Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝ N
-
Anup Biswas
and
Prasun Roychowdhury
Abstract
We study the generalized eigenvalue problem in
1 Introduction
This article contributes to the study of eigenvalue problem of the form
where F is a fully nonlinear, convex, positively 1-homogeneous elliptic operator with measurable coefficients. We establish the
existence of half-(or demi-)eigenvalues and characterize the set of all eigenvalues with positive and negative
eigenfunctions. This generalizes a recent work of Berestycki and Rossi [12] which considers
linear elliptic operators. We also derive necessary and sufficient conditions for the validity of maximum
principles in
It has long been known that certain types of positively homogeneous operators possess
two principal eigenvalues (one corresponds to a positive eigenfunction and the other one
corresponds to a negative eigenfunction). In fact, it first appeared in the work of Pucci [24]
who computed these eigenvalues explicitly for certain extremal operators in the unit ball. Later
it also appeared in a work of Berestycki [8] while studying the bifurcation phenomenon for some nonlinear Sturm–Liouville problem and Berestycki referred them as half eigenvalues. In connection to this work of Berestycki,
Lions used a stochastic control approach in [22] to characterize
these eigenvalues (he called it demi-eigenvalues) of operators which are the supremum of linear operators with
All the above-mentioned works deal with bounded domains. It is then natural to ask how the eigentheory
changes for unbounded domains. In fact, the study of eigenvalue problems in
The rest of the article is organized as follows: In the next section we introduce our model and state our main results. We also motivate the model by providing a discussion in Section 2.3. Proofs of the main results are given in Section 3.
2 Statement of main results
In this section we introduce our model and state the main results. We also provide a motivation in Section 2.3 for considering these eigenvalue problems.
2.1 Model and assumptions
Let
Choosing
where
Our operator F is a Borel measurable function
F is positively 1-homogeneous in the variables
In particular,
F is convex in the variables
There exist locally bounded functions
for all
The function
Throughout this article we assume conditions (H1)–(H4) without any further mention. Also, observe that due to our hypotheses the operator F satisfies the conditions in [25] which studies the Dirichlet eigenvalue problem for F in bounded domains. Therefore the results of [25] holds for F in smooth bounded domains.
Let us now define the principal eigenvalues of F in a smooth domain
By sub- or supersolution we always mean
Using the convexity of F and [25, Proposition 4.2] it follows that
Remark 2.1.
We can replace the
2.2 Main results
We now state our main results. Most of the results obtained here are generalization of its linear counterpart in [12]. Recall from [12, Theorem 1.4] that for F linear and
Definition 2.1.
We say
We denote the collection of all eigenvalues with positive eigenfunctions by
Our first result generalizes [12, Theorem 1.4].
Theorem 2.1.
We have
It is well known that for bounded domains it is also possible to define principal eigenvalues through subsolutions (cf. [25, Theorem 1.2]). However, this situation is bit different for unbounded domains. To explain, we introduce the following quantities:
We remark that in case of bounded domains one has
and
provided we required the subsolution
(supersolution) to vanish on
Theorem 2.2.
The following hold:
We have
Suppose that
(2.1)Then we have
In view of Theorem 2.2 we see that
Example 2.1.
Consider two linear elliptic operators of the form
for
Now define a nonlinear operator
It is then easily seen that
and
Combining we obtain
Next we list a few class of operators for which these three eigenvalues coincide (compare them with [12, Theorem 1.9]). We only provide the result for “plus” eigenvalues and the analogous result for “minus” eigenvalues are easy to guess.
Theorem 2.3.
The equality
Assume that
There exists a
for some ball
Now we turn our attention towards maximum principles. It was observed in the seminal work of Berestycki, Nirenberg and Varadhan [10] that the sign of the principle eigenvalue determines the validity of maximum principles in bounded domains. Extension of this result for nonlinear operators are obtained by Quaas and Sirakov [25] and Armstrong [5]. Further generalization in smooth bounded domains for a class of degenerate, nonlinear elliptic operators are obtained by Berestycki, Capuzzo Dolcetta, Porretta and Rossi [9] and Birindelli and Demengel [13]. Recently, Berestycki and Rossi [12] establish the maximum principles in unbounded domains for linear elliptic operators. Here we extend their results to our nonlinear setting.
Definition 2.2 (Maximum principles).
We say that the operator F satisfies
we have
We say that the operator F satisfies
we have
Note that
or
Generalizing [12, Definition 1.2], we now consider the following quantities.
Definition 2.3.
Given a positive function
Our maximum principles would be established under the following growth conditions on the coefficients:
Next we state our maximum principle
Theorem 2.4.
Suppose that either (2.2) and (2.4) or (2.3) and (2.5) hold. Then the following hold:
The operator F satisfies
The operator F satisfies
As a consequence of Theorem 2.4 we obtain the following corollaries.
Corollary 2.1.
Suppose that either (2.4) or (2.5) holds. Then we have:
The operator F satisfy
The operator F satisfy
Suppose that
Corollary 2.2.
Suppose that F satisfies
Then we have
Proof.
Denote by
Hence the result follows from
Generalizing
As a necessary condition for the validity of maximum principles we deduce the following.
Theorem 2.5.
The following hold:
If F satisfies the
If F satisfies the
Finally, we discuss about simplicity of the principal eigenvalues. For linear F uniqueness of principal eigenfunctions can be established imposing Agmon’s minimal growth condition at infinity [12, Definition 8.2] on the eigenfunctions. But such criterion does not seem to work well for nonlinear F. Recently, in [3, Theorem 2.1] it is shown that Agmon’s minimal growth criterion is equivalent to monotonicity of the principal eigenvalue on the right. Our next result establish simplicity of principal eigenvalue under certain monotonicity condition of principal eigenvalue at infinity.
Theorem 2.6.
Suppose that there exists a positive
for some compact ball K and
We remark that (2.6) is equivalent to
Our next result is about simplicity of
Theorem 2.7.
Suppose that there exists a positive
for some compact ball K and
2.3 Motivation
One of the important examples of F comes from the control theory. In particular, we may consider
where α varies over some index set
Then under reasonable hypothesis, one can show that Λ is an eigenvalue of F (i.e.,
3 Proofs of main results
In this section we prove Theorems 2.1–2.7. Let us start by recalling the following
Harnack inequality from [25, Theorem 3.6] which will be crucial for our proofs. The
result in [25, Theorem 3.6] is stated for
Theorem 3.1.
Let
Then for any compact set
for some constant C dependent on
Next we prove Theorem 2.1. The idea is the following: we show using the Harnack inequality and stability
estimate that the Dirichlet principal eigenpair in
Lemma 3.1.
It holds that
Proof.
Let
Step 1. We show that
for all
Thus applying [25, Theorem 3.3], we obtain, for
Since K is arbitrary, using a standard diagonalization argument we can find a non-negative
Again, applying Theorem 3.1, we have
Step 2. We show that
Denote by
By the strong maximum principle [25, Lemma 3.1] it follows that
Clearly,
Now we continue as in Step 1 and extract a subsequence of
This gives us
Next lemma concerns the eigenvalues with negative eigenfunctions.
Lemma 3.2.
It holds that
Proof.
Idea of the proof is similar to Lemma 3.1. Let
We divide the proof of into two steps.
Step 1. We show that
Normalize each
for
Step 2. As discussed in Lemma 3.1, it is enough to show that for any
Denote by
Since G satisfies (H3) we can apply the strong maximum principle [25, Lemma 3.1] to obtain that
The following (standard) existence result will be required.
Lemma 3.3.
Suppose that
Proof.
The proof is based on monotone iteration method. See also [25, Lemma 4.3] for a similar argument.
Define the operator
Also, note that
Existence of
Applying Lemma 3.3, we obtain the following.
Theorem 3.2.
It holds that
Proof.
We divide the proof into two steps.
Step 1. We show that
where
Now we plan to find a bounded, positive solution of
where
Let
The proof of Lemma 3.3 also reveals that
Step 2. We next show that
Normalize
where
Thus, using Lemma 3.3 and the arguments of Step 1, we obtain a negative, bounded solution
This, of course, implies
Theorem 2.2 (ii) will be proved using Theorem 2.4. Thus we prove Theorem 2.4 first.
Theorem 3.3.
Suppose that either (2.2) and (2.4) or (2.3) and (2.5) hold.
Then F satisfies
Proof.
Let
Also, since
Multiplying ψ with a suitable constant, we may assume that
For this proof we follow the idea of [12, Theorem 4.2].
Choose a smooth positive function
Using (H3) and an easy computation, we obtain, for
Hence for both the cases, using (2.4) and (2.5) accordingly on
Modifying C, if required, we can assume (3.5) to hold in
Hence there exist
Let us now estimate the term
which implies
Hence for each natural number n there exist a small positive
On the other hand, using convexity of F with (3.5) and (3.6), we get
Since
Now note that
By the strong maximum principle [25, Lemma 3.1] we then obtain
Therefore,
In the same spirit of Theorem 3.3 we can also prove
Theorem 3.4.
Suppose that either (2.2) and (2.4) or (2.3) and (2.5) hold for the function β.
Then F satisfies
Proof.
As done in Theorem 3.3, we choose
Let
We need to show that
Then using convexity, (3.5) and above estimate, we obtain
in
Note that
This, of course, implies
Now we prove Theorem 2.5.
Proof of Theorem 2.5.
First we consider (i). To the contrary, suppose that
This, of course, implies
Next we consider (ii). Again, we suppose that
This gives
Now we can prove Theorem 2.2 (ii).
Theorem 3.5.
Assume that either (2.4) or (2.5) holds. Then we have
Proof.
Let us first show that
We prove the second claim. To the contrary,
suppose that there exists λ with
Our next result should be compared with [12, Theorem 7.6]. Recall that for a smooth domain Ω,
Theorem 3.6.
It holds that
Proof.
Notice that the function
Also, from definition we already have
Let us now show that the above inequality cannot be strict. That is, for every
we have
Let us first complete the proof assuming (3.9). By Morrey’s inequality we see that
From the construction we can immediately say that
for ϵ small enough, where we have again used convexity of F. This, of course, implies
To complete the proof, we only need to show (3.9). To this end, we may assume that
and
Now we would like to see if a result analogous to Theorem 3.6 holds for
It is easily seen that G is a concave operator and
Theorem 3.7.
Suppose that
Then we have
Proof.
It is easy to see that
As done in Theorem 3.6, we show that the above inequality can be strict. So we consider any
and show that
Hence repeating the arguments of Theorem 3.6, we can find
repeating a calculation analogous to Theorem 3.6, we find that for some ϵ small
To this end, we define
Proposition 3.1.
Define
We need a small lemma to prove Proposition 3.1.
Lemma 3.4.
The following hold for any smooth domain Ω:
Proof.
Part (i) follows from convexity property of F. Note that for
Now we prove Proposition 3.1
Proof of Proposition 3.1.
First consider (i). Assume that
By Theorem 3.6, we obtain
Recall the Dirichlet principal eigenfunction
for all large n,
and therefore, by strong maximum principle [25, Lemma 3.1],
Proof for (ii) would be analogous. ∎
Next we prove Theorem 2.3.
Proof of Theorem 2.3.
(i) From the definition it follows that
and then letting r towards infinity we have
Since
Applying Theorem 3.6, we obtain
(ii) Using Lemma 3.4 and the given hypothesis, we find
Hence, by Theorem 3.6, we get
(iii) We show that under the given condition we have (ii). It is enough to show that
if
on the ball
For
Since all the notions of eigenvalues of F coincide in bounded domains (cf. [25]), using (3.13) we deduce
Thus we only need to establish (3.13). For a different way to construct such subsolutions we refer to [26]. Using (H3), we see that
Given ε, we choose R such that
We now compute (3.14) in two steps.
Step 1. Suppose
Now we can choose small positive δ, independent of ε, so that
Step 2. Now we are left with the part
Using (3.15), we can choose ε small enough so that the right-hand side becomes positive.
Combining the above steps, we obtain (3.13), completing the proof of part (iii).
(iv) This follows from Theorem 3.6. Let us also provide a more direct proof.
Let
By using convexity and homogeneity, it follows that
using (3.16). Hence
Thus it remains to prove Theorems 2.6 and 2.7. Let us first attack Theorem 2.6.
Proof of Theorem 2.6.
Without any loss of generality, we assume that
Let
for large n using (2.6) and (3.17).
Thus, if w vanishes in
We now show that any other principal eigenfunction is a multiple to φ.
For η, a small positive number, we define
provided we choose n large (depending on η). Let ψ be any principal eigenfunction satisfying
Define
Then
we can choose
Thus, δ remains bounded and positive as
Furthermore, there exists
By the strong maximum principle we must have
Finally, we prove Theorem 2.7.
Proof of Theorem 2.7.
The main idea of the proof is the same as that of the proof of Theorem 2.6. Without any loss in generality, we
assume that
By Lemma 3.2,
Note that G satisfies (H1), (H2) and (H3) but it is a concave operator. So need some extra care to apply the proof of Theorem 2.6. Since F is convex, it follows from (2.7) that
As done in Theorem 2.6, using (3.19), we can normalize
It is enough to show that φ agrees with any other positive eigenfunction (up to a multiplicative constant) of G with eigenvalue 0.
Next we define
for all large n. Now consider any positive eigenfunction
and let
Then
we can
apply the maximum principle [25, Theorem 1.5] in
We conclude the paper with a remark on the eigenvalue problem in a general smooth unbounded domain.
Remark 3.1.
For the case of an unbounded domain with smooth boundary all the results developed here hold true and the proofs would be somewhat similar. As mentioned in [12], in case of general unbounded domains, one needs the boundary Harnack property to control the behavior of eigenfunctions near the boundary. For the operator F, the boundary Harnack property has been obtained recently by Armstrong, Sirakov and Smart in [7, Appendix A]. Therefore one can easily adopt the techniques of [12] along with our results to deal with general unbounded domains.
Funding statement: The research of Anup Biswas was supported in part by DST-SERB grants EMR/2016/004810 and MTR/2018/000028. Prasun Roychowdhury was supported in part by Council of Scientific & Industrial Research (File no. 09/936(0182)/2017-EMR-I).
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Articles in the same Issue
- Frontmatter
- Borderline regularity for fully nonlinear equations in Dini domains
- Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions
- Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝ N
- A note on Kazdan–Warner equation on networks
- Integral representation for energies in linear elasticity with surface discontinuities
- Minkowski inequalities and constrained inverse curvature flows in warped spaces
- Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian
- Pairings between bounded divergence-measure vector fields and BV functions
- Equivalence between distributional and viscosity solutions for the double-phase equation
- Vanishing John–Nirenberg spaces
- Extremals in nonlinear potential theory
- On the structure of divergence-free measures on ℝ2
- Solutions of the (free boundary) Reifenberg Plateau problem
- Equilibrium measure for a nonlocal dislocation energy with physical confinement
Articles in the same Issue
- Frontmatter
- Borderline regularity for fully nonlinear equations in Dini domains
- Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions
- Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝ N
- A note on Kazdan–Warner equation on networks
- Integral representation for energies in linear elasticity with surface discontinuities
- Minkowski inequalities and constrained inverse curvature flows in warped spaces
- Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian
- Pairings between bounded divergence-measure vector fields and BV functions
- Equivalence between distributional and viscosity solutions for the double-phase equation
- Vanishing John–Nirenberg spaces
- Extremals in nonlinear potential theory
- On the structure of divergence-free measures on ℝ2
- Solutions of the (free boundary) Reifenberg Plateau problem
- Equilibrium measure for a nonlocal dislocation energy with physical confinement
