A logistic equation with nonlocal interactions

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a L\'evy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: bounded domains, periodic environments, and transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.


Introduction
In this paper we study stationary solutions for a logistic equation. The solution u can be interpreted, from the point of view of mathematical biology, as the density of a population living in some environment Ω ⊆ R n .
In the classical logistic equation (see e.g. [Ver45,MP12,PR20]), the population is supposed to increase proportionally to the resource of the environment (the growing effect being modeled by a nonnegative function σ) and to die when the resources get extinguished (the dying effect being described by a nonnegative function µ). The population is also assumed to diffuse randomly (the random diffusion being modeled by the Laplace operator). These considerations lead to a detailed study of the evolution equation ∂ t u = ∆u + (σ − µu) u and to the stationary case of equilibrium solution described by the elliptic equation ∆u + (σ − µu) u = 0.
In this paper we will consider two variants of the latter equation, motivated by the nonlocal features of the population.
First of all, the diffusion operator of the population is considered to be nonlocal, that is, we replace the Gaussian diffusion by the one induced by Lévy flights. These types of nonlocal dispersal strategy have been observed in nature and may be related to optimal hunting strategies and adaptation to the environment stimulated by the natural selection, see e.g. [VAB + 96, HQD + 10] for experimental results and [Alu14] for divulgative explanations of these phenomena in popular magazines. From the mathematical point of view, taking into account this kind of nonlocal diffusion translates in our setting into the analysis of logistic equations driven by fractional Laplace operators.
Moreover, we take into account the possibility that also the increasing rate of the species has a nonlocal character. This feature is motivated in concrete cases by the fact that a population takes advantage not only of the resources that are exactly in the area in which they permanent settle, but also of the ones that are "at their reach" (say, a "giraffe's neck" effect). This nonlocal feature will be modeled for us by the convolution with an integrable kernel (from the mathematical point of view, we remark that the two types of nonlocal operators considered are very different, since the fractional Laplacian causes a loss of differentiability on the function, while the convolution produces a regularizing effect).
The precise mathematical formulation that we consider is the following. Given s ∈ (0, 1), we consider the fractional Laplacian (−∆) s u(x) := 2s (1 − s) P V R n u(x) − u(y) |x − y| n+2s dy. (1) The notation "P V " denotes, as customary, the singular integral taken in the "principal value" sense, that is The constant s (1 − s) in (1) is just a normalizing factor, to allow ourselves to consider the case s = 1 as a limit. Indeed, with this choice, for a suitable normalizing constant c ⋆ > 0, only depending on n, for any u ∈ C 2 (R n ) ∩ L ∞ (R n ). The stationary logistic equation that we study is then −(−∆) s u + (σ − µu) u + τ (J * u) = 0, where σ, µ and J are nonnegative functions, τ ≥ 0 is a constant and s ∈ (0, 1]. As usual, J * u denotes the convolution between two functions, that is, for any x ∈ R n , We also assume that the convolution kernel is even and normalized with total mass 1, that is and J(−x) = J(x) for any x ∈ R n .
(3) We consider two types of setting for our equation: the bounded domain with Dirichlet datum (corresponding to a confined environment with hostile surrounding areas) and the periodic case. These two cases will be discussed in detail in the forthcoming subsections.
1.1. Bounded domains with Dirichlet data. The environment with hostile borders is modeled in our case by the following equation: in Ω, u = 0 outside Ω, u ≥ 0 in R n . (4) We will present an existence theory for nontrivial solutions and we will compare local and nonlocal behaviors of the population, analyzing their effectiveness in terms of the resource and of the domain.
In further detail, we consider the (possibly fractional) critical Sobolev exponent 2 * s := 2n/(n − 2s) and we state a general existence result as follows: , +∞], and that (σ + τ ) 3 µ −2 ∈ L 1 (Ω). Then, there exists a solution of (4). To study the solutions obtained by Theorem 1.1 it is useful to compare them to the domain using a spectral analysis. For this, we denote by λ s (Ω) the first Dirichlet eigenvalue for (−∆) s in Ω, i.e.
and the infimum is taken under the conditions that u L 2 (R n ) = 1 and u = 0 outside Ω, if s ∈ (0, 1), and, as classical, For a detailed study of these eigenvalues (also in the nonlocal case) see for instance Appendix A in [SV13]. The existence of nontrivial solutions to (4) can be characterized in terms of these first eigenvalues: roughly speaking, when the resource σ is too small, the only solution of (4) is the one identically zero, i.e. all the population dies; viceversa, if the resource σ is large enough, there exists a positive solution.
A consequence of Theorem 1.2 is that nonlocal species can better adapt to sparse resources. For instance, there exist examples of disjoint domains Ω 1 and Ω 2 such that the resource in each single Ω i is not sufficient for the species to survive, but the combined resources in the union of the domains can be used by a nonlocal population efficiently enough. A formal statement goes as follows: Theorem 1.3. Let s ∈ (0, 1). Let Ω 1 be a domain in R n , and Ω 2 be a domain congruent to Ω 1 , with Ω 1 ∩ Ω 2 = ∅. Then, there exists σ ∈ (0, +∞) such that the only solution of is the trivial one, for any i ∈ {1, 2}, but the equation Also, in light of Theorem 1.2 it is interesting to determine for which s positive solutions of (4) may occur. Roughly speaking, when Ω is "small", the strongly diffusive species corresponding to small values of s may be favored. Viceversa, when Ω is "large", the species corresponding to small s may be annihilated. As a prototype example we present the following two results: Proposition 1.4. Let Ω be a bounded Lipschitz domain and set Then the equation
Theorem 1.5. Fix s, S ∈ (0, 1], with s < S. Let Ω be a bounded Lipschitz domain and set Let also J be a nonnegative function satisfying (2) and (3).
The biological interpretation of Theorem 1.5 is that "large" environments are "more favorable" to "local" populations (namely, the population with faster diffusion related to (−∆) s is extinguished, while the population with slower diffusion related to (−∆) S is still alive); viceversa, "small" environments are "more favorable" to "nonlocal" populations (namely, in this case it is the population with slower diffusion (−∆) S that is extinguished, while the population with faster diffusion (−∆) s persists).
Another relevant question in this framework is whether or not the population fits the resources. An easy observation is that, if τ = 0, the population never overcomes the maximal available resource. This follows from the more general result: in Ω, It is conceivable to think that large resources in a given region favor, at least locally, large density populations. We show indeed that there is a linear dependence on the largeness of the resource and the population density (independently on how large the resource is), according to the following result: in Ω, Next result stresses the fact that nonlocal populations can efficiently plan their distribution in order to consume and possibly beat the given resources in a given "strategic region" (up to a small error). That is, fixing a region of interest, say the ball B 1 , one can find a solution of a (slightly perturbed by an error ε) logistic equation in B 1 which exhausts the resources in B 1 and which vanishes outside B Rε , for some (possibly large) R ε > 1. The "strategic plan" in this framework consists in the fact that, in order for the population to consume all the given resource in B 1 , the distribution in B Rε \ B 1 must be appropriately adjusted (in particular, the logistic equation is not satisfied in B Rε \ B 1 , where the population needs to be "artificially" settled from outside). The detailed statement of such result goes as follows: Theorem 1.8. Let s ∈ (0, 1) and k ∈ N, with k ≥ 2. Assume that and that σ, µ ∈ C k (B 2 ). Fix ε ∈ (0, 1). Then, there exist a nonnegative function u ε , R ε > 2 and σ ε ∈ C k (B 1 ) such that and In light of Lemma 1.6 and Theorems 1.7 and 1.8, a relevant question is also whether or not the population can beat the resource, i.e. whether or not the set {u > σ} is void. Notice indeed that Lemma 1.6 says that, if τ = 0, this does not occur for constant resources σ. Nevertheless, when the resource is oscillatory, then this phenomenon occurs, thanks to the diffusive terms which allow the species to somewhat attains resources "from somewhere else". Namely we have the following result: for which {u > σ m } is nonvoid.
1.2. Periodic environments. We now turn our attention to a periodic environment, i.e. we suppose that σ and µ are periodic with respect to translations in Z n and we look for periodic solutions. In this framework, the equation that we take into account is We suppose here that σ and µ are bounded and periodic functions (with respect to the lattice Z n ), that µ is positive and bounded away from zero and that J is compactly supported.
In this setting, we obtain the following existence result for periodic solutions: Theorem 1.10. Assume that either σ is not identically zero or τ > 0.
Then, there exists a solution of (13).
We remark that the solutions obtained in Theorem 1.10 are in general not constant (for instance, when µ is constant and σ is not). But when both σ and µ are constant then the periodic solutions need also to be constant, according to the following result: Theorem 1.11. Let u be a positive solution of (−∆) s u = (σ −µu)u+τ (J * u) in R n . Assume that u is periodic with respect to Z n and that σ ∈ (0, +∞), µ ∈ (0, +∞) and τ ∈ [0, +∞) are all constant.

1.3.
A transmission problem. Now, inspired by the recent work in [Kri15], we consider a transmission model in which the population is made of two species (or of one population that adapts to two different environments), one with a local behavior in a domain Ω 1 , and one with a nonlocal behavior in a domain Ω 2 , with Ω 1 ∩Ω 2 = ∅. The transmission problem occurs between Ω i and its complement, for i ∈ {1, 2}, and it is modeled by positive parameters ν i . More precisely, we take two disjoint, bounded and Lipschitz domain Ω 1 and Ω 2 ⊂ R n . We define Ω := Ω 1 ∪ Ω 2 and Here, s, s 1 , In this setting, we have the following existence result: Theorem 1.12. The functional T attains its minimum among the functions u ∈ L 2 (Ω) for which and such that u = 0 a.e. outside Ω. Also, such minimizer is nonnegative.
It is worth to point out that minimizers of T satisfy the equations in the weak sense (and also pointwise, by Theorem 5.5(3) in [Kri15] and Theorem 1 in [SV14]). The biological interpretation of equation (16) is that the population has local behavior in Ω 1 , with nonlocal interactions outside Ω 1 , and a nonlocal transmission between the domains Ω 1 and Ω 2 takes place. See also [Kri15] for additional comments and motivations.
The existence/nonexistence of nontrivial solutions in dependence of the spectral analysis of the domain will be addressed in the following result. To this end, we define λ ⋆ (Ω) the first Dirichlet eigenvalue for the operator in (15). Namely, we set where and the infimum in (17) is taken under the conditions that u L 2 (R n ) = 1 and u = 0 a.e. outside Ω. In this setting, we obtain a result similar to Theorem 1.2 for the transmission problem in (15): Theorem 1.13. In the setting above, with strict inequality on a set of positive measure then (16) possesses a solution u such that u > 0 in Ω 1 ∪ Ω 2 .
1.4. Organization of the paper. The rest of the paper is organized as follows: in Section 2 we discuss the existence of a solution by energy minimization and we prove Theorem 1.1. Then, in Section 3, we discuss the qualitative properties of the solution and we present a proof of Theorem 1.2.
In Sections 4, 5 and 6 we discuss how the population adapts to the resources and we give the proof of Theorem 1.3, Proposition 1.4, Theorem 1.5, Lemma 1.6 and Theorem 1.7.
The strongly nonlocal diffusive strategy is considered in Section 7, where we prove Theorem 1.8.
The case in which the population actually beats the resource is discussed in Section 8, where Theorem 1.9 is proved.
The existence/nonexistence of nontrivial periodic solutions in a periodic environment is taken into account in Section 9 with the proofs of Theorems 1.10 and 1.11.
Then, in Section 10, we consider the transmission problem and we prove Theorems 1.12 and 1.13.

Existence theory and proof of Theorem 1.1
The proof of Theorem 1.1 is based on a minimization argument. More precisely, in order to deal with problem (4), if s ∈ (0, 1), given u ∈ L 1 loc (R n ) with u = 0 a.e. outside Ω, we consider the energy functional where Q Ω is defined in (5). When s = 1, instead we consider the standard energy functional with condition u ∈ H 1 0 (Ω). It is worth to point out that solutions of (4) are strictly positive, unless they vanish identically: Lemma 2.1. Let u be a nonnegative solution of (−∆) s u = (σ − µu) u + τ (J * u) in Ω. Then either u > 0 in Ω or it vanishes identically.
Proof. Suppose that u(z) = 0 for some z ∈ Ω and, by contradiction, that u > 0 in a set of positive measure. Then u(z + x) − u(z) = u(z + x) ≥ 0 for any x ∈ R n , and in fact strictly positive in a set of positive measure. Accordingly, (−∆) s u(z) < 0. Nevertheless, from (4), we have that which is a contradiction. Equation (4) has a variational structure, according to the following observation: Lemma 2.2. The Euler-Lagrange equation associated to the energy functional E at a nonnegative function u is (4).
Proof. We denote by If φ ∈ C ∞ 0 (Ω) and ǫ ∈ (−1, 1), we have that As a consequence, Now we recall that u and φ vanish outside Ω and we use (3) to see that Using this into (18) we obtain that With this, the case s = 1 is standard, so we consider the case s ∈ (0, 1).
which gives the desired result.
In the light of Lemma 2.2, to prove existence of solutions, it is useful to look at the minimizing problem for E. We first show the following useful inequality: Proof. By the Hölder Inequality with exponents equal to 2 and the Young Inequality for convolutions with exponents 1 and 2, we have that where (2) was also used. This shows (19).
Then the following existence result holds: Then E attains its minimum among the functions u ∈ L p (Ω) for which and such that u = 0 a.e. outside Ω.
Moreover, there exists a nonnegative minimizer. Finally, if u is such minimizer, it is a solution of (4).
Proof. We deal with the case s ∈ (0, 1), since the case s = 1 is similar, and simpler. The proof is by direct methods. First, we notice that p ∈ [2, 2 * s ) and By (19) (used here with v := u and w := u) we have that Furthermore, we use the Young Inequality, with exponents 3/2 and 3, to see that As a consequence of this and (21), This implies that for κ := (σ + τ ) 3 µ −2 L 1 (Ω) /6. So we can take a minimizing sequence u j . We may suppose that We obtain that Hence, by compactness, up to a subsequence u j converges to some u in L p (Ω) and a.e. in R n . So we recall (20) and we find that Now, by (19) with v := u j − u and w := u j we obtain Moreover, making again use of (19) with v := u and w := u j − u, we have that So, from (23), (24) and (25), we conclude that thanks to the Fatou Lemma. These inequalities imply that hence u is the desired minimum. Also, E(|u|) ≤ E(u), so we can suppose in addition that u is nonnegative. Furthermore, u is a solution of (4) thanks to Lemma 2.2.
The claim in Theorem 1.1 now follows directly from the one in Proposition 2.4.
3. Qualitative properties and proof of Theorem 1.2 The proof of Theorem 1.2 is based on energy arguments, by using the functional introduced in Section 2. The details are the following: Proof of Theorem 1.2. Assume that sup Ω σ + τ ≤ λ s (Ω). Suppose, by contradiction, that there exists a nontrivial solution to (4). Then, by Lemma 2.1, we have that u > 0 in Ω.
Therefore, using Lemma 2.3 (with v := u and w := u) and recalling (26), we see that Now, we test (4) against u itself and we use (27) to see that This is a contradiction and it establishes the first claim in Theorem 1.2. Now we show the second claim. For this, we suppose inf Ω σ ≥ λ s (Ω) with strict inequality on a set of positive measure and we remark that it is enough to show that 0 is not a minimizer. To this goal, we take e to be the first eigenfunction of (−∆) s with Dirichlet datum and ǫ > 0. We recall that e > 0 in Ω and it is bounded. Then The proof of Theorem 1.3 is based on a spectral analysis and on the use of Theorem 1.2. The details are the following.
Then the claim in Theorem 1.3 follows from Theorem 1.2.

Scaling arguments and proof of Proposition 1.4 and Theorem 1.5
The proof of Proposition 1.4 follows by a simple scaling argument, which we present here for the sake of completeness: Proof of Proposition 1.4. By scaling, we have that λ s (Ω r ) = r −2s λ s (Ω).
Also, by Theorem 1.2, a nontrivial solution exists if and only if 1 > λ s (Ω r ). These considerations imply the desired claim.
The proof of Theorem 1.5 combines scaling arguments and spectral analysis and it is presented here below.
From Theorem 1.2, we have that in this case equation (6) has a nontrivial solution, while (7) only has the trivial solution.
In this case, Theorem 1.2 gives that (7) has a nontrivial solution, while (6) has only the trivial solution.
6. Fitting the resources and proof of Lemma 1.6 and Theorem 1.7 The proof of Lemma 1.6 is a simple maximum principle, whose details are presented here below for completeness: Proof of Lemma 1.6. Suppose by contradiction that there exists which is a contradiction. Now we show that u always fits the "abundant" resources (up to a multiplicative constant): Proof. We take e o to be the first Dirichlet eigenfunction of B R . Then we have The latter quantity is negative if M ≥ M o , for large values of M o , therefore the energy of the minimizer u is negative and u is not the trivial function. Consequently, from Proposition 2.4 and Lemma 2.1, we can define So we take the first η for which a contact point in Ω occurs (of course, if ηe o ≤ u for all η > 0, we obtain the desired result by taking η as large as we wish, hence we can assume that such contact point exists). That is, we have that ηe o ≤ u and there existsx ∈ Ω such that ηe o (x) = u(x). Since e o vanishes outside B R , we have thatx ∈ B R . Therefore Accordingly, as long as M ≥ M o and M o is large enough. This says that In particular η ≥ M/(2 e o L ∞ (R n ) ) and therefore, for any x ∈ B r , Now, Theorem 1.7 follows plainly from Proposition 6.1.

7.
Fitting the resources in a nonlocal setting and proof of Theorem 1.8 Now we prove Theorem 1.8, by exploiting a result in [DSV15], joined to a minimization argument.
More precisely, we make use of Theorem 1.1 in [DSV15], which we state here for the convenience of the reader: Theorem 7.1. Fix k ∈ N. Then, given any function f ∈ C k (B 2 ) and any ε > 0, we can find R ε > 2 and a function u ε ∈ C s 0 (B Rε ) such that The details of the proof of Theorem 1.8 now go as follows: Proof of Theorem 1.8. First of all, we use Theorem 7.1 to find a function w ε and a radius R ε > 2 such that Let W ε := |w ε | and σ ε := µw ε .
Notice that for some C k > 0, possibly depending on µ C k (B 1 ) , and this proves (10) (up to renaming ε).
if we take ε > 0 small enough, therefore Accordingly, for any x ∈ B 1 , for any x ∈ B 1 . As a consequence, for any x ∈ B 1 . By (34), we get that (−∆) s W ε ∈ L ∞ (B 1 ), and consequently Now we introduce the energy functional and we aim to minimize G among all the functions that vanish outside B 1 . For this, we observe that G(0) = 0 and we take a minimizing sequence v j , namely where the infimum is taken among the functions v such that v = 0 in R n \ B 1 . We observe that, by (10), we know that inf Also, by Lemma 2.3, and, by (22) (used here with σ = 1), Using these considerations, we find that for some C ε > 0 that does not depend on j. As a consequence, This gives that v j is precompact in L 2 (B 1 ) (see e.g. Theorem 7.1 in [DNPV12]) and so we may suppose, up to a subsequence, that it converges to some v ⋆ in L 2 (B 1 ) and a.e. in R n , with v ⋆ = 0 outside B 1 . Therefore, by Fatou Lemma, Also, by weak convergence in L 2 (B 1 ), In addition, by Lemma 2.3, that are infinitesimal as j → +∞. Using this, (37), (38) and (39), we obtain that v ⋆ is a minimizer for G.
8. Beating the resources and proof of Theorem 1.9 The proof of Theorem 1.9 is based on a contradiction and limit argument.
Proof of Theorem 1.9. Let u m be the solution of (12) provided by Proposition 2.4. If the desired claim were false, we would have that u m ≤ σ m . Then Hence, using Lemma 1.6 with τ = 0, Notice that the latter quantity does not depend on m. Thus, by fractional elliptic regularity (see e.g. Proposition 1.1 in [ROS14] and Lemma 4.3 in [CS11]) we have that u m converges uniformly in Ω to some u 0 as m → 0, and u 0 solves in Ω. By Theorem 1.7, we know that u 0 > 0 in B r . In particular u 0 is not the trivial solution, and so u 0 > 0, thanks to Lemma 2.1. Then we have which is a contradiction.
9. Periodic solutions and proof of Theorems 1.10 and 1.11 To prove Theorem 1.10, we consider an auxiliary minimization problem. The functional is tailored in order to be compatible with integer translations and produce solutions of (13) via an Euler-Lagrange equation, tested against periodic test functions.
Here, we assume that J is supported in some ball B ρ and we let We define the energy functional Then we consider the space X of functions v ∈ L 2 (Q), with v(x + k) = v(x) for any k ∈ Z n and a.e. x ∈ R n . We have that F attains a minimum in X, according to the following result: Proof. First of all, we notice that F (0) = 0, so we take a minimizing sequence v j ∈ X such that lim and we may suppose that F (v j ) ≤ 0.
(44) Our goal is to obtain estimates that are uniform in j.
Letting w j := |v j |χ B ρ+ √ n and recalling Lemma 2.3, we see that for some C > 0, possibly depending on ρ and n. Hence, Using this and (22) (with Cτ in the place of τ ), we get where κ > 0 depends on σ, τ , µ, ρ and n. As a consequence of this and (44), we obtain In addition, utilizing (44) and (45), we have that and so, by Hölder Inequality, . Accordingly, v j L 3 (Q) is bounded uniformly in j and therefore v j L 2 (Q) is also bounded uniformly in j. From this and (46), it follows that v j is precompact in L 2 (Q) (see e.g. Theorem 7.1 in [DNPV12]). Thus, up to a subsequence, we may assume that v j → v * in L 2 (Q) and a.e. in Q (and thus, by periodicity, a.e. in R n ), as j → +∞. Notice also that v * is periodic, since so is v j . This gives that v * ∈ X. Furthermore, using the convergence of v j and Fatou Lemma, Moreover, thanks to Lemma 2.3, and the latter quantity is infinitesimal as j → +∞. These considerations and (43) give that so the desired result follows.
Now we can complete the proof of Theorem 1.10 by considering the minimizer produced by Lemma 9.1 and by checking that periodic perturbations indeed give (13) as Euler-Lagrange equation.
Proof of Theorem 1.10. Let v * be as in Lemma 9.1 and u := |v * |. Then Now we take ψ ∈ C ∞ 0 (Q) and we consider its periodic extension in R n , that is φ(x) := k∈Z n ψ(x + k).
Using v := u + ǫφ as test function in (47), we obtain that Now we write and thus, using the substitutionsx := x − k andỹ := y − k, Similarly, So, we insert (49) and (50) into (48) and we obtain that This gives that u is a solution of the desired equation in Q (and thus in the whole of R n , by periodicity). We also claim that u > 0 in R n . (51) The proof is by contradiction: if there exists x o for which u(x o ) = 0, then, by Lemma 2.1, we see that u vanishes identically. In particular, by (47), where ǫ > 0 is a fixed constant. On the other hand, where Notice that c 3 := c 2 2 + τ 2 > 0, thanks to (14), and thus F (ǫ) = c 1 ǫ 3 3 −c 3 ǫ 2 < 0 for small ǫ. This is in contradiction with (52) and so it proves (51). This completes the proof of Theorem 1.10. Now we establish Theorem 1.11 via some algebraic and analytical identities.
Proof of Theorem 1.11. Let Q be as in (42). We define Notice that m > 0, (54) due to the sign of u, and Also, since u is periodic, there exists a minimal point x o , that is Thus, since u and v differ by a constant, it follows that Now we point out that, for any y ∈ R n , due to (53) and the periodicity of u. Therefore, if we fix δ > 0, we see that and so, by taking δ → 0, Moreover, using again (58), we find that Using this, (55), (59) and the equation for u, we conclude that This says that Now, we observe that, thanks to (56). In addition, from (56) we also deduce that for every x ∈ R n . Hence, we compute the equation at x o and we find that Therefore, since u(x o ) > 0, we conclude that We insert this into (60) and we recall (54), in order to obtain that Thus, by (57), which implies that v vanishes identically. Accordingly, by (53), we obtain that u is constant and constantly equal to m. We insert this information into the equation and we obtain that 0 = (σ − µm)m + τ (J * m) = (σ − µm)m + τ m = (σ + τ − µm)m.

10.
A transmission problem and proof of Theorems 1.12 and 1.13 Now we consider the transmission problem introduced in (15) and we prove the existence of minimizers.
The following is a maximum principle related to the transmission problem (15): Lemma 10.1. Let u be a nonnegative solution of (16). Then either u > 0 in Ω 1 ∪ Ω 2 or it vanishes identically.
As a consequence, which is a contradiction. This establishes the first claim in Theorem 1.13, so we can now focus on the second claim. To this goal, we now assume that inf Ω σ ≥ λ ⋆ (Ω) with strict inequality on a set of positive measure. Therefore, recalling (64), we have that, in this case,