ELLIPTIC APPROXIMATION OF FORWARD-BACKWARD PARABOLIC EQUATIONS

. In this note we give existence and uniqueness result for some elliptic problems depending on a small parameter and show that their solutions converge, when this parameter goes to zero, to the solution of a mixed type equation, elliptic-parabolic, parabolic both forward and backward. The aim is to give an approximation result via elliptic equations of a changing type equation.

1. Introduction. In [8] and [7] existence results for mixed type equations, in particular forward-backward parabolic equations, are given. The simplest examples are the two following: given T > 0, Ω open subset of R n , r ∈ L ∞ (Ω × (0, T )) consider (∆ p denotes the p-Laplacian for p 2 and ν the outside normal to ∂Ω) (1) with f, ϕ, ψ suitable data and, at least for r assuming both positive and negative sign, suitable assumptions on the two sets {x ∈ Ω|r(x, 0) > 0} × {0} and {x ∈ Ω|r(x, T ) < 0} × {T }. Equations of such type arise in the study of some stochastic differential equation, in the kinetic theory, in some physical models like electron scattering or neutron transport. For some references one can see [3] or the much less recent papers [6,1,2] (or the references contained therein and in [8] and [7]) where simple equations like sgn(x)|x| m u t − u xx = f are considered, being m ∈ N.
The aim of the present note is to give an approximation result for abstract forward-backward parabolic equations via elliptic problems (see Theorem 4.1). For the problems (1) the result may be stated as follows. In accordance with the equation and with the boundary condition considered in (1)  This result is similar to that contained in [4], even if our purpose is different: the result of Lions aims to give an existence result for linear parabolic equations with boundary conditions depending on time, we only want to give an approximation result, via more standard equations, of a mixed type equation, even if the technique can be used also in other different environments.
2. Notations, hypotheses and preliminary results. Consider the following family of evolution triplets where H(t) is a separable Hilbert space, V (t) a reflexive Banach space which continuously and densely embeds in H(t) and V (t) the dual space of V (t), and we suppose there is a constant C 0 which satisfies for every w ∈ H(t), v ∈ V (t) and every t ∈ [0, T ].
The framework seems to be similar to the one considered in [4], but we have in mind something different (see the example in the last section). We will suppose the existence of a Banach space U such that and define, for some p 2, the set Moreover we will suppose that the functions are measurable for every u ∈ U and we define the spaces as the completion of U with respect to the natural norms Finally by V we denote the dual space of V endowed with the norm Given a family of linear operators R(t) such that R depends on a parameter t ∈ [0, T ] and R(t) ∈ L(H(t)), being L(H(t)) the set of linear and bounded operators from H(t) in itself, instead of (6) we sometimes will write improperly Now consider an abstract function R : [0, T ] −→ L(H(t)). We say that R belongs to the class E(C 1 , C 2 ), C 1 , C 2 > 0, if it satisfies what follows for every u, v ∈ U : Now, given two positive constants C 1 and C 2 , consider R ∈ E(C 1 , C 2 ). For every t ∈ [0, T ] we consider the spectral decomposition of R(t) (see, e.g., Section 8.4 in [5]) and define R + (t), and respectively R − (t), the operator connected to the positive, respectively negative, part of the spectrum, so that R(t) = R + (t) − R − (t) and R + (t) • R − (t) = R − (t) • R + (t) = 0 and R + (t) and R − (t) turn out to be invertible. Equivalently one can define R + (t) and R − (t) as follows: since R(t) is self-adjoint we get that R(t) 2 = R * (t) • R(t) is a positive operator; then we can define the square root of R(t) 2 (see, e.g., Chapter 3 in [5]), which is a positive operator, and then define the two positive operators By this decomposition we can also write Finally we denoteH 0 (t) = H 0 (t) = KerR(t) and with respect to the norm Clearly the operation depends on R. Moreover we consider P + (t) and P − (t) the orthogonal projections fromH(t) onto H + (t) and H − (t) respectively, P 0 (t) the projection defined in H(t) onto H 0 (t).
Given an operator R ∈ E(C 1 , C 2 ) it is possible to define two other linear operators. First we can define the derivative of R which, unlike R, is valued in L(V (t), V (t)), i.e. the set of linear and bounded operators from V (t) to V (t): since R ∈ E(C 1 , C 2 ) we can define a family of equibounded operators By the density of U in V (t) we can extend R (t) to V (t). Then we can also define which turn out to be linear and bounded by the constant C 1 and, by density of U in V, an operator which turns out to be linear, self-adjoint and bounded by C 2 . For a function u : [0, T ] → U we denote by u the distributional derivative, i.e. the function such that . We maintain the same notation for functions belonging to V.
We now could consider for R ∈ E(C 1 , C 2 ) the two operators u → (Ru) and Ru defined respectively in the two spaces Since R admits a derivative one has (see [7]) that (Ru) = R u + Ru and that W 1 = W 2 even if we will endow the two spaces respectively with the norms Because of that, it will not always be necessary to specify which of the two spaces we are talking about and in those cases we will simply refer to them as As done before we can define, in a way analogous to that done for the spaces (8), with respect to the norm w H = |R| 1/2 w H , where |R| = R + + R − . Analogously, we define H + and H − and P + and P − the orthogonal projections fromH onto H + and H − respectively. H 0 is the kernel of R and P 0 the projection defined in H onto H 0 . Now we recall a result which can be found in [7] (see also [8]).
. Then we have that for every u, v ∈ W R the following holds: Moreover the function t → (R(t)u(t), v(t)) H(t) is continuous and there exists a constant c, which depends only on T , such that Finally we recall a classical result (see, e.g., Section 32.4 in [11], in particular Corollary 32.26) for which we need some definitions, which we remind.
We say that an operator Q : The same operator Q is hemicontinuous if the map A monotone and hemicontinuous operator Q is of type M if (see, for instance, Basic Ideas of the Theory of Monotone Operators in volume B of [11] or Lemma 2.1 in [10]), i.e. it satisfies what follows: for every sequence (u j ) j∈N ⊂ X such that If, moreover, M is strictly monotone the solution is unique.
3. The approximating problems. In this section we want to give an existence and uniqueness result for a family of elliptic problems defined below (see (28)). Before we introduce another functional space, denoted by V * below. To do that first consider another family of reflexive Banach spaces K(t) such that where V (t) continuously embeds in K(t) and K(t) continuously embeds in H(t) and there is a positive constant, which for simplicity we suppose to be C 0 , such that Then we suppose that the functions are measurable for every u ∈ U and we define the space K as the completion of U with respect to the natural norm To see that it is sufficient to adapt Proposition 3.4 in [7]. Then we consider the space (the orthogonal projection operators P + , P 0 , P − are defined in Section 2) We will suppose that We now consider, besides the operator R, two operators A and B A : the two following family of problems (ε > 0 is a parameter which, in the following, we will let go to zero) (with suitable boundary conditions we will specify below) where f ∈ V . These equalities are to be intended in V * as follows: We suppose there are four positive constants α 1 , α 2 , β 1 , β 2 and a function b such that: and suppose that the operator B is defined as in such a way that and, if we consider problems (17)-(I), we require that for every u, v ∈ V; if we consider problems (17)-(II) we require for p = 2 for every u, v ∈ V. If we denote by while, similarly, by (21) one gets For p > 2 by (20) and (22) (c p being a constant depending only on p) one gets Notice that the operators P ε u := A ε u + Ru and Q ε u := A ε u + (Ru) defined in V * with above assumptions are strictly monotone in V * . Indeed if (19) in the case p = 2 or (20) in the case p > 2 holds then Similarly if (21) in the case p = 2 or (22) in the case p > 2 holds then for every We now want to apply Theorem 2.3. First we state the following result. Consider the space  Proof. We prove the lemma for L 1 , being the other proofs similar and, indeed, simpler.
From Proposition 2.2 we have that for every u ∈ V 0 * , and then L 1 is monotone. To see that it is maximal monotone fix w ∈ (V 0 * ) and v ∈ V 0 * and suppose w − L 1 u, v − u V * ×V * ≥ 0 for every u ∈ V 0 * . We want to show that v ∈ V 0 * and w = L 1 v. Choose u = ϕz with ϕ ∈ C 1 0 ([0, T ]) and z ∈ U and get w, v ≥ L 1 u, v − u + w, u that is, since L 1 u, u = 0, R and R are linear and self adjoint, the following equivalent inequalities: Since this holds for each z ∈ U we can consider λz with λ ∈ R and get Since this holds both for λ > 0 and λ < 0 we derive that Rϕ v, z + 1 2 R ϕv, z + ϕw, z = 0 and since this holds for every z ∈ U we get that where p is a polynomial with coefficients in U , i.e.
z k t k for some N ∈ N and z k ∈ U.
Since the space of such polynomials is dense in U and then in V * we finally get that Remark 3.3. -Notice that in Theorem 3.6 we consider f ∈ V , even if, a priori, in (28) one could consider a datum F ∈ V * . If one consider F ∈ V * there are f ∈ V and g ∈ K such that If one confines to consider F ∈ V * such that g = 0 (or, more generally, g ∈ V ) F does not act directly on v . In the following theorem we will confine to consider This is needed to have the estimates in Theorem 3.6 with a constant c independent of ε.

II) Consider
A and suppose A and R satisfy (21) for p = 2 and (22) for p > 2. Moreover suppose that A and B are hemicontinuous. Then for every f ∈ V and φ ∈ {u ∈ V u ∈ K} there exists a unique u ∈ V N * satisfying (28)-(II) and there is c > 0, depending only on α 1 , β 1 , α 2 , β 2 , p, such that (for ε ∈ (0, 1]) Proof. Estimates -Consider point I) in the case p > 2, being the proof in the other cases very similar. Then, as observe in Remark 3.3, we stress that the estimate we are going to show would not be true uniformly in ε for a general f ∈ V * . Precisely, consider u ∈ V N * and suppose that P ε u := εBu + Ru + Au ∈ V .
Existence and uniqueness -Consider first ϕ = 0 and ψ = 0. By assumptions we have that, both in case I) and in case II), and for every p 2, the operator A ε is strictly monotone, coercive, bounded and hemicontinuous. By Lemma 3.1 the operator u → Ru in case (I) and the operator u → (Ru) in case (II) are maximal monotone in V 0 * , and then in {v Applying Theorem 2.3 we conclude. Now consider ϕ, ψ ∈ U , any δ ∈ (0, T /2) and φ defined as In this way φ ∈ V N * . Then a function u satisfies (28)-(I) if and only if the function v = u − φ satisfies we have the following problem It is not difficult to verify thatB andÃ are bounded, coercive, strongly monotone and hemicontinuous, so arguing as before we get a unique solution v ∈ V 0,N * satisfying (35), and then a unique u ∈ V N * satisfying (28)-(I). Now we use the a priori estimates previously obtained to get the thesis for every admissible datum. Consider now ϕ ∈H + (0) and ψ ∈H − (T ) and two sequences (ϕ n ) n , (ψ n ) n ⊂ U such that (this is possible thanks to assumption (29)) ϕ n → ϕ inH + (0), ψ n → ψ inH − (T ).
In this way the function φ n defined in a way analogous to (33) belong to V N * . Similarly as done above to get the a priori estimate one gets (for instance, in case for every n, m ∈ N, and then there is a function u ∈ V N * such that u n → u in V, We also get that Bu n V * c, Au n V c for some positive constant c. Up to select a subsequence we get that Au n weakly converge to some b ∈ V and then Au n , u n V ×V → b, u V ×V . Since A is type M we conclude that b = Au. In the same way one has that Bu n → Bu. Since for every subsequence (u nj ) j∈N we can extract a further subsequence (u nj k ) k∈N such that Au nj k → Au and Bu nj k → Bu we conclude that all the sequence satisfies Au n → Au and Bu n → Bu and u is the solution looked for. 4. Taking the limit for ε → 0. In this section we want to prove the result which is the goal of the paper: to show that the solutions of problems (28)-(I) (respectively of problems (28)-(II)) converge, in a suitable way, to the solution of (36)-(I) (respectively of (36)-(II)). We recall that the existence of a solution of the following problems has already been proved in [8] and [7]: In the following three steps we will consider the problem (28)-(I) for p > 2. The proofs in other cases, problem (28)-(I) for p = 2 and problem (28)-(II) both for p > 2 and p = 2, are very similar.
Limit in the equation -Consider some f ∈ V , ϕ ∈H + (0) and ψ ∈H − (T ) and denote by u ε ∈ V * the solution of (28)-(I), p > 2. By Theorem 3.6 and boundedness of A we get that (up to select a sequence ε j → 0 which we will still denote by ε for sake of simplicity) letting ε go to 0 Notice that, by (18) and since ε 1/p u ε is bounded in K, we also get that Moreover for every η ∈ C 1 c ([0, T ]; U ) Ru ε , η V ×V = − Ru ε , η H − R u ε , η V ×V and taking the limit for ε → 0 one gets With these informations we consider the limit in the equation of problem (28)-(I) and get The goal now is to show that g = Au. (39) Now we consider problems (28) and multiply by u ε the equations of problem (28). First observe that Au ε → g and then We get Observe that, since u → Ru is monotone in V 0 * and u ε − u ∈ V 0 * , and taking the limit and using (40) and since g = f − Ru we get Since we suppose A to be hemicontinuous and, as already observed, A is of type M we get that Limit in the boundary conditions -By Lemma 3.19 in [7] given u ∈ W R we have that In particular for the family of solutions u ε of problems (28)-(I) since (u ε ) ε are bounded in V and Ru ε are bounded in V we have (C 0 is defined in (3)) Ru ε (s)ds Notice that, since R is linear and continuous and (u ε ) ε converge to u in V, we have that R u ε → R u. Since we also have that Ru ε → Ru we derive that the quantity R u ε V + Ru ε V is bounded with respect to ε and then we got that the family is equibounded and equicontinuous in [0, T ] with respect to the topology of H(0) and then R(t)u ε (t) >0 is weakly relatively compact in H(0) uniformly in time.
Precisely, since Ru ε → Ru in H we get that for every η ∈ H(0) The same argument can be used to get that for every η ∈ H(T ) In particular we get that and also but for these we loose the propery to belong to V N . Summing up, we have that there exists a sequence of the family of the solutions (u ε ) ε > 0 of problems (28)-(I) with p > 2 which converge to a function u which satisfies (36)-(I).
The proofs of the other cases are completely similar.
Convergence of the whole family -Since for every subfamily of (u ε ) ε>0 satisfying (28)-(I) one can repeat the same argument as above and get a limit function u satisfying (36)-(I) by the uniqueness of the solution just of (36)-(I) we get that from every subfamily one can select a sequence converging to the same u. Then we conclude that we do not need to select a sequence, but all the family of solutions is converging.
Summing up, we have proved the following result.
where u and v are the unique solutions respectively of the problems (36)-(I) and (36)-(II) and satisfy with C = C(p, α 1 , α 2 ).
As an immediate consequence we have the following corollaries. Corollary 4.2. As a consequence of the previous result, u ε , u, v ε , v as above, we also get that Proof. The proof follows immediately from Theorem 3.6 and Proposition 3.4 in [7] (see also Theorem 2.14 and Proposition 2.6 in [8]). Proof. Consider u ε , the solution of (28)-(I), and u its limit in W R satisfying (36)-(I). Since u ε weakly converge in V and εBu ε strongly converge to zero (see (38)) we immediatly conclude. Similarly one proves the convergence for εBv ε , v ε V * ×V * .
Corollary 4.4. As a consequence of the previous corollary, u ε and v ε as above, we also get that Av ε = Av in V -strong.
By Theorem 4.1 one derives that and then (u ε ) ε>0 is a Cauchy family in V and then lim ε→0 + u ε = u strongly in V.
Since εBu ε + Ru ε + Au ε = f and εBu ε strongly converge to zero (see (38)) in V * (and in V ) we also get that Ru ε + Au ε → f strongly in V .
By the continuity of A we get that Au ε → Au and consequently lim ε→0 + Ru ε = Ru strongly in V .

5.
Examples. In this section we present just two examples, since many examples of forward-backward parabolic equations are already given in the two papers [8] and [7]. Before exposing these examples we stress that, obviously, the two simple cases R ≡ 0 and R = Id are admitted. In the first case we approximate an elliptic problem in dimension n with an analogous elliptic problem in dimension n + 1, while in the second case the limit problem is a parabolic equation (completely forward).