Uniform Hölder Bounds for Competition Systems with Strong Interaction on a Subdomain

We prove the uniform Hölder bounds of solutions to a singularly perturbed elliptic system arising in competing models in population dynamics. In this system, two species compete to some extent throughout the whole domain but compete strongly on a subdomain. The proof relies upon the blow up technique and the monotonicity formula by Alt, Caffarelli and Friedman.


Introduction
A central problem in population ecology is the understanding of spatial behavior of interacting species, in particular in the case when the interactions are large and of competitive type. Spatial segregation may occur when two or more species interact in a highly competitive way. Such phenomenon has been studied using competition models (or its parabolic case) with positive parameter k → +∞ : Here Ω is a smooth bounded domain in n  , 2 n ≥ , i u denotes the density of the i-th population, whose internal dynamics is prescribed by , , , M u u u u =  , the limit satisfies 0 i j u u = for i j ≠ , which is called the spatial segregation (cf. [1]). Segregation systems arise in different applicative contests, from biological models for competing species to the phase segregation phenomenon in Bose-Einstein condensation of the form: In recent years, people show a lot of interests in segregation phenomenon, and abroad literature is present: starting from [2]- [9], in a series of recent papers [10]- [22], also in the fractional diffusion case [23] [24] [25] [26]. Among the others, the following results are known: the uniform Hölder bounds [7] [12] [15] [24]) and the optimal Lipschitz bound [16], the Lipschitz regularity of the limiting profiles and the regularity of the free boundaries, which is defined as the nodal set ( ) { } 0 u u Γ = = of the singular limit. It is proved that the free boundary consists of two parts: a regular set, which is 1, C α locally smooth hypersurface, and a singular set of Hausdorff dimension less then 2 n − , see [11] [17], for the nondivergence system, [10] [14] [17] for the variational one. Further information about the structure of the singular set has been provided in [27].
Among the models proposed so far, the species compete strongly on the whole of Ω. However, in some heterogeneous environment, species may compete to some extent in the whole of a region Ω, but compete strongly on a subdomain A.
To analysis the corresponding spatial segregation phenomenon governed by strong competition on A, Crooks and Dancer [28] proposed the following k-dependent system: where k is again a positive competition parameter, u and v denote the densities of two species, the self-interaction functions f and g are assumed to be continuously differentiable and such that ( ) ( ) u v , u and v segregate on A but not necessarily on Ω\A. The limit problem is a system on Ω\A and a scalar equation on A.
The objective of this paper is to improve the convergence result of [28], we shall establish the uniform Hölder bounds for solutions to system (1.3). To begin with, we define 2 3, when 2, 1 2, when 3.
n n Due to the apparent of subdomain, we can not expect boundedness for every Hölder exponent. In fact we have the following.
be nonnegative solutions of (1.3), and α * be defined in (4). Assume that for every k, there exists 0 M > , independent of k, such that Notations Throughout the paper, we denote by the open ball with center 0 x and radius 0 R > .
. We assume that any n x ∈  can be written as and n x ∈  . In this way, we denote The proof of Theorem 1.1 mainly follows the blow up method, developed by Terracini and her coauthors in [7] [15]. This method is a blow up analysis and need us to establish some Liouville type results, which can be achieved by some monotonicity formulas of Alt-Caffarelli-Friedman type. Compared with [7] [15], the segregation occurs only in the subdomain A , and we lack the essential information both of the location of A and the boundary conditions on A ∂ . In the blow up procedure, the entire solutions may segregate only on the half space. Thus the Liouville type theorems established in [7] are no longer valid in the current situation. To attack this problem, new ACF type monotonicity formulas and corresponding Liouville type theorems are needed.
The rest of this paper is organized as follows: in Section 2, we establish a monotonicity formula of ACF type, and by utilizing this monotonicity formula, we prove a Liouville type theorem for entire solutions to a semilinear system. In Section 3, we perform the blow up procedure and complete the proof of Theorem 1.1.

Liouville-Type Results
In this section, we prove some nonexistence result in n  . The main tools will be the monotonicity formula by Alt, Caffarelli, Friedman originally stated in [29], as well as some generalizations made by Conti, Terracini, Verzini [7], Dancer, Wang, Zhang [12], and Terracini, Verzini, Zilio [23] [24]. The validity of ACF type formula depends on optimal partition problems involving spectral This completes the proof of Lemma 2.1.  In the following, we shall prove an ACF type monotonicity formula associated with the following system , in , , in .
χ is the characteristic function on T. As in [15], we introduced an auxiliary function: . In this setting, we note that Under the previous notations, we can prove the following monotonicity formula.
 be positive solutions of (2.1) and let 0 ε > be fixed. Then there exists Proof The proof is inspired by [15]. In order to simplify notations we shall denote Let us first evaluate the derivative of ( ) By testing the equation for u in (2.1) with ( ) , we obtain Journal of Applied Mathematics and Physics Then for every δ ∈  , by Hölder inequality and Young's inequality, there holds After some calculation, we obtain Then a change of variables gives Notice first of all that there exists a constant 0 C > such that Then one of the functions is identically zero and the other is a constant. Proof We first note that, by (2.6) and Lemma 2.3, if one of the functions is identically zero or a positive constant, then the other must be a constant or 0 respectively. Hence we may assume by contradiction that neither u nor v is constant. Then by the maximum principle u and v are positive, and Theorem 2.2 ensures the existence of a constant 0 C > such that for r sufficiently large. Let

The Uniform Hölder Bounds
In this section, we shall establish the uniform Hölder bounds for . In order to improve the uniform convergence result obtained in [28], it suffices to establish the uniform C α bounds on subdomain A . We now state the main results in this section. , , The proof of Theorem 3.1 is inspired from the work of [15]. We assume by contradiction that, for some ( ) * 0, α α ∈ , up to a subsequence, it holds We can assume that k L is achieved, say, by k u at the pair ( ) where 0 k r → will be chosen later. By direct calculation, ( k u  and k v  ) satisfy We note that n k Ω →  as k → +∞ , and depending on the asymptotic behavior of the distance ( ) , 0 k r → and k L → +∞ , by diect calculations it is easy to see that In the following, we need to make different choices of the sequence k r . Once k r is chosen, we will use Ascoli-Arzelà's Theorem to pass to the limit on compact sets. Now since the , In order to simplify the notation, let and for each compact set K K ′  , we choose a cut-off function . Then by testing (3.3) with 2 k v η  on K, we obtain So, there exist two positive constants 1 2 , C C such that From the first equation, we can also obtain a uniform Lipschitz estimate of Let k → ∞ in (3.10) and (3.11), we obtain