On the global bifurcation diagram of the Gel’fand problem
For domains of first kind [BLin3, CCL] we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gel’fand problem crossing the origin. At least to our knowledge this is the first result about the exact monotonicity of the branch of non-minimal solutions which is not just concerned with radial solutions [Kor] and/or with symmetric domains [HK]. Toward our goal we parametrize the branch not by the -norm of the solutions but by the energy of the associated mean field problem. The proof relies on a carefully modified spectral analysis of mean field type equations.
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Keywords: Global bifurcation, Gelfand problem, Mean field equation.
We are concerned with the global bifurcation diagram of solutions of,
where is any smooth, open and bounded domain and . Problem , also known as the Gel’fand problem [Gel], arises in many applications, such as for example the thermal ignition of gases [beb], the dynamics of self-interacting particles [bav] and of chemiotaxis aggregation [suzC], the statistical mechanics of point vortices [clmp2] and of self-gravitating objects with cylindrical symmetries [KLB], [Os]. A basic question seems unanswered so far concerning the qualitative behavior of the unbounded continuum [Rab] of solutions of ,
emanating from the origin . Under which conditions on , takes the same form (see Fig. 1) as that corresponding to a disk ? Here and in this case solutions are radial [gnn] and can be evaluated explicitly, see for example [suz].
For a general domain, classical results [KK] show that is a monotonic increasing function of as far as the first eigenvalue of the associated linearized problem is strictly positive. This is the so called branch of minimal solutions which is well understood and naturally described in the
-plane for . In fact, this nice behavior breaks down
at some positive value , which is the least upper bound of those such that has solutions for any , see
[KC]. In particular, the first eigenvalue of the linearized problem at is zero and is known to be a bending point, see [suzC].
The situation for non-minimal solutions on , i.e. after the first bending point, is more involved. Besides classical facts (see [Ban1], [suz] and [Lions] for a complete discussion and references), at least to our knowledge there are only two rather general result concerning this problem. The first one is in [suz], where it is shown that, for a certain class of simply connected domains (see Remark 1.3 below), is a smooth curve with only one bending point which makes 1-point blow up [NS90] as . The second one is an unpublished but straightforward corollary of some results in [BLin3], which implies that the result in [suz] holds even for a large class of domains with holes, see Remark 1.3 below. Therefore it is natural in this situation to guess that takes the same form shown in Fig. 1. However, a subtle point arises since after the bending, even in this situation where we know that the bifurcation curve cannot bend back to the right, the first eigenvalue of the linearized equation for is negative (while the second eigenvalue is positive [suz, BLin3]) and then the monotonicity of , depending in a tricky way on certain changing sign quantities, cannot be taken for granted. Of course, one expects that, as is the case for radial solutions, is still a monotone function of , and well known pointwise estimates ([CLin1, yy]) for blow up solutions suggest that this is the case for small enough. At least to our knowledge there are no proofs of this fact. Actually, under some symmetry assumptions on , by the result in [HK], for any there exists one and only one solution of such that and is a smooth curve which contains all solutions of . Therefore, for these symmetric domains, the results in [suz] and [HK] together show that indeed is monotone along , which answer to our question in this case. Finally, it seems that there is no gain in replacing with other seemingly natural quantities, as for example or either , as one is always left with the problem of possibly sign changing terms.
We attack this problem here by a new method based on some ideas recently introduced in [Bons], that is, to parametrize the curve not by but with the energy of the associated mean field equation, naturally arising in the Onsager description of two-dimensional turbulence [clmp2]. Our proof works for domains of ”first kind” (see Definition 1.2 below), initially introduced in statistical mechanics [clmp2] and then sharpened and fully characterized in [CCL] and in [BLin3]. For any pair solving we define,
where is the Green function for with Dirichlet boundary conditions. For later use let us set,
We say that , where is an open set and is a Banach space, is real analytic [but] if for each it admits a power series expansion in , which is totally convergent in the -norm in a suitable neighborhood of . Our main result is the following:
Let be a domain of first kind (see Definition 1.2). For any , the equation
admits a unique solution .
In particular, and are real analytic functions of and is a parametrization of . Finally has the following properties:
as , , as ;
for , , for , where is uniquely defined by , that is .
Therefore, on domains of first kind, we have found a global parametrization of ,
which takes the form depicted in Fig. 2, as claimed. At least to our knowledge, this is the first global result (i.e. including non-minimal solutions) about the monotonicity of the bifurcation diagram for an elliptic equation with superlinear growth in dimension , which is not just concerned with radially symmetric solutions [Kor], [KLO], and/or with domains sharing some kind of symmetries [HK]. The situation in higher dimension is far more subtle, see for example [JL] and more recently [Dan2, Dan3].
Although its definition in the context of the Gel’fand problem looks rather unnatural, it turns out that indeed is just the energy in the Onsager mean field model, when expressed as a function of . For we consider the mean field equation [clmp2],
To simplify the notations, for fixed , here and in the rest of this paper we set,
The energy associated with the density is by definition (see [clmp2]),
Clearly, if solves , then is a solution of , for some which satisfies,
and then from we see that,
Therefore, in particular the energy is just the Dirichlet energy of when expressed in terms of .
The uniqueness of solutions of is easy to prove for , see Proposition 2.1 below. On the other side it is well known that admits a unique solution for any (see [suz] for simply connected domains and [BLin3] for general domains). The existence/non existence problem for is a subtle issue, since is critical with respect to the Moser-Trudinger inequality [moser]. For a complete discussion of this problem see [CCL] for simply connected domains and [BLin3] for general domains. This is why we need the following definition,
A domain is of first kind if has no solution for and is of second kind otherwise.
It has been proved in [CCL] and [BLin3] that is of first kind if and only if the unique solutions of for blow up [bm, NS90], as . As a consequence, it turns out that the domains considered in Theorem 2 in [suz] are exactly the simply connected domains of first kind. However the estimates about the first and second eigenvalue in [suz] has been extended in [BLin3] to the case of any connected domain. As an immediate consequence, the result in [suz] hold for any domain of first kind. It is well known that any disk is of first kind. Actually any regular polygon is of first kind, see [CCL]. It has been proved in [BdM2] that there exists a universal constant such that any convex domain whose isoperimetric ratio is larger than is of second kind. If is a rectangle of sides , then there exists such that is of first kind if and only if , see [CCL]. If with , , then there exists such that is of first kind for any , see [BLin3]. We refer to [CCL] and [BLin3] for other equivalent characterizations of domains of first kind and a complete discussion concerning this point. Among other things it is proved there that the set of domains of first kind is closed in the -topology. Therefore, for example domains of first kind need not be symmetric.
It is well known [CLin2],[KMdP] that if is not simply connected, then there exist countably many distinct families of blow up solutions of as . These families of blow up solutions are unique [BJLY3] and nondegenerate [GOS] under suitable nondegeneracy assumptions. Therefore, it is not true in general that contains all solution of . However, as remarked above, this is the case for domains with certain symmetries, see [HK].
Let us sketch the argument of the proof. We describe by using the unique solutions of for . Any solution of yield a solution of with . This might seem not a good point of view since, being a constrained problem, the associated linearized equation is more difficult to analyze. In general the first eigenvalue is not simple and the first eigenfunctions may change sign, see [B2] or the Appendix below for an example of this sort. Therefore, even if we know that (see [CCL] and [BLin3]), it is not clear how to use this information to establish the monotonicity of . However, as recently observed in [Bons], it is possible to modify the standard spectral theory relative to the linearization of , see (2.2) below, and build a complete set of eigenfunctions which span the space of functions of vanishing mean.
All the main steps of the proof rely on this modified spectral setting. The modified first eigenvalue is strictly positive for and the set of solutions of can be shown to be locally an analytic curve with no bifurcation points. A crucial fact which follows from is that the energy is a (real analytic) strictly increasing function of . Therefore, to understand the monotonicity of as a function of , it is enough to evaluate the sign of . The difficult part is to show that there exists such that . Indeed, a major problem arises in the proof of along the non-minimal branch of solutions, that is for . We solve this problem by two non-trivial facts about the quantity which controls the sign of , which is in Lemma LABEL:lem1.3 below. First of all, still by exploiting , we obtain a version of the maximum principle based on the sign of . This is not at all obvious since does not imply that the maximum principle holds for the linearized problem relative to . The second fact is a remarkable formula for : it turns out that it satisfies a first order non-homogeneous O.D.E. (see (LABEL:cruxg) below). In particular we conclude that changes sign only once in .
The description of on domains of second kind is more difficult. Indeed, solutions on a certain part of correspond to solutions of with , a region where solutions are not unique and is not anymore positive. Therefore it is not easy to understand the monotonicity of , see [Bons] for further details concerning this point.
This paper is organized as follows. In section 2 we first introduce the modified spectral analysis and collect some important preliminary results concerning the linearized mean field equation. In section 3 as a first step toward the proof of the main result we deduce the monotonicity of the energy (3.1). Then, in section LABEL:sec:proof.thm we prove the main Theorem 1.1 postponing the proof of the key Lemma LABEL:lem1.3 to section LABEL:sec:proof.lem. Finally, further discussion on the modified spectral analysis with an explicit example is given in the Appendix.
2. Spectral decomposition of linearized mean field type equations
For any solving , we introduce the linearized operator,
where we recall that
We say that is an eigenvalue of the linearized operator (2.1) if the equation,
admits a non-trivial weak solution . This definition of the eigenvalues requires some comments. Let be a fixed solution of and let us define,
where denotes the scalar product . Let us also define
By the results in [bm, yy] and standard elliptic regularity theory we see that is a smooth function for , so these definitions are well posed. Clearly is an Hilbert space, and, since , then it is not difficult to see that the linear operator,
is self-adjoint and compact. As a consequence, standard results concerning the spectral decomposition of self-adjoint, compact operators on Hilbert spaces show that is the Hilbertian direct sum of the eigenfunctions of , which can be represented as
for some , . In fact is an eigenfunction whose eigenvalue is , that is,
if and only if the function ,
is in and weakly solves,
At this point, standard arguments in the calculus of variations show that,
The ratio in the right hand side of (2.5) is well defined in by the Jensen inequality, which implies that
where the equality holds if and only if a.e. in . Higher eigenvalues are defined inductively via the variational problems,
Obviously a base of can be constructed to satisfy,
The eigenvalues form a countable nondecreasing sequence , where in particular,
the last inequality being an immediate consequence of (2.5).
Obviously, by the Fredholm alternative,
Finally, any admits the Fourier series expansion,
If we let be the standard first eigenvalue defined by,
then we see that, as far as , we have,
and then, in view of the results in [suz], as later improved in [CCL] and [BLin3], we have the following:
Theorem A.([BLin3], [CCL], [suz]) Let be a smooth and bounded domain. For any , the first eigenvalue of (2.1) is strictly positive, that is, , for each .
from an open interval is said to be analytic if for each , admits a power series expansion as a function of , which is totally convergent in the -norm in a suitable neighborhood of . We have the following,
Let be a smooth and bounded domain.
For any it holds .
If solves for some , then there exists an open neighborhood of in the product topology, , such that the set of solutions of in is an analytic curve , for suitable neighborhoods of in and of in .
For any there exists a unique solution of .
If solves for some , then Theorem A, (2.12) and (2.8) immediately imply that , which proves . As a consequence of , (2.9) and the analytic implicit function theorem [bdt, but], it
can be shown by standard arguments that holds, see Lemma 2.4 in [Bons]. Therefore also is proved.
Putting , then solutions of are critical points of
The existence of at least one minimizer for for any is a well known consequence of the Moser-Trudinger inequality [moser]. The uniqueness of solutions of for is trivial, while for it has been proved in [BLin3], [CCL], [suz]. For the second variation of is strictly positive definite. Indeed, by (2.11), the first eigenvalue of the associated quadratic form, which is , is strictly positive. Therefore admits at most one critical point, which concludes the proof of .
3. Monotonicity of the energy
The proof of Theorem 1.1 relies on Proposition 2.1 and on the following result about the global structure of the bifurcation diagram of for and the asymptotic behavior of the energy. This is an extension of some ideas first introduced in [Bons].
Let be a domain of first kind. For each there exists a unique solution of and
is an analytic curve in . In particular
is real analytic in and satisfies