Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions
In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and unfavorable regions for a species to survive. The mathematical formulation of the model leads to an indefinite weight linear eigenvalue problem in a fixed box and we consider the general case of Robin boundary conditions on . It is well known that it suffices to consider bang-bang weights taking two values of different signs, that can be parametrized by the characteristic function of the subset of on which resources are located. Therefore, the optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to , under a volume constraint. By using symmetrization techniques, as well as necessary optimality conditions, we prove new qualitative results on the solutions. Namely, we completely solve the problem in dimension 1, we prove the counter-intuitive result that the ball is almost never a solution in dimension 2 or higher, despite what suggest the numerical simulations. We also introduce a new rearrangement in the ball allowing to get a better candidate than the ball for optimality when Neumann boundary conditions are imposed. We also provide numerical illustrations of our results and of the optimal configurations.
Keywords: extremal eigenvalue problem, shape optimization, symmetrization technique.
AMS classication: 49J15, 49K20, 49R05, 49M05.
1. Introduction: Elliptic Problem with indefinite weight
1.1. The optimal design problem
In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by a model of population dynamics. The question is to determine the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. We prove new qualitative results on the optimizer, using rearrangement techniques on the one hand, first order optimality conditions on the other hand.
More precisely, the following linear eigenvalue problem with indefinite weight is formulated in :
where is a bounded domain (open and connected set) in with a Lipschitz boundary , is the outward unit normal vector on , and the weight is a bounded measurable function which changes sign in (meaning that has a measure strictly between 0 and ) and satisfies
where is a given constant.
As we are motivated by a biological problem, we focus in this article on varying sign weights , but most if not all our techniques can be applied to the case of positive , as soon as a positive principal eigenvalue exists. Additional comments on positive weights can be found in Section 2.3.
It is said that is a principal eigenvalue of (1) if the corresponding eigenfunction is positive. The existence of principal eigenvalues of (1) with respect to the parameter was discussed in [1, 8]. More precisely,
in the Dirichlet case (“”), there are exactly two principal eigenvalues , respectively associated with the eigenfunctions and satisfying
the case is similar to the Dirichlet case,
in the critical case , which corresponds to Neumann boundary conditions, there are two principal eigenvalues, and , respectively associated with the eigenfunctions and ; moreover if and only if , in which case we have
In the rest of the paper , we focus on the case which is relevant for applications in the context of species survival. Therefore, assuming that , and besides that if , there exists a unique positive principal eigenvalue for Problem (1), denoted . Moreover, rewrites also as
whenever . This has been proved for Neumann boundary conditions () in  and the reader can check that the extension to is straightforward. Moreover, is simple, the infimum is reached, the associated eigenfunctions do not change sign in , and any eigenfunction belonging to and that do not change sign is associated with .
In the Dirichlet case, where the boundary condition on is replaced by on , this formulation becomes
Indeed, according to Proposition 2 below, the Dirichlet eigenvalue can be obtained from the Robin eigenvalues by letting .
Throughout this paper, we will analyze the following optimization problem, modeling the optimal arrangement for a species to survive.
Optimal arrangement of ressources for species survival. Let be a bounded domain of . Given and if or if , we consider the optimization problem
It is well known (see for example [28, 37] and Section 2.4) that Problem (6) has a solution , and moreover there exists a measurable subset such that, up to a set of zero Lebesgue measure, there holds
In addition, one has , which is a direct consequence of a comparison principle111Indeed, by comparing the Rayleigh quotients for and , one gets One can refer for instance to [37, Lemma 2.3].. In other words, the minimizer saturates at the same time the pointwise and integral constraints on . As a consequence, the optimal design problem above can actually be rewritten as a shape optimization problem:
Shape optimization formulation of Problem (3). Using the same notations as above, let . We investigate the optimal design problem
where denotes the set of Lebesgue measurable sets such that .
Therefore and to sum up, given and , it is equivalent to choose either the parameter (with, in addition, when ) or the parameter (with if ) and the solution is then naturally a function of three parameters (either or ), once is given.
1.2. New results
In this paper, we obtain three new qualitative results on the shape optimization problem we described in the previous section.
In our first result, we provide a complete description of the optimal sets in the one-dimensional case.
Assume and without loss of generality, let us consider . Let , , and whenever or whenever . Define
if , the unique222Here the uniqueness must be understood up to some subset of zero Lebesgue measure. In other words if is optimal then the union of with any subset of zero measure is also a solution. solution of (7) is the interval of length and centered at ,
if , the solutions of (7) are exactly and ,
if , the solutions of (7) are exactly all intervals of length ,
This result is clear for (i.e. for the Dirichlet case) using symmetrization, and is proven in  for . In the more general situation , the minimizer among intervals has been computed in  when and in  for . Therefore the previous result rests upon the fact that the optimal set is an interval. We prove this in Section 3. Our method is based on a symmetrization argument, and is therefore closer to the case than the method of . We cannot use Steiner/Schwarz symmetrization since it may not decrease the gradient term (except if in which case ). Therefore we use a sort of two sided decreasing rearrangement, whose center is chosen appropriately so that symmetrized functions are still admissible, and which decreases every term in the Rayleigh quotient. Notice that even in the case , this gives a new proof of the result of , which is more straightforward.
Our second result deals with the case , and disproves the commonly stated conjecture that the ball is a minimizer for certain domains and certain values of the parameters , and . This conjecture was also suggested by numerical computations and results (see ).
We prove that the conjecture is false, except maybe for very particular choices of the parameters such as the box . In particular, if is not a ball, a minimizer for (7) cannot be a ball, whatever the value of the parameters , and are. This means in particular that the optimal set does not minimize the surface area of its boundary.
More precisely, we have the following general result.
Let , a domain of such that its boundary is connected and of class , with if , and an open subset of of measure . Assume that either or is rotationally symmetric (i.e. a union of concentric rings, whose center is denoted ) and has a finite number of connected components.
If the set is a critical point333This means that satisfies the necessary first order optimality conditions of Problem (7), in other words that is an upper level set of the eigenfunction associated with the principal eigenvalue , more precisely that there exists such that , see also Section 2.4. of the optimal design problem (7) then is a ball of center .
There exists such that if , and if solves Problem (7), then and are concentric balls.
To our best knowledge, this result is completely new, even if or . Theorem 2 lets open the issue of knowing whether the optimal configuration is rotationally symmetric.
Note that when is disconnected, it is likely that the result is not valid anymore. For instance, if is an annulus, one would expect the existence of a rotationally symmetric critical set .
The assumption on the finite number of connected components for ensures that is analytic, which is crucial in our proof. It could thus be replaced by an analyticity assumption on .
Note that our result is also interesting if is a ball. It asserts that the only rotational symmetric domain which is a candidate for optimality is the centered ball whenever the parameter is large enough. It implies in particular that an annulus cannot be a minimizer, even if .
The proof of Theorem 2 uses the first order optimality condition, namely that is constant on , to infer that is necessarily radial (i.e. is a function of ) on the whole domain . To that end, we built particular test functions that can be interpreted as angular derivatives of the function .
Then, we rewrite the problem as an optimization problem bringing into play only functions of the polar variable . This allows to conclude that must be a ball, proving that the associated eigenvalue on the largest inscribed ball and the smallest circumscribed ball are the same.
The second part of the result is proven by using a symmetrization argument, which, as for Theorem 1, works for large values of the parameter and despite the lack of the usual hypotheses for this kind of argument.
We also underline here that the converse of Theorem 2 is not true. More precisely, the radial symmetry of does not imply that a similar symmetry will hold for the minimizing set . Indeed, an analytical example which shows that a radially symmetric set cannot be a minimizer has been provided in [29, Theorem 2.5] when is a thin and large annulus, for Neumann boundary conditions. Even for Dirichlet boundary conditions, symmetry breaking can occur. In , this phenomenon is observed and explicit examples are provided for a closely related problem.
Finally, let us highlight that we prove in the second step of Theorem 2 the following interesting byproduct: among the set of rotationally symmetric open subsets of of prescribed measure, the centered ball is the only minimizer for large enough.
Let , be the -dimensional unit ball of centered at the origin, with if . Let be a rotationally symmetric and concentric open subset of of measure . Then, any eigenfunction associated with is radial. Moreover, there exists such that there holds for every , where denotes the centered ball of volume , and if and only if .
Our third result is motivated by Theorem 2 which asserts in particular that a ball is a candidate for optimality only if itself is a concentric ball. It remains to decide, in the case where is a ball, whether the centered ball is optimal or not. In the case it is actually the case (classically, by using the so-called Schwarz symmetrization), but we expect that it is not the case for every values of . We prove that the centered disk is not optimal in the case (Neumann boundary condition) and for .
Let , , and . Assume and is the disk of radius centered at the origin. Then the centered ball of volume is not a minimizer for Problem (7).
This result is a particular case of the more general result stated in Theorem 5: we use a non-local deformation that decreases strongly the value of , more precisely if is a centered ball (or more generally a radially symmetric set), we build a set that “sticks” on the boundary of and satisfies
We compute which yields Theorem 3. For , we have so we cannot conclude from the estimate (9). For , the situation is unclear: first, it seems our strategy cannot be adapted, even if is small, though it is reasonable to expect that the centered ball is not a solution in that case. For large, we do not know whether the situation is similar to the 1-dimensional case (that is there exists , possibly depending on and , such that for the solution is a centered ball, in other words the same as if ) or if it can be proven that the centered ball is a solution only if .
The article is organized as follows: in Section 2, we provide some explanations about the biological model motivating our study, as well as a short survey on several existing results related to the problem we investigate and similar ones. Section 3 is devoted to the proof of Theorem 1, solving completely Problem (7) in the case where with Robin boundary conditions. The whole section 4 is devoted to proving Theorem 2. Finally, in Section 5, we provide qualitative properties of the minimizers of Problem (7) in the particular cases where Neumann boundary conditions are considered, is a -orthotope or a two dimensional euclidean disk. In this last case, we prove in Theorem 5 a quantitative estimate showing symmetry-breaking for the minimizers. This allows in particular to derive Theorem 3. All these results are illustrated by numerical simulations.
2. Preliminaries and State of the art
In this section, we gather several known facts about Problem (6), from the biological motivation of the model to deep and technical results about minimizers, mainly for two reasons. First of all, we will use several known results in our proofs, therefore we want to recall them for the convenience of the reader. Second, we want to highlight the novelty of our results, even when we will be led to state results for certain choices of parameters (such as , , the dimension, and so on).
2.1. Biological model
The main biological motivation for studying extremal properties of the principal eigenvalue with respect to the weight comes from the diffusive logistic equation
introduced in , where represents the density of a species at location and time , and is a positive parameter. Concerning the boundary conditions on , the case corresponds to Neumann or no-flux boundary condition, meaning that the boundary acts as a barrier, i.e. any individual reaching the boundary returns to the interior. The case corresponds to Dirichlet conditions and may be interpreted as a deadly boundary, i.e. the exterior environment is completely hostile and any individual reaching the boundary dies. For intermediate values , we are in the situation where the domain is surrounded by a partially inhospitable region, where inhospitableness grows with . The weight represents the intrinsic growth rate of species: it is positive in the favorable part of habitat () and negative in the unfavorable one (). The integral of over measures the total resources in a spatially heterogeneous environment.
The logistic equation (10) plays an important role in studying the effects of dispersal and spatial heterogeneity in population dynamics; see, e.g. [9, 10, 12] and the references therein. It is known that if , then uniformly in as for all non-negative and non-trivial initial data, i.e., the species go to extinction; if, however, , then uniformly in as , where is the unique positive steady solution of (10), i.e., the species survives.
Since the species can be maintained if and only if , we see that the smaller is, the more likely the species can survive. With this in mind, the following question was raised and addressed by Cantrell and Cosner in [9, 10]: among all functions , which will yield the smallest principal eigenvalue , whenever it exists? From the biological point of view, finding such a minimizing function is equivalent to determining the optimal spatial arrangement of the favorable and unfavorable patches of the environment for species to survive. This issue is important for public policy decisions on conservation of species with limited resources.
2.2. Other formulation
In this section, we address a closely related optimal design problem. Let such that . For , the classical reaction-diffusion model in homogeneous environments of Fisher, Kolmogorov et al. [20, 2] generalizes as:
where represents the population density at time and position . The function stands for the intrinsic grow rate of the species whereas the function is the susceptibility to crowding and is chosen in and such that .
It is notable that this condition does not depend on the function .
The principal eigenvalue of (12) is unique, nonnegative and given by
Moreover, is simple, and the infimum is attained only by associated eigenfunctions that do not change sign in .
As previously, a similar analysis of the biological model leads to the study of the following optimal design problem.
Optimal rearrangement of species problem, equivalent formulation. Given such that and , we are interested in
Note that Problems (6) and (14) have been considered independently in the literature on optimal arrangement of ressources for species survival. In  these two similar problems have been investigated and it is shown that they are equivalent in a sense recalled below. The main difference with our case is, roughly speaking, that the weight is positive in . However as far as the equivalence of (6) and (14) is concerned, the proof is the same and we recall the result here, for mainly two reasons: firstly because it allows us to use certain results from both literatures, and secondly, because while our statements and proofs deal with formulation (6), our new results are actually also valid for solutions of (14).
([13, Theorem 13], Equivalence between the two formulations) Let a bounded domain and .
2.3. About the class of admissible weights
In this section, we gather several comments related to the choice of constraints on the weight : the pointwise one and the global (mass constraint) one.
As a first remark, there exists a wide literature concerning problems similar to (6), where one aims at minimizing the first eigenvalue of the operator where denotes the Dirichlet-Laplacian operator, with respect to functions satisfying the pointwise constraint a.e. in with as well as a global integral constraint. Such problems are motivated by optimal design issues with respect to structural eigenvalues. We refer for instance to [18, 32, 33, 34, 35, 36] where Dirichlet boundary conditions are considered, and to [24, Chapter 9] for a survey on these problems.
We also mention the case of non-homogeneous membranes, similar to (6) with Neumann boundary conditions, but with the positive weight (also called density) and without a mass constraint on the weight, see [22, Theorem 1.4.2] and [3, 4]. See also  for a similar problem in the context of Riemannian manifolds with a mass preservation constraint. Notice however that in the case where the weight is positive and where Neumann boundary conditions are imposed on the eigenfunction , there is no positive principal eigenvalue: indeed, assuming there exists a positive eigenfunction associated to , we obtain by integration by parts that
which is a contradiction since this yields if .
All the results presented in this section and in Section 2.4 (i.e. the monotonicity of eigenvalues, the bang-bang property of minimizers) were established in  and  in one dimension for Neumann conditions (i.e. ) for Problem (6), and in  for periodic boundary conditions for Problem (14). We claim that they can be straightforwardly extended to Robin conditions. Therefore we do not reproduce here the proof but rather refer to [37, Theorem 1.1] or [41, Appendix A] for details. We also mention , for an extension of these results to principal eigenvalues associated to the one dimensional -Laplacian operator.
Finally, concerning the constraint , or equivalently (see Section 1.1), we claim that it is active and therefore, it is similar to deal with the same optimal design problem where the inequality constraint is replaced by the equality one
Indeed, it is a consequence of the comparison principle (see [37, Lemma 2.3])
This comparison principle is obtained in an elementary way, by comparing the Rayleigh quotient for and (resp. and ).
2.4. First order optimality conditions and bang-bang property of minimizers
The minimizing set is a level surface of the principal eigenfunction . Indeed, this is proved in  for Dirichlet boundary conditions but the arguments can be straightforwardly extended to Robin boundary conditions. Let us briefly recall the main steps. We denote by the eigenfunction associated to the minimal principal eigenvalue . First, note that the optimal design problem , has a solution given by , where . This is the so-called “baththub principle”, see e.g. [39, Theorem 1]. Using a direct comparison argument and arguing by contradiction, one shows that and therefore, is a minimizing set for Problem (6). Next, it is standard, as that a.e. on , which implies a.e. on , which is impossible if is not negligible. Hence, one infers that up to a set of measure zero.
2.5. Regularity theory
Proving the regularity of the free boundary is a very difficult question in general. It follows from classical elliptic regularity that the principal eigenfunction is for every . Hence, as up to a set of Lebesgue measure zero (see Section 2.4), the boundary is -smooth at any point where and therefore, using a bootstrap argument, one infers the local analytic regularity of in this case, see . The regularity problem is thus reduced to the one of the degeneracy of the eigenfunction on its level line .
When Dirichlet conditions are imposed on the boundary (in other words, when “”), then it has been proved in , when , that , that does not hit the boundary and consists of finitely many disjoint, simple and closed real-analytic curves. We believe that the arguments involved in  could be extended to our framework. Indeed, for Neumann boundary conditions, we expect to hit the boundary, but most of the arguments of  are local and do not see the Dirichlet boundary conditions. However, this is not the main topic of the present paper and we will thus leave this question open, since we do not need these results to obtain Theorems 1, 2 and 3. In higher dimensions, it is only known that is smooth up to a closed set of Hausdorff dimension . However, the situation is much more complicated since one could expect, as for some other free boundary problems, the emergence of stable singularities.
2.6. Dirichlet boundary conditions
When Dirichlet boundary conditions are imposed on , it is possible to derive qualitative properties on from that of . Symmetrization techniques apply () and allow to show that, if is symmetric and convex with respect to some hyperplane, then so is . Notice nevertheless that symmetry breaking phenomenon might arise if the convexity property with respect to the hyperplane is not satisfied, for example for annuli or dumbbells .
For particular sets of parameters, it has been proved that is connected if is simply connected and is convex if is convex .
We also mention  where the authors investigate the related optimization problem of locating an obstacle of given shape, namely a ball, inside a domain so that the lowest eigenvalue of the Dirichlet-Laplacian operator is minimized. Numerous symmetry results have been derived from the moving plane method.
It is interesting to note that the Dirichlet case can be recovered by letting the parameter tend to .
As an infimum of real affine functions, is concave. Moreover, taking and comparing the principal eigenvalues thanks to the Rayleigh definition (3) shows that is monotone non-decreasing. Let us prove that is moreover increasing. For that purpose, we argue by contradiction and consider and two eigenpairs solving (3) with respectively and and such that . Since the mapping is concave and non-decreasing, we infer that is constant on . Notice that
whenever does not vanish identically on . Hence, it follows that necessarily on and , . Hence, by simplicity of the principal eigenvalue, one has also on . In particular, according to the Robin boundary condition on , one also has . One gets a contradiction by observing that the Neumann principal eigenfunction is positive in (see e.g. ).
Choosing test functions in in (3) proves that . As a monotone non-decreasing bounded function, has a finite limit as . Hence, the family of eigenpairs , where denotes a solution of (3), maximizes as . Assuming moreover that (by homogeneity of the Rayleigh quotient), one has and therefore showing that the -norm of is uniformly bounded with respect to . Thus, there exists with such that, up to a subsequence, converges to weakly in and strongly in , by using the Rellich-Kondrachov Theorem. Writing then
and letting tend to shows that , or in other words that . Moreover, by weak convergence of in , there holds
Since , one has necessarily and therefore the quotient above is well defined. The expected conclusion follows.
2.7. Periodic boundary conditions
When is embedded with periodic boundary conditions, then the optimal set is Steiner symmetric, that is, convex and symmetric with respect to all the hyperplanes . It follows that the restriction of the set to is a minimizer for problem (14) for the set embedded with Neumann boundary conditions . Hence, there is a bijection between the minimization problem for the periodic principal eigenvalue in the square and for the Neumann principal eigenvalue in the restricted square , and thus, one derives easily corollaries of our results to the periodic framework.
We will apply our results to the framework where is a rectangle in dimension in Section 5.1, and prove in particular that cannot have a part of its boundary with constant curvature when .
As the minimizing set is a level set of the principal eigenfunction , thresholding methods based on the so-called bathtub principle provide very fast algorithms in order to compute . Indeed, starting with an arbitrary set of measure and computing the eigenfunction associated with the principal eigenvalue, one then defines recursively where is a positive number chosen in such a way that ([13, 28]), and so on. Note that is unique since one shows in particular that the level sets have zero Lebesgue measure for every . This algorithm converges to a critical point in a small number of iterations for reasonable parameters.
This method has been used to compute optimal sets for Dirichlet boundary conditions , Neumann boundary conditions in squares and ellipses , Robin boundary conditions in squares . In general these solutions look like stripes, balls, or complementary of balls, depending on the parameters. However, as already underlined above, very few analytical results confirmed these simulations. In particular, it was not clear whether balls could be minimizing sets or not and we provide a negative answer to this problem in the present paper. We also refer to Section 5 where we provide numerical investigations and illustrations of our results in the particular cases where is either the two-dimensional unit square or the unit disc.
3. The one-dimensional case (Proof of Theorem 1)
In , the authors solve the one-dimensional version of Problem (6) in the particular case where . The proof methodology was to first exhibit the solutions when the sets are intervals, and then to show that the optimizers must be intervals. Subsequently, in , a much simpler proof of the same result was obtained using increasing or decreasing rearrangements. Here we generalize this result to the case of Robin boundary conditions, in other words for every . Throughout this section we will assume without loss of generality that .
Take , and . Let with , in order to highlight the dependence of the eigenvalue on . The function is symmetric with respect to , and moreover, with defined in (8), we have:
if , then is strictly decreasing on ; in particular its minimum is reached for .
if , then is strictly increasing on ; in particular its minimum is reached for and .
if , then is constant and any is a global minimum for .
for given parameters and . The next result is devoted to proving this claim.
If , then any optimal set for Problem (7) is an interval.
Proof of Proposition 4..
Assume in a first step that . As recalled in the sections 1 and 2.4, we know there exists solution of (7). We denote by the associated weight and the corresponding eigenfunction, solution of (1). The Rayleigh quotient defined by (4) rewrites in this case
one has . Since we have and . We also have and , so reaches his maximum inside . Let
Introduce the function defined on by
where denotes the monotone increasing rearrangement of on and
denotes the monotone decreasing rearrangement of on (see for instance ). Thanks to the choice of , it is clear that this symmetrization does not introduce discontinuities, and more precisely that .
Similarly, we also introduce the rearranged weight , defined by
with the same notations as previously. In other words, (resp. ) is bang-bang, equal to or almost everywhere, such that and (resp. and ).
We aim at proving now that is admissible for the optimal design problem (6), that is an admissible test function of the Rayleigh quotient , and that the Rayleigh quotient decreases for the symmetrization, in other words that .
First it is clear that . Indeed, every integral on can be written as the sum of the integral on and , and we use the equimeasurability property of monotone symmetrizations on each of these intervals. Then, writing and using Hardy-Littlewood inequality (again on and ), we also have , and the expected conclusion follows.
Also, we easily see that
Using now Polyà’s inequality twice: and , we obtain that
Investigating the equality case of Polyà’s inequality, it follows that first increases up to its maximal value and then decreases on (see for example  and references therein). As a consequence, since is a upper level-set of , it is necessarily an interval, which concludes the proof.
Finally, the case is simpler. A direct adaptation of this proof shows that the claim remains valid in that case.
In this section, we investigate the optimality of a ball, or more generally of rotationally symmetric sets (i.e. a union of concentric rings). This question naturally arises, in particular according to numerical results in  for dimension , where for some values of the parameters, the solutions seem to take the shape of a ball; also in the case of periodic boundary condition, H. Berestycki stated that the solution might be a ball, for some values of the volume constraint . We prove that for every and , the ball is not optimal, except possibly if itself is a ball having the same center as .
Let us first assume that is a critical point of the optimal design problem (6). We recall that “ is a critical point of the optimal design problem (6)” means that satisfies the necessary first order optimality conditions (see Section 2.4), in other words that the associated principal eigenfunction is constant on .
Part 1: the function is radially symmetric. Assume that is rotationally symmetric, and is a critical point. In what follows, we will denote by the principal eigenvalue and by the associated eigenfunction that solves (1). We follow the following steps: we first prove that is radial in , then is also radial in , and we conclude that must be a centered ball. Generalizing the methods used in [25, 26], take with , and define
This function lies in for all and a straightforward computation yields that verifies in the sense of distributions the partial differential equations
Let us now prove that vanishes in .
As a consequence and according to (15), since the set is rotationally symmetric, the function vanishes on . We now assume and leave the case at the end of this part. Moreover, since is rotationally symmetric, a straightforward computation shows that,