On a constrained reaction-diffusion system related to multiphase problems

We solve and characterize the Lagrange multipliers of a reaction-diffusion system in the Gibbs simplex of R^{N+1} by considering strong solutions of a system of parabolic variational inequalities in R^N. Exploring properties of the two obstacles evolution problem, we obtain and approximate a N-system involving the characteristic functions of the saturated and/or degenerated phases in the nonlinear reaction terms. We also show continuous dependence results and we establish sufficient conditions of non-degeneracy for the stability of those phase subregions.


Introduction
This paper is motivated by the vector-valued reaction-diffusion equation for U = U (x, t), defined from Q = Ω × (0, T ) into R N +1 , with homogeneous Neumann condition on ∂Ω × (0, T ), where Ω is a bounded domain of R n and T > 0 is arbitrary. We are interested in the case when every component u i = u i (x, t) is nonnegative and the system is subject to the multiphase non-voids condition with J = (1, . . . , 1) ∈ R N +1 : From the equation (1) it is clear that the constraint (2) implies F (x, t, U ) · J = 0 in Q and so the reaction vector F should satisfy the necessary and very restrictive condition For instance, in replicator dynamics describing the evolution of certain frequencies in a population, one possible definition of the reaction term with this compatibility condition consists in choosing in Q, i = 1, ..., N + 1, where v i represents the i-frequency of the population and φ i the respective fitness (see, for instance, [10] and [11]), the constraint (2) is essential to describe mixed strategies in evolutionary game theory in spatially homogeneous population dynamics (see [18] and its references) or to model the non-voids condition in biological tissue growing [15,14]. In phase fields models, the condition (2) arises naturally in simulation of multiphase flows (ref 13 ) and multiphase systems with diffuse phase boundaries, as in solidification of alloys or in grain boundary motion (see [9] or [3]). Of course, in the case (3), in particular, if F = 0, the problem becomes a simple one if the initial data U (0) = U 0 also satisfies the constraint (2). However the situation is entirely different in the general case of non trivial reactions, specially in multiphase problems where at least one phase "i" in a subregion of Q is absent (i.e. u i = 0), or fulfils another subregion (when u i = 1).
Here K denotes the convex subset of the Sobolev space H 1 (Ω) N defined by where v = (v 1 , . . . , v N ).
The main part of this work is the analysis of the new unilateral problem (6)- (7) under general assumptions on f : only continuity on u and integrability in (x, t) ∈ Q. In particular, we prove that its solution u = u(x, t), which each component u i satisfies a double obstacle problem where j =i u j denotes the sum of all N − 1 components but u i is, in fact, also the solution of a reaction-diffusion system in the form denotes the summation over all the subsets {i 1 , . . . , i k } of {1, . . . , N } to which i belongs, in particular, k varies from 1 to N . We also denote g + = g ∨ 0 and g − = −(g ∧ 0) the positive and negative parts of a scalar function g = g + − g − , χ A the characteristic function of the set A, (i.e., χ A = 1 in A and χ A = 0 in Q \ A) and χ i 1 ...i k the characteristic function of the set In particular  (10) we see that, in general, the respective reaction terms are coupled not only through the semilinear term f (u) but also through the characteristic functions of the saturation sets of I i 1 ...i k . In this way, by setting for i = 1, . . . , N, u j ), we can solve the system (1) under the constraint (2) and identify To illustrate the meaning of the system (10), that contains 2 N − 1 + N characteristic functions, in general, we may consider the cases N = 1, 2 or 3. Denoting, for simplicity, f i = f i (u), χ i = χ {u i =1} , we may write the Lagrange multipliers as Ignoring the third equation and all the terms involving the third component, we may obtain the case N = 2. The first two terms of the right hand side of the first equation correspond, in the case N = 1, to the scalar two obstacles problem that has been proposed for phase separations in [4,5].
The mathematical treatment of this unilateral system is done in the following three sections. In section 2, we consider the semilinear approximation of the unique solution of (6)- (7) in the case of the reaction f is in L 2 (Q) N and independent of the solution. Although there exists a large literature on parabolic variational inequalities (see, for instance, [16], [6], [12], [7] or [8]), the direct approach of the bounded penalization used for the two obstacles problem in [22] (see also [19]), extended here for the system (10), allows the use of monotone methods. This yields a direct way of obtaining Lewy-Stamppachia inequalities (26), obtained first by [7] for parabolic problems, implying the W 2,1 p and Hölder regularity for the solution to (6). Similar results for the N -membranes stationary problem have been obtained in [1,2]. We note in our case the simplification due to homogeneous Neumann condition.
In section 3, we extend the existence result to general nonlinear reaction f = f (u) taking values in L 1 (Q) N . Here we explore the fact that the convex set (8) lies in the unit disc and we extend the direct technique of [20]. We show also a continuous dependence result and, in the case of λI − f being monotone non-decreasing, in particular if f is Lipschitz continuous in u, also the uniqueness of solution and their strong approximation by the penalized solutions.
Finally, in the last section, we characterize the solution of the variational inequality (6) as solutions of the reaction-diffusion system (10), by extending some remarks of [23] to the two obstacles parabolic problem. We also show that and . . , N and we can modify the system (10) (see (77)) and show that the a.e. pointwise nondegeneracy assumptions k j=1 f i j (u) = 0, 1 ≤ i 1 < · · · < i k ≤ N, k = 1, . . . , N, are sufficient conditions for the local stability of the characteristic functions χ {u i =0} and χ i 1 ...i k with respect to the perturbation of the nonlinear reaction terms f .

Approximation of strong solutions by semilinear problems
In this section we consider the case where f = (f 1 , . . . , f N ) depends only on (x, t) and is given in To prove existence of solution of the variational inequality (6)-(7), we consider a family of approximating semilinear systems of equations. We define, for each ε > 0, θ ε : and we denote where ∂ t u = (∂ t u 1 , . . . , ∂ t u N ) and ∆u = (∆u 1 , . . . , ∆u N ). We also denote P u i = ∂ t u i −∆u i , i = 1, . . . , N . The approximating problems are given by the following weakly coupled parabolic system with Neumann condition where ∂ ∂n is the outward normal derivative on ∂Ω × (0, T ), the meaning of was explained in the introduction and Defining the penalization operator Θ ε by we formulate (12)- (13) in variational form for a.e. t ∈ (0, T ), associated with the initial condition (14).
the problem (18)- (14) has a unique solution Proof. We begin by proving the monotonicity of the penalization operator Θ ε . In fact, recalling that θ ε is monotone nondecreasing and the definition (15) we have The existence and uniqueness of solution u ε ∈ L 2 (0, T ; H 1 (Ω) N ) is immediate by applying the theory of monotone operators ( [16], [25])).
Letting, formally, v = ∂ t u ε in (18) (in fact in the respective Faedo-Galerkin approximation) and integrating in time, we get Proposition 2.2. Assuming (19), the solution u ε of the problem (18)- (14) satisfies Proof. In fact, we are going to prove the following more general set of inequalities and the proof of the right hand side inequalities will be done by induction on r.
We remark that no regularity of the boundary ∂Ω has been required in (18) and, in fact, the Neumann boundary condition (13) is only formal. In the proof of Theorem 2.3 we have used the compactness of the sequence (u ε ) ε in L 2 (Q) N . This holds, for instance, for domains with Lipschitz boundaries, but also, since the sequence (u ε ) ε is uniformly bounded in L ∞ (Q) N , for a larger class of bounded open subsets of R N +1 . However, the approximation by semilinear parabolic equations yields immediately an additional regularity of these strong solutions.
These spaces satisfy the Sobolev imbeddings, for p > (n + 2)/(2 − k), with k = 0, 1, where C k,0 α (Q) denotes the spaces of Hölder continuous functions v in Q, with exponent α in the x-variables and α/2 in the t-variable and, in the case k = 1, with ∇v satisfying the same property (see [17], p. 80). Therefore, as a consequence of (27), we conclude.
Theorem 2.4. Assume that ∂Ω is smooth, say of class C 2 and with each component u 0i satisfying the compatibility condition ∂u 0i ∂n = 0 on ∂Ω if p > 3.
Then the unique solution u of the variational inequality (6)- (7) is such that and, in particular, is Hölder continuous in Q if p > (n + 2)/2 and has ∇u also Hölder continuous if p > n + 2.

Existence and uniqueness of variational solutions
In this section, requiring the compactness of the inclusion of H 1 (Ω) into L 2 (Ω) by assuming a Lipschitz boundary ∂Ω, we show how we can still solve the variational inequality (25) for a more general initial condition and for general nonlinear f = f (u) defining a continuous operator from L 2 (0, T ;K) in However, now the solution has less regularity, namely and its derivative may not be a function, since we only have Hence the first term in the variational inequality (25) should be interpreted in the duality sense between L 1 (Q) N + L 2 0, T ; H 1 (Ω) N and for arbitrary t ∈ (0, T ] since, as we shall see, (27) yields P u ∈ L 1 (Q) N .
Proof. We consider the closed convex subset of L 2 (Q) N K = L 2 (0, T ;K) = {v ∈ L 2 (Q) N : u i ≥ 0, i = 1, . . . , N, and we define Φ : K → K as the nonlinear operator that associates to each w ∈ K the solution u w = Φ(w) of the variational inequality (25) with f replaced by g = f (x, t, w) and fixed initial data u 0 ∈K. By showing that Φ is a continuous and compact operator, a fixed point u = Φ(u), given by Schauder Theorem, will provide a solution with the required properties.
Indeed, first we observe that if we consider any sequence K (31) and (32), the Lebesgue Theorem implies Next, for any g ∈ L 1 (Q) N and any u 0 ∈K we consider sequences g ν ∈ L 2 (Q) N and u 0ν ∈ K such that and we denote by u ν ≡ S(u 0ν , g ν ) the unique solution of (25)-(7) given by Theorem 2.3, for each g ν and u 0ν . We observe that each component of P u ν satisfies the inequality (27) with f i replaced by (g ν ) i . From (25) for u µ and u ν , we easily find, for a.e. t ∈ (0, T ) and, integrating in time, we obtain This estimate shows that {u ν } ν is a Cauchy sequence in the Banach space with respect to the norm and, hence, there exists a function u g ∈ W In addition, u g ∈ L 2 (0, T ; K) ∩ C([0, T ];K) and P u g ∈ L 1 (Q) N , which implies, by (35), that ∂ t u g satisfies (34). Hence, using (35), we may pass to the limit in ν in for an arbitrary v ∈ L 2 (0, T ; K) ⊂ L ∞ (Q) N , and using the formula we conclude that u g = S(u 0 , g) is the (unique) solution of the variational inequality (25) (or equivalently (36)) with data g ∈ L 1 (Q) N and u 0 ∈K. In particular, from (37), we also obtain that, for fixed u 0 ∈K, the operator Σ : g → u g = S(u 0 , g) is Hölder continuous of order 1/2, from L 1 (Q) N into W . Since ∂ t u g satisfies the property (34), it is in fact in L 1 (0, T ; H −s (Ω) N ), for s sufficiently large and, by a well known compactness embedding (see [24] or Theorem 3.11 of [25]), the compactness of H 1 (Ω) ⊂ L 2 (Ω) implies that, in fact, Σ regarded as an operator from L 1 (Q) N into K ⊂ L 2 (Q) N is, therefore, completely continuous. Hence, Φ = Σ • f fulfils the requirements of the Schauder fixed point theorem and the proof is complete.

Remark 3.2.
It is clear that if u 0 ∈ K and, in (32), ϕ 1 ∈ L 2 (Q), we obtain in Theorem 3.1 the existence of a strong solution satisfying (23) and (24). Of course, if we have the regularity assumptions of Theorem 2.4, i.e., ϕ 1 ∈ L p (Q), implying by the inequalities (27) that P u ∈ L p (Q) N , we also obtain solutions in W 2,1 p (Q) N , in particular Hölder continuous solutions if p > (n + 2)/2.
In general (36) may have more than one solution, but if we assume, in addition, that for some λ > 0, λ I − f is monotone non-decreasing in [0, 1] N , i.e.
in particular, if f is Lipschitz continuous in v, then there exists at most one solution u of the variational inequality (25) in the class (33) and initial condition u 0 ∈K.
In order to prove the uniqueness of solution, we suppose that u 1 and u 2 are two solutions of the variational inequality (25) with initial condition u 0 ∈K and f = f (u 1 ), f = f (u 2 ) respectively. Then, choosing u 2 and u 1 as test functions, respectively, using (40) we find and so, by Grownvall inequality u 1 = u 2 a.e. in Q, since u 1 (0) = u 2 (0) = u 0 . We redefine the variational formulation of the approximating problem (18) in the framework of this section with Θ ε defined in (16) and with initial condition only in L 2 (Ω) N , (41) Arguing as in Theorem 3.1 we may prove the existence of a solution of the approximating problem (12), with initial condition u 0 ∈K as long as f satisfies (31) and (32). We also have uniqueness if we assume (40).
Proof. We choose in (41) v = u ε − u as test function. Since u ∈ K, then Choosing, as test function in (25) and subtracting (44) from (43) we get With similar arguments we may give a continuous dependence result for solutions of the variational inequality (36).
Suppose we have a sequence f ν − −−− → ν f in the following sense In addition, the assumption (32) is satisfied for all f uniformly in ν, i.e., there is a common ϕ 1 such that (32) holds for all ν, and the initial data are such that Hence, by Theorem 3.1, it is clear that there are solutions {u ν } ν∈N to the corresponding problems associated with f ν and u ν 0 and, moreover, they satisfy (33) and (34) uniformly in ν, i.e., their norms in those spaces are bounded by a constant independent of ν. Therefore, we have a function u in the same class (33) and (34), and a subsequence, still denoted by ν, such that By assumption (46) and Lebesgue Theorem, we conclude first that a.e. in Q and in L 1 (Q) N , as well as since, in particular, |u ν | ≤ 1 and |u| ≤ 1 a.e. in Q.
Recalling (27) for each ν, we may take the limit in for a fixed v ∈ L 2 (0, T ;K). Using (50) and (48), that in particular imply we conclude that u is a solution of (36) with initial condition u 0 .
Using v = u χ (0,t) + u ν χ (t,T ) in (52) and v = u ν χ (0,t) + u χ (t,T ) in (36) we find, for a.e. t ∈ (0, T ), and, by (51), we conclude that u ν − −−− → ν u strongly in W . Therefore, we have proved the following result Theorem 3.4. If u ν denotes the solution to the variational inequality (36) with f ν satisfying the assumptions (46) and (32) uniformly in ν and initial condition satisfying (47), then there exists a subsequence {u ν } ν∈N such that where u is a solution to (36) corresponding to the limit f and the limit initial condition u 0 . In addition, if f satisfies (40), by uniqueness of u, the whole sequence {u ν } ν∈N converges.

The multiphases system and its characterization
In this section we consider a variational solution u of (25) obtained in Theorem 3.1, i.e., satisfying (33) and (34). Setting each component u i satisfies a double obstacle problem For an arbitrary nonnegative and bounded function ϕ = ϕ(x, t) defined for (x, t) ∈ Q, such that and for a given g ∈ L 1 (Q), we may introduce the parabolic double obstacle scalar problem u ∈ K ϕ 0 : subject to a given compatible initial condition For each i = 1, . . . , N , we have u i ∈ K w i 0 and, by choosing in (25) v ∈ L 2 (0, T ; K), such that v j = u j for j = i and v i = v ∈ K w i 0 arbitrarily, it is clear that u i is a solution of the scalar double obstacle problem (56) with ϕ = w i and g = f i (u). Hence we can obtain further properties of our solution by applying the general theory of the obstacle problem. For the sake of completeness we prove here the result below. and We observe that (59) means that ϕ satisfies the formula and it satisfies the parabolic semilinear equation Proof. Using the function θ ε given by (11) and defining we can consider the approximating problem, for ε > 0, with the initial condition u ε (0) = u 0 in Ω. Since ϑ ε is monotone and ϕ is bounded, arguing as in Theorem 3.1, the problem (64) has a unique solution u ε in the class (61). Moreover, it satisfies −ε ≤ u ε ≤ ϕ + ε a.e. in Q, as we can show by choosing, in (64), v = (−u ε − ε) + and v = (u ε − ϕ − ε) + , respectively. Indeed, in the first case we have since ϑ ε (u ε ) = −1 and ϑ ε (ϕ − u ε ) = 0, because u ε < −ε and ϕ − u ε > ε, and, in the second case, Hence, using the monotonicity argument, we easily conclude that u = lim ε→0 u ε ∈ K ϕ 0 is the unique solution of the variational inequality (56). Remarking that, from (63) we have from (64) we deduce in the limit the Lewy-Stampacchia inequalities e. in Q.
In particular, this yields P u ∈ L 1 (Q) and (56) implies that u also solves . Let O ⊂ Q be an arbitrary measurable set and set v = u in Q \ O and v = δϕ in O, with δ ∈ [0, 1], in (66). Since O is arbitrary, we conclude the pointwise inequality which implies, up to null measure subsets of Q, On the other hand, arguing as in Lemma 2 of [23] and noting that V = (u, −∇u) ∈ L 1 (Q) n+1 and D · V = P u ∈ L 1 (Q), with D = (∂ t , ∂ x 1 , . . . , ∂ xn ), we have P u = 0 a.e. in {u = 0} and P u = P ϕ a.e. in {u = ϕ}.
where χ i 1 ...i k = χ I i 1 ...i k , for k = 1, . . . , N , denotes the characteristic function of Proof. We notice that the regularity (58) and the condition (70) will follow if we show that and these sets are a.e. disjoint. Here the union is taken also over all the subsets {i 1 , . . . , i k } of {1, . . . , N } that include i and over all k = 1, . . . , N . We remark that P w i = P u i in that subset and • in the sets I i = {u i = 1}, P w i = 0 and (P w i − f i (u)) − = f i (u) + , for i = 1, . . . , N ; • in each set I i 1 ...i k , for k ≥ 2, as we shall see, and this fact concludes the proof.
Returning to the inequality (36) and setting S j = P u j − f j (u), we get Since ω ⊃ {(x 0 , t 0 )} was taken arbitrarily in O and (x 0 , t 0 ) is a generic point of I i 1 ...i k , we conclude that a.e. in I i 1 ...i k , for any j ∈ {2, . . . , k}.
Recalling that N j=1 P u j = P u i 1 ...i k = 0, in the set I i 1 ...i k we get, using (74), that where, for simplicity, we set f j = f j (u), and so But in I i 1 ...i k we have S i ≤ 0 (recall that u i = w i and (68)) and so Then, denoting by |A| the (n + 1)-Lebesgue measure of A ⊂ Q, we have for each partial coincidence subset I i 1 ...i k , as well as Proof. Being I i 1 ...i k defined in (71), using the equation (70), we obtain, for each i j with j = 1, . . . , k, denoting f i j = f i j (u), a.e in I i 1 ...i k .
As a consequence of this corollary the semilinear system (70) can, in fact, be written in the equivalent form for i = 1, . . . , N ,  But each u i also solves the equation (77), so, by subtraction, we obtain a.e. in Q, Noticing that χ {u ν i =0} u ν i = 0, passing to the limit, we get χ * i,0 u i = 0, which means that χ * i,0 = 0 whenever u i > 0. To conclude that χ * i,0 = χ {u i =0} we only need to prove that χ * i,0 = 1 if u i = 0.