8) in
Hence, the function v, given by
is a solution to (4.8) in
Moreover, v is convex in a left neighborhood of 1 and in a right neighborhood of -1, whereas it is concave in a right neighborhood of 1 and in a left neighborhood of -1.
Finally, in a neighborhood
for
is a convex solution to (4.8) in
One can verify that, if β is sufficiently large,
Thereby, the function η can be continued to the whole of
One can verify that, if
Now, define the function
and the function
Thus,
One has that
Let us show that the function φ satisfies assumptions (4.1), (4.2) and (4.18)–(4.20).
Assumptions (4.1) and (4.18) are satisfied by the very definition of φ. This definition also tells us that
Therefore, (4.19) and (4.20) hold as well, and
and
Assume next that
Hence, in the same interval the function v, given by
solves equation (4.8).
In the interval
The function
and the function
The function
The conclusion then follows as in the case when
(ii) Let φ be a function as in the definition of manifolds of revolution introduced at the beginning of the present section.
By Proposition 4.3, if ν is as in the statement, then the function φ can be chosen in such a way that the associated n-dimensional manifold of revolution
and
Now, recall that the function φ satisfies condition (4.2). Hence,
Moreover,
since
Owing to Theorem 4.2 (ii), condition (4.40) is equivalent to
and, by Theorem 4.2 (iii), condition (4.41) is equivalent to
The conclusion will follow if we exhibit a number
Let ψ and η be the functions defined in terms of φ, p and n as in (4.6) and (4.7), respectively. Owing to condition (4.43),
the function ψ fulfills
is compact, where I denotes either
The existence of an eigenfunction of problem (4.8) could hence be established via the general Ljusternik–Schnirelman principle, as hinted at the end of Section 1. However, we also give a direct, more elementary proof, exploiting the one-dimensional nature of the problem at hand. Let J be the functional given by
for
In particular, by embedding (4.47),
It remains to show that the function v
fulfills the Euler–Lagrange equation (4.10) for all test functions
Consider, for the time being, test functions
is convex and tends to
i.e. condition (4.48) is actually satisfied with v replaced by
Next, there exists
Since we are assuming that the function ϕ is bounded, there exist a sequence
As a consequence of the minimizing property of the function v, one can thus infer that
Equation (4.10) hence follows under the assumption
To verify this assertion, we first show that the space
Inasmuch as
On the other hand,
the space
It remains to show that any eigenfunction v of problem (4.8) is unbounded. By [33, Theorem 3],
condition (4.12) entails that equation (4.8) is nonoscillatory at infinity, and hence that every solution has constant sign at infinity. Thus, we may assume that
and hence v is concave near
Integration of equation (4.8) and this limit yield
Hence, there exists
Integration of this inequality over
This is impossible, since condition (4.45), rewritten in terms of the function η, reads
The proof is complete. ∎
Proof of Theorem 1.2.
(i) Let
(ii) Let
5 Applications
We conclude with applications of our results to some special
instances. The first three examples are concerned with problem (1.2) in customary classes of sets
5.1 Hölder domains
Consider the eigenvalue problem (1.2)
in a connected bounded open set
near 0, for some positive constant c. The first and third lines of equation (5.1) follow via the Sobolev–Poincaré embedding of [35, Theorem] and the equivalence of Sobolev embeddings and isocapacitary inequalities [46, Theorem 6.4.3/2]. The second one can be established via a variant in the proof of [35, Theorem]. Hence, Theorem 1.3 implies that any eigenfunction of problem (1.2) is bounded in Ω.
Let us also mention that
for some positive constant c. This inequality is a consequence of
the equivalence of the Sobolev embedding [35, Theorem] with
5.2 Cusp-shaped domains
Here we deal with cusp-shaped sets of the form (see Figure 3)
where
A cusp-shaped domain.
By [46, Example 6.3.6/1],
Thus,
The boundedness of all eigenfunctions of problem (1.2) hence follows, via Theorem 1.3.
The same conclusion can be derived from Theorem 1.1 and the inequality
which holds by [46, Example 5.3.3/1].
5.3 γ-John domains
A bounded open set
If
then the Sobolev–Poincaré inequality from [31, Theorem 2.3] and its equivalence to the isocapacitary inequality [46, Theorem 6.4.3/2] ensure that
Theorem 1.3 then enables one to infer that every eigenfunction of problem (1.2) in Ω is bounded.
5.4 A family of manifolds of revolution with borderline decay
Consider a one-parameter family of manifolds of revolution
is such that
and fulfills the assumptions of Theorem 4.2. This theorem tells us that
and
An application of Theorem 1.3 (i) ensures, via (5.3),
that all eigenfunctions of problem (1.2), with
On the other hand, from Theorem 1.3 (ii) and equation (5.3) one can infer that the relevant eigenfunctions are bounded under the more stringent assumption that
The same conclusions can be derived via Theorem 1.1. Thus, like for any other manifold of revolution of the kind considered in Theorem 4.2, isoperimetric and isocapacitary methods lead to equivalent results for this family of noncompact manifolds.
In both cases, the existence of eigenfunctions is guaranteed by Theorem 1.5.
5.5 A family of manifolds with clustering submanifolds
Here, we are concerned with a class of noncompact surfaces
Let us emphasize that the submanifolds
The isoperimetric and isocapacitary functions of
Proposition 5.1.
Let
for some
If
then
(5.4)If
then
(5.5)
Owing to equation (5.5), one can derive the following conclusions from Theorem 1.3,
involving the isocapacitary function
Then any eigenfunction of problem (1.2) with
then any eigenfunction is in fact bounded.
Conditions (5.6) and (5.7) are weaker than parallel conditions which are obtained from an application of Theorem 1.1 and equation (5.4), and read
and
respectively. For instance, if
then (5.6) and (5.7) amount to
Compliance with ethical standards.
Conflict of interest: The authors declare that they have no conflict of interest.
Funding source: Ministero dell’Istruzione, dell’Università e della Ricerca
Award Identifier / Grant number: 2017AYM8XW
Award Identifier / Grant number: 201758MTR2
Funding statement: This research was partly funded by: (I) Research Project 2017AYM8XW of the Italian Ministry of Education, University and Research (MIUR) Prin 2017 “Nonlinear Differential Problems via Variational, Topological and Set-valued Methods” (G. Barletta). (II) Research Project 201758MTR2 of the Italian Ministry of Education, University and Research (MIUR) Prin 2017 “Direct and inverse problems for partial differential equations: Theoretical aspects and applications” (A. Cianchi). (III) GNAMPA of the Italian INdAM – National Institute of High Mathematics (grant number not available) (G. Barletta and A. Cianchi).
Acknowledgements
We wish to thank the referees for their careful reading of the paper and for their valuable comments.
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Articles in the same Issue
- Frontmatter
- An inequality for the normal derivative of the Lane–Emden ground state
- Homogenization of high-contrast composites under differential constraints
- Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds
- The first Grushin eigenvalue on cartesian product domains
- Lipschitz bounds for integral functionals with (p,q)-growth conditions
- Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains
- Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets
- Functional inequalities and applications to doubly nonlinear diffusion equations
- Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case
- Stability of the ball under volume preserving fractional mean curvature flow
Articles in the same Issue
- Frontmatter
- An inequality for the normal derivative of the Lane–Emden ground state
- Homogenization of high-contrast composites under differential constraints
- Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds
- The first Grushin eigenvalue on cartesian product domains
- Lipschitz bounds for integral functionals with (p,q)-growth conditions
- Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains
- Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets
- Functional inequalities and applications to doubly nonlinear diffusion equations
- Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case
- Stability of the ball under volume preserving fractional mean curvature flow
