Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations
We investigate quantitative properties of nonnegative solutions to the semilinear diffusion equation , posed in a bounded domain with appropriate homogeneous Dirichlet or outer boundary conditions. The operator may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian () in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity is increasing and looks like a power function , with .
The aim of this paper is to show sharp quantitative boundary estimates based on a new iteration process. We also prove that, in the interior, solutions are Hölder continuous and even classical (when the operator allows for it). In addition, we get Hölder continuity up to the boundary.
Particularly interesting is the behaviour of solution when the number goes below the exponent corresponding to the Hölder regularity of the first eigenfunction . Indeed a change of boundary regularity happens in the different regimes , and in particular a logarithmic correction appears in the “critical” case .
For instance, in the case of the spectral fractional Laplacian, this surprising boundary behaviour appears in the range .
Keywords. Nonlocal equations of elliptic type, nonlinear elliptic equations, bounded domains, a priori estimates, positivity, boundary behavior, regularity, Harnack inequalities.
Mathematics Subject Classification. 35B45, 35B65, 35J61, 35K67.
Matteo Bonforte. Departamento de Matemáticas, Universidad
Autónoma de Madrid,
Campus de Cantoblanco, 28049 Madrid, Spain. e-mail address: email@example.com
Alessio Figalli. ETH Zürich, Department of Mathematics,
8092 Zürich, Switzerland. E-mail: firstname.lastname@example.org
Juan Luis Vázquez. Departamento de Matemáticas, Universidad
Autónoma de Madrid,
Campus de Cantoblanco, 28049 Madrid, Spain. e-mail address: email@example.com
In this paper we address the question of obtaining a priori estimates, positivity, upper and lower boundary behaviour, Harnack inequalities, and regularity for nonnegative solutions to Semilinear Elliptic Equations of the form
where is a bounded domain with smooth boundary, , is a monotone nondecreasing function with , and is a linear operator, possibly of nonlocal type (the basic examples being the fractional Laplacian operators, but the classical Laplacian operator is also included). Since the problem is posed in a bounded domain we need boundary conditions, or exterior conditions in the nonlocal case, that we assume of Dirichlet type and will be included in the functional definition of operator . This theory covers a quite large class of local and nonlocal operators and nonlinearities. The operators include the three most common choices of fractional Laplacian operator with Dirichlet conditions but also many other operators that are described in Section 2, see also [9, 6]. In fact, the interest of the theory we develop lies in the wide applicability. The problem is posed in the context of weak dual solutions, which has been proven to be very convenient for the parabolic theory, and is also convenient in the elliptic case.
The focus of the paper is obtaining a priori estimates and regularity. The a priori estimates are upper bounds for solutions of both signs and lower bounds for nonnegative solutions. A basic principle in the paper is that sharp boundary estimates may depend not only on but also on the behaviour of the nonlinearity near . For this reason we assume that the nonlinearity looks like a power with linear or sublinear growth, namely for some when , and in that case we identify the range of parameters where the more complicated behaviour happens.
We point out that, for nonnegative solutions, our quantitative inequalities produce sharp behaviour in the interior and near the boundary, both in the case (the eigenvalue problem) and when , (the sublinear problem). Our upper and lower bounds will be formulated in terms of the first eigenfunction of , that under our assumptions will behave like for a certain characteristic power , cf. Section 2. This constant plays a big role in the theory.
Apart from its own interest, the motivation for this paper comes from companion papers, [9, 6]. In  a theory for a general class of nonnegative very weak solutions of the parabolic equation is built, while in  we address the parabolic regularity theory: positivity, sharp boundary behaviour, Harnack inequalities, sharp Hölder continuity and higher regularity. The proof of such parabolic results relies in part on the elliptic counterparts contained in this paper.
In this paper we concentrate the efforts in the study of the sublinear case , since we are motivated by the study of the Porous Medium Equation of the companion paper , see also Subsection 6.1.1. The boundary behaviour when is indeed the same as for .
Notation. Let us indicate here some notation of general use. The symbol will always denote . We also use the notation whenever there exist constants such that . We use the symbols and . We will always consider bounded domains with smooth boundary, at least . The question of possible lower regularity of the boundary is not addressed here.
2 Basic assumptions and notation
In view of the close relation of this study with the parabolic problem, most of the assumptions on the class of operators are the same as in  and . We list them for definiteness and we refer to the references for comments and explanations.
Basic assumptions on . The linear operator is assumed to be densely defined and sub-Markovian, more precisely satisfying (A1) and (A2) below:
is -accretive on ,
If then .
The latter can be equivalently written as
If is a maximal monotone graph in with , , , , , a.e., then
Such assumptions are the starting hypotheses proposed in the paper  in order to deal with the parabolic problem . Further theory depends on finer properties of the representation kernel of , as follows.
Assumptions on . In other to prove our quantitative estimates, we need to be more specific about operator . Besides satisfying (A1) and (A2), we will assume that it has a left-inverse with a kernel such that
and that moreover satisfies at least one of the following estimates, for some :
- There exists a constant such that for a.e. :
- There exist constants , such that for a.e. :
where we adopt the notation . Hypothesis (K2) introduces an exponent , which is a characteristic of the operator and will play a big role in the results. Notice that defining an inverse operator implies that we are taking into account the Dirichlet boundary conditions.
- The lower bound of assumption (K2) is weaker than the best known estimate on the Green function for many examples under consideration; a stronger inequality holds in many cases:
The role of the first eigenfunction of . Under the assumption (K1) it is possible to show that the operator has a first nonnegative and bounded eigenfunction , satisfying for some , cf. Proposition 5.1. As a consequence of (K2), we show in Proposition 5.3 that the first eigenfunction satisfies
hence it encodes the parameter , which takes care of describing the boundary behaviour, as first noticed in .
We will also show that all possible eigenfunctions of satisfy the bound , cf. Proposition 5.4. Recall that we are assuming that the boundary of the domain is smooth enough, for instance .
In view of (2.1), we can rewrite (K2) and (K4) in the following equivalent forms: There exist constants , such that for a.e. :
2.1 Main Examples
The theory applies to a number of operators, mainly nonlocal but also local. We will just list the main cases with some comments, since we have already presented a detailed exposition in [9, 6] that applies here. In all the examples below, the operators satisfy assumptions and and .
As far as fractional Laplacians are concerned, there are at least three different and non-equivalent operators when working on bounded domains, that we call Restricted Fractional Laplacian (RFL) , the Spectral Fractional Laplacian (SFL) and the Censored Fractional Laplacian (CFL), see Section 3 of  and Section 2.1 of . A good functional setup both for the SFL and the RFL in the framework of fractional Sobolev spaces can be found in .
For the application of our results to these cases, it is important to recall that for the RFL , for the CFL and , while for SFL and .
There are a number of other operators to which our theory applies: (i) Fractional operators with more general kernels of RFL and CFL type, under some assumptions on the kernel; (ii) Spectral powers of uniformly elliptic operators with coefficients; (iii) Sums of two fractional operators; (iv) Sum of the Laplacian and a nonlocal operator of Lévy-type; (v) Schrödinger equations for non-symmetric diffusions; (vi) Gradient perturbation of restricted fractional Laplacians; (vii) Relativistic stable processes, and many other examples more. These examples are presented in detail in Section 3 of  and Section 10 of . Finally, it is worth mentioning that our arguments readily extend to operators on manifolds for which the required bounds on hold.
3 Outline of the paper and main results
In this section we give a overview of the results that we obtain in this paper. Although the first two examples (the linear problem and the eigenvalue problem) are easier and rather standard, some of the results proved in these settings are preparatory for the semilinear problem , which is the main focus of this paper. In addition, since we could not find a precise reference for (i) and (ii) below in our generality, we present all the details.
(i) The linear equation. We consider the linear problem with with , and we show that nonnegative solutions behave at the boundary as follows
(ii) Eigenvalue problem. We prove a set of a priori estimates for the eigenfunctions, i.e. solutions the Dirichlet problem for the equation . We first prove that, under assumption (K1), eigenfunctions exist and are bounded, see Proposition 5.1 and Lemma 5.2. Then, under assumption (K2), we show the boundary estimates
see Section 5 for more details and results. Boundary estimates have been proven in various settings, especially for the common fractional operators (RFL and SFL), see for instance [4, 7, 13, 18, 17, 20, 23, 28, 29, 30, 31, 33].
(iii) Semilinear equations. This is the core of the paper, and our main result concerns sharp boundary behaviour. In Section 6 we show that all nonnegative solutions to the semilinear equation (1.1) with , , satisfy the following sharp estimates whenever :
Here depend only on , and the exponent is given by
Note that (i.e., it is independent of ) whenever . In particular the “exceptional value” does not appear neither when or when , nor in the case of the RFL or CFL. See also the survey . When (i.e. in the limit case ) a logarithmic correction appear, and we prove the following sharp estimate:
(iv) Regularity. In Section 7 we prove that, both in the linear and semilinear case, solutions are Hölder continuous and even classical in the interior (whenever the operator allows it). In addition we prove that they are Hölder continuous up to the boundary with a sharp exponent. Regularity estimates have been extensively studied: as far as interior Hölder regularity is concerned, see for instance [18, 25, 2, 15, 16, 32, 34]; for boundary regularity see [17, 23, 28, 29, 30, 31]; for interior Schauder estimates, see [3, 22].
Remark. The results apply without changes in dimension when .
Method and generality. The usual approach to prove a priori estimates for both linear and semilinear equations, relies De Giorgi-Nash-Moser technique, exploiting energy estimates, and Sobolev and Stroock-Varopoulos inequalities. In addition, extension methods à la Caffarelli-Silvestre  turn out to be very useful. However, due to the generality of the class of operators considered here, such extension is not always possible. Hence, we develop a new approach where we concentrate on the properties of the properties of the Green function of . In particular, once good linear estimates for the Green function are known, we proceed through a delicate iteration process to establish sharp boundary behaviour of solutions even in a nonlinear setting, see Propositions 5.3 and 6.5, and Lemmata 6.6, 6.7 and 6.8.
4 The linear problem. Potential and boundary estimates.
In this section we prove estimates on the boundary behaviour of solutions to the linear elliptic problem with zero Dirichlet boundary conditions
The solution to this problem is given by the representation formula
whenever with . This representation formula is compatible with the concept of weak dual solution that we shall use in the semilinear problem, see Section 6; this can be easily seen by using the definition of weak dual solution and approximating the Green function by means of admissible test functions, analogously to what is done in Subsection 6.3. In the case of SFL and/or of powers of elliptic operators with continuous coefficient, boundary estimates were obtained in [17, 18, 33], and for RFL and CFL see [25, 27] and references therein. See also Section 3.3 of  for more examples and references.
The main result of this section is the following theorem.
Let be the kernel of , and assume . Let be a weak dual solution of the Dirichlet Problem (4.1), corresponding to with . Then there exist positive constants depending on such that the following estimates hold true
where , and is defined as follows:
Remark on the existence of eigenfunctions. Under assumption (K1) on the kernel of we have existence of a positive and bounded eigenfunction , see Subsections 5.1 and 5.2; if we further assume (K2) then , cf. Subsection 5.3 for further details.
The proof of the theorem is a simple consequence of the following Lemma
Lemma 4.2 (Green function estimates I)
Let be the kernel of , and assume that holds. Then, for all , there exist a constant such that
Moreover, if (K2) holds, then for the same range of there exists a constant such that, for all ,
where is defined as in (4.4). Finally, for all , implies that
The constants , , depend only on , and have an explicit expression given in the proof.
Proof of Lemma 4.2. We split the proof in three steps.
Step 1. Proof of estimate (4.5). As consequence of assumption (K1) we obtain
where we used that and the notation (recall that by assumption ).
We next prove the upper bounds of inequality (4.6). Let us fix , and define
so that for any we have .
Notice that it is not restrictive to assume , since we are focusing here on the boundary behaviour, i.e. when (note that when we already have estimates (4.5)).
Recall now the upper part of (K2) estimates, that can be rewritten in the form
We consider three cases, depending whether is positive, negative, or zero.
- We first analyze the case when ; recalling that ,we have
- Next we analyze the case when ; using again that , we get
- Finally we analyze the case when ; again since , it holds
(note that, since we are assuming , ). The proof of the upper bound (4.6) is now complete.
Step 3. Proof of estimates (4.7). For all , and , the lower bound in (K2) implies
5 The eigenvalue Problem
In this section we will focus the attention on the eigenvalue problem
for . It is clear, by standard Spectral theory, that the eigenelements of and are the same. We hence focus our study on the “dual” problem for .
We are going to prove first that assumption (K1) is sufficient to ensure that the self-adjoint operator is compact, hence it possesses a discrete spectrum. Then we show that eigenfunctions are bounded. Finally, as a consequence of the stronger assumption (K2), we will obtain the sharp boundary behaviour of the first positive eigenfunction , and also optimal boundary estimates for all the other eigenfunctions, namely we prove also that .
5.1 Compactness and existence of eigenfunctions.
Let be the kernel of , and assume that holds. Under this assumption we show that is compact, hence it has a discrete spectrum.
Assume that satisfies and , and that its inverse satisfies . Then the operator is compact. As a consequence, possesses a discrete spectrum, denoted by , with as . Moreover, there exists a first eigenfunction and a first positive eigenvalue , such that
As a consequence, the following Poincaré inequality holds:
Proof of Proposition 5.1. The proof is divided in several steps. We first prove that the self-adjoint operator is bounded.
Step 1. Boundedness of . We shall prove the following inequality: there exists a constant such that for all we have
For this, we have
The last inequality holds because we know that
where is the inverse of , the transpose isomorphism of the restriction operator ; we recall that , so that gives the isomorphism
We refer to  for further details, see also Section 7.7 of .
Next we recall the Poincaré inequality that holds for the Restricted Fractional Laplacian, cf. [4, 19, 31] and also [7, 9]. For all such that we have that there exists a constant such that
We apply the above inequality to , to get
Step 2. The Rayleigh quotient is bounded below: Poincaré inequality. We can compute
Step 3. Compactness. Fix small and set
Note that, by (K1), we can bound
Thus, for all and all we have
Now, by Young’s convolution inequality, the last two terms can be bounded by
Also, by Hölder inequality we have
Note that, because of , is bounded and therefore it belongs to . Thus, for fixed, it holds that as , therefore
Recalling (5.9), this proves that
and since is arbitrary we obtain
Since is linear, thanks Riesz-Fréchet-Kolmogorov Theorem we have proved that the image of any ball in is compact in with respect to the strong topology. Hence the operator is compact and has a discrete spectrum.
5.2 Boundedness of eigenfunctions.
We now show that, under the only assumption (K1), all the eigenfunctions are bounded, namely there exists depending only on and , such that
Recall that we are considering eigenfunctions normalized in . The key point to obtain such bounds is that the absolute value of eigenfunctions satisfies an integral inequality:
Thus satisfies the hypothesis of Lemma 5.2 below.
Assume that satisfies and , and that its inverse satisfies . If is nonnegative and satisfies
then there exists a constant such that the following sharp upper bound holds true: