Quasi-linear elliptic problems in L 1. with non homogeneous boundary conditions

Transcription

1 Rendiconti di Matematica, Serie VII Volume 26, Roma (2006), Quasi-linear elliptic problems in L 1 with non homogeneous boundary conditions K. AMMAR F. ANDREU J. TOLEDO Abstract: We study quasi-linear elliptic problems with L 1 data and non homogeneous boundary conditions. Existence and uniqueness of entropy solutions are proved. 1 Introduction Let be a bounded domain in IR N with smooth boundary and 1 <p<, and let a : IR N IR N beacaratheodory function such that (H 1 ) there exists λ>0such that a(x, ξ) ξ λ ξ p for a.e. x and for all ξ IR N, (H 2 ) there exists c>0 and g L p () such that a(x, ξ) c(g(x)+ ξ p 1 ) for a.e. x and for all ξ IR N, where p = p p 1,(H 3)(a(x, ξ) a(x, η)) (ξ η) > 0 for a.e. x and for all ξ,η IR N,ξ η. We are interested in the quasi-linear problem { div a(., Du)+u = φ in (S) a(., Du) η + β(u) ψ on, where ψ L 1 ( ),φ L 1 () and β is a maximal monotone graph in IR 2 such that 0 β(0). The main difficulties in the study of this problem are related to the non regularity of the data (see [4]) and to the condition on the boundary which is more general than the classical Dirichlet condition or the Neumann one. Key Words and Phrases: Quasi-linear elliptic problem Non homogneous boundary condition Entropy solution Accretive operator. A.M.S. Classification: 35J60, 35D02

2 292 K. AMMAR F. ANDREU J. TOLEDO [2] We solve problem (S) for φ L 1 () and ψ L 1 ( ) when a is smooth or D(β) isclosed in the entropy sense introduced in [4] for problem (S) with homogeneous Dirichlet condition. The homogeneous case (that is ψ 0) was studied in [2] for particular graphs β. Inthe present paper, we overcome these restrictions on β using similar techniques than the ones employed in [2] and monotonicity arguments. We also study the quasi-linear problem { div a(., Du) =0 in (P) a(., Du) η + u = ψ on, where ψ L 1 ( ). We introduce a capacity operator which will be used to study parabolic problems with dynamical boundary conditions. 2 Notations As usual, λ N denotes the Lebesgue measure in R N.For1 p<+, L p () and W 1,p () denote respectively the standard Lebesgue and Sobolev spaces, and W 1,p 0 () is the closure of D() in W 1,p (). For u W 1,p (), we denote by u or γ(u) the trace of u on inthe usual sense and by W 1 p,p ( ) the set γ(w 1,p ()). In [4], the authors introduce the set T 1,p () = {u : IR measurable such that T k (u) W 1,p () k >0}, where T k (s) =sup( k, inf(s, k)). They also prove that given u τ 1,p (), there exists a unique measurable function v : IR N such that DT k (u) =vχ { v <k} k>0. This function v will be denoted by Du for the function u T 1,p (). It is clear that if u W 1,p (), then v L p () and v = Du in the usual sense. As in [2], T 1,p tr () denotes the set of functions u in T 1,p () satisfying the following condition, there exists a sequence u n in W 1,p () such that (a) u n converges to u a.e. in, (b) DT k (u n ) converges to DT k (u) inl 1 () for all k>0, (c) there exists a measurable function v on, such that γ(u n ) converges a.e. in tov. The function v is the trace of u in the generalized sense introduced in [2]. In the sequel we use the notations u or τ(u)todesignate the trace of u T 1,p tr () on. Let us recall that in the case u W 1,p (), τ(u) coincides with γ(u), the trace of u in the usual sense. Moreover γ(t k (u)) = T k (τ(u)) for every u T 1,p tr () and k>0, and if u T 1,p tr () and φ W 1,p () L (), then u φ T 1,p tr () and τ(u φ) =τ(u) γ(φ).

3 [3] Quasi-linear elliptic problems in L 1 with etc Existence and uniqueness of solutions of problem (S) We will prove existence and uniqueness of an entropy solution of problem (S) in the case D(β) isclosed or a is smooth, that is, for all φ L (), there exists g L 1 ( ) such that the solution of the homogeneous Dirichlet problem { div a(., Du) =φ in u =0 on, is a solution of the Neumann problem { div a(., Du) =φ in a(., Du) η = g on. Functions a corresponding to linear operators with smooth coefficients and p- Laplacian type operators are smooth. Definition 3.1. A measurable function u in is an entropy solution of problem (S) if u L 1 () T 1,p tr () and there exists w L 1 ( ), w(x) β(u(x)) a.e. on, such that a(., Du) DT k (u v)+ ut k (u v)+ wt k (u v) (3.1) ψt k (u v)+ φt k (u v) k >0, for all v L () W 1,p (), v(x) D(β) a.e. in. As we will see in the existence results, when a is smooth it is possible to remove the condition v(x) D(β) a.e. in for the test functions in the above definition. We prove the following result of existence and uniqueness of entropy solutions of problem (S). Theorem 3.2. Let D(β) be closed or a smooth. (i) For any φ L 1 (), ψ L 1 ( ), there exists a unique entropy solution of problem (S). (ii) If u 1 is the entropy solution of problem (S) corresponding to φ 1 L 1 () and ψ 1 L 1 ( ) and u 2 is the entropy solution of problem (S) corresponding to φ 2 L 1 () and ψ 2 L 1 ( ) then there exist w 1 L 1 ( ), w 1 (x) β(u 1 (x)) a.e. in, and w 2 L 1 ( ), w 2 (x) β(u 2 (x)) a.e. in, such that a(., Du i ) DT k (u i v)+ u i T k (u i v)+ w i T k (u i v) ψ i T k (u i v)+ φ i T k (u i v) k >0,

4 294 K. AMMAR F. ANDREU J. TOLEDO [4] for all v L () W 1,p (), v(x) D(β) a.e. in, i =1, 2. Moreover (u 1 u 2 ) + + (w 1 w 2 ) + (ψ 1 ψ 2 ) + + (φ 1 φ 2 ) +. To prove the above theorem we will proceed by approximation. Theorem 3.3. Let D(β) be closed and m, n IN, m n. (i) For φ L () and ψ L ( ), there exist u = u φ,ψ,m,n W 1,p () L () and w = w φ,ψ,m,n L ( ), w(x) β(u(x)) a.e. on, such that (3.2) (3.3) a(., Du) D(u v)+ + 1 m ψ(u v)+ u + (u v) 1 n u(u v)+ w(u v)+ u (u v) φ(u v), for all v W 1,p (), v(x) D(β) a.e. on, and all k>0. Moreover, u + w ψ + φ. (ii) If m 1 m 2 n 2 n 1, φ 1,φ 2 L (), ψ 1,ψ 2 L ( ) then (u φ1,ψ 1,m 1,n 1 u φ2,ψ 2,m 2,n 2 ) + + (w φ1,ψ 1,m 1,n 1 w φ2,ψ 2,m 2,n 2 ) + (ψ 1 ψ 2 ) + + (φ 1 φ 2 ) +. Proof. Observe that 1 m s+ 1 n s = 1 m s + ( 1 m ) 1 n s = ( 1 m n) 1 s n s. For r IN, itiseasy to see that the operator B r : W 1,p () (W 1,p ()) defined by B r u, v = a(x, D(u)) Dv + T r (u)v + 1 u p 2 uv+ r (3.4) + T r (β r (u))v + 1 T r (u + )v 1 T r (u )v m n ψv φv,

5 [5] Quasi-linear elliptic problems in L 1 with etc. 295 where β r is the Yosida approximation of β, isbounded, coercive, monotone and hemicontinuous. On the other hand, since D(β) is closed, W 1,p β () := {u W 1,p (),u(x) D(β) a.e. on } is a closed convex subset of W 1,p (). Then, by a classical result of Browder ([9]), there exists u r = u φ,ψ,m,n,r W 1,p (), u r (x) D(β) a.e. on, such that (3.5) a(x, Du r ) D(u r v)+ T r (u r )(u r v)+ 1 u r p 2 u r (u r v)+ r + T r (β r (u r ))(u r v)+ 1 T r ((u r ) + )(u r v) m 1 T r ((u r ) )(u r v) ψ(u r v)+ φ(u r v), n for all v W 1,p (), v(x) D(β) a.e. in. Taking v = u r T k ((u r mm) + )in(3.5), where M = φ + ψ, dropping nonnegative terms, dividing by k, and taking limits as k goes to 0, we get 1 T r (u r )sgn + (u r mm)+ 1 T r (u r )sgn + (u r mm) m m ψsgn + (u r mm)+ φsgn + (u r mm), consequently (T r (u r ) mm)sgn + (u r mm)+ (T r (u r ) mm)sgn + (u r mm) (mψ mm)sgn + (u r mm)+ (mφ mm)sgn + (u r mm) 0, therefore, for r large enough, u r (x) mm a.e in. Similarly, taking v = u r + T k ((u r + nm) )in(3.5), we get u r (x) nm a.e in. Consequently, for r large enough, and taking into account that m n, (3.6) u r nm.

6 296 K. AMMAR F. ANDREU J. TOLEDO [6] Taking v =0as test function in (3.5) and using (H 1 ) and (3.6), it follows that (3.7) Du r p 1 ( ) λ nm ψ + φ. As a consequence of (3.6) and (3.7) we can suppose that there exists a subsequence, still denoted u r, such that u r converges weakly in W 1,p () to u W 1,p (), u r converges in L q () and a.e. on to u, for any q 1, u r converges in L p ( ) and a.e. to u. Next we show that T r (β r (u r )) is weakly convergent in L 1 ( ). Since u r (x) D(β), β r (u r )(x) inf{ r,r β(u r (x))}. If D(β) =IR, sup{β( nm)} β r (u r ) inf{β(mm)}. In the case D(β) is a bounded interval [a, b], a<b, If D(β) =[a, + ),a 0, sup{β(a)} β r (u r ) inf{β(b)}. sup{β(a)} β r (u r ) inf{β(m)}. The case D(β) =(,a], a 0 can be treated similarly. Consequently, for r large enough, T r (β r (u r )) = β r (u r )isuniformly bounded and there exists a subsequence, denoted in the same way, L 1 ( )-weakly convergent to some w L ( ). From here, since u r u in L 1 ( ), applying [7, Lemma G], it follows that w β(u) a.e. on. Let us see now that Du r converges in measure to Du. Wefollow the technique used in [8] (see also [2]). Since Du r converges to Du weakly in L p (), it is enough to show that Du r is a Cauchy sequence in measure. Let t and ɛ>0. For some A>1, we set C(x, A, t) :=inf{(a(x, ξ) a(x, η)) (ξ η) : ξ A, η A, ξ η t }. Having in mind that the function ξ a(x, ξ) iscontinuous (since ψ denotes a datum) for almost all x and the set {(ξ,η) : ξ A, η A, ξ η t } is compact, the infimum in the definition of C(x, A, t) isaminimum. Hence, by (H 3 ), it follows that (3.8) C(x, A, t) > 0 for almost all x.

7 [7] Quasi-linear elliptic problems in L 1 with etc. 297 Now, for r, s IN and any k>0, the following inclusion holds { Du r Du s >t} (3.9) { Du r A} { Du s A} { u r u s k 2 } {C(x, A, t) k} G, where G = { u r u s k 2,C(x, A, t) k, Du r A, Du s A, Du r Du s >t}. Since the sequence Du r is bounded in L p () we can choose A large enough in order to have (3.10) λ N ({ Du r A} { Du s A}) ɛ for all r, s IN. 4 By (3.8), we can choose k small enough in order to have (3.11) λ N ({C(x, A, t) k}) ɛ 4. On the other hand, if we use u r T k (u r u s ) and u s + T k (u r u s )astest functions in (3.5) for u r and u s respectively, we obtain a(x, Du r ) DT k (u r u s )+ r T k (u r u s )+ u 1 u r p 2 u r T k (u r u s )+ r (3.12) + β r (u r )T k (u r u s )+ 1 u + r T k (u r u s ) m 1 u r T k (u r u s ) ψt k (u r u s )+ φt k (u r u s ), n and a(x, Du s ) DT k (u r u s ) u s T k (u r u s ) 1 u s p 2 u s T k (u r u s ) s (3.13) β s (u s )T k (u r u s ) 1 u + s T k (u r u s )+ m + 1 u s T k (u r u s ) ψt k (u r u s ) φt k (u r u s ). n Adding (3.12) and (3.13), we get (a(x, Du r ) a(x, Du s )) DT k (u r u s ) ( 1 r u r p 2 u r 1 ) s u s p 2 u s T k (u r u s ) (β r (u r ) β s (u s )) T k (u r u s ).

8 298 K. AMMAR F. ANDREU J. TOLEDO [8] Consequently, there exists a constant ˆM independent of r and s such that (a(x, Du r ) a(x, Du s )) DT k (u r u s ) k ˆM. Hence (3.14) λ N (G) λ N ({ u r u s k 2, (a(x, Du r ) a(x, Du s )) D(u r u s ) k}) 1 (a(x, Du r ) a(x, Du s )) D(u r u s )= k { u r u s <k 2 } = 1 (a(x, Du r ) a(x, Du s )) DT k 2(u r u s ) 1 ɛ k k k2 ˆM 4 for k small enough. Since A and k have been already chosen, if r 0 is large enough we have for r, s r 0 the estimate λ N ({ u r u s k 2 }) ɛ 4.From here, using (3.9), (3.10), (3.11) and (3.14), we can conclude that λ N ({ Du r Du s t}) ɛ for r, s r 0. From here, up to extraction of a subsequence, we also have a(., Du r ) converges in measure and a.e. to a(., Du). Now, by (H 2 ) and (3.7), a(., Du r ) converges weakly in L p () N to a(., Du). Finally, letting r + in (3.5), we prove (3.2). In order to prove (ii), let us put u 1,r =u φ1,ψ 1,m 1,n 1,r and u 2,r =u φ2,ψ 2,m 2,n 2,r. Taking u 1,r T k ((u 1,r u 2,r ) + ), with r large enough, as test function in (3.5) for u 1,r, m = m 1 and n = n 1,weget (3.15) a(., Du 1,r ) DT k ((u 1,r u 2,r ) + )+ u 1,r T k ((u 1,r u 2,r ) + )+ + 1 u 1,r p 2 u 1,r T k ((u 1,r u 2,r ) + )+ β r (u 1,r )T k ((u 1,r u 2,r ) + )+ r + 1 u + 1,r m T k((u 1,r u 2,r ) + ) 1 u 1,r 1 n T k((u 1,r u 2,r ) + ) 1 ψ 1 T k ((u 1,r u 2,r ) + )+ φ 1 T k ((u 1,r u 2,r ) + ),

9 [9] Quasi-linear elliptic problems in L 1 with etc. 299 and taking u 2,r + T k (u 1,r u 2,r ) + as test function in (3.5) for u 2,r, m = m 2 and n = n 2,weget a(., Du 2,r ) DT k ((u 1,r u 2,r ) + ) u 2,r T k ((u 1,r u 2,r ) + ) 1 u 2,r p 2 u 2,r T k ((u 1,r u 2,r ) + ) β r (u 2,r )T k ((u 1,r u 2,r ) + ) r (3.16) 1 u + 2,r m T k((u 1,r u 2,r ) + )+ 1 u 1,r 2 n T k((u 1,r u 2,r ) + ) 2 ψ 2 T k ((u 1,r u 2,r ) + ) φ 2 T k ((u 1,r u 2,r ) + ). Adding these two inequalities, dropping some nonnegative terms, dividing by k, and letting k 0, we get (u 1,r u 2,r ) + + (β r (u 1,r ) β r (u 2,r )) + (3.17) (ψ 1,r ψ 2,r ) + + (φ 1,r φ 2,r ) +. From here, taking into account the above convergences, (ii) can be obtained. Finally, observe that when φ 2 =0and ψ 2 =0,taking v =0in (3.5) for φ = φ 2 and ψ = ψ 2,wegetu 2,r =0. Therefore, from (3.17) we get (3.3). Theorem 3.4. Let a be smooth and m, n IN, m n. (i) For φ L () and ψ L ( ), there exist u = u φ,ψ,m,n W 1,p () L () and w = w φ,ψ,m,n L 1 ( ), w(x) β(u(x)) a.e. on, such that a(., Du) D(u v)+ u(u v)+ w(u v)+ + 1 u + (u v) 1 u (u v) m n ψ(u v)+ φ(u v), for all v W 1,p () and all k>0. Moreover, u + w ψ + (ii) If m 1 m 2 n 2 n 1, φ 1,φ 2 L (), ψ 1,ψ 2 L ( ) then (u φ1,ψ 1,m 1,n 1 u φ2,ψ 2,m 2,n 2 ) + + (w φ1,ψ 1,m 1,n 1 w φ2,ψ 2,m 2,n 2 ) + (ψ 1 ψ 2 ) + + (φ 1 φ 2 ) +. φ.

10 300 K. AMMAR F. ANDREU J. TOLEDO [10] Proof. Applying Theorem 3.3 to β r, the Yosida approximation of β, there exists u r = u φ,ψ,m,n,r W 1,p () L (), such that (3.18) a(., Du r ) D(u r v)+ + 1 m ψ(u r v)+ u + r (u r v) 1 n u r (u r v)+ β r (u r )(u r v)+ u r (u r v) φ(u r v), for all v W 1,p (). Moreover, u r is uniformly bounded by n ( φ + ψ ). Let û be the solution of the Dirichlet problem { div a(x, Dû)+û = φ in û =0 on. Since a is smooth, there exists ˆψ L 1 ( ) such that (3.19) a(., Dû) D(û v)+ û(û v) = ˆψ(û v)+ φ(û v), for any v W 1,p () L (). Taking v = u r ρ(β r (u r û)) as test function in (3.18), where ρ C (IR), 0 ρ 1, supp(ρ )iscompact and 0 / supp(ρ) (supp(ρ) being the support of ρ), and û + ρ(β r (u r û)) as test function in (3.19), and adding both inequalities we get, after dropping nonnegative terms, that β r (u r )ρ(β r (u r )) (ψ ˆψ)ρ(β r (u r )), which implies, see [6], that lim r + β r(u r )=w weakly in L 1 ( ). Now, arguing as in the proof of Theorem 3.3, we obtain (i). To prove (ii), by Theorem 3.3 applied to β r,wehave that, denoting u i,r = u φi,ψ i,m i,n i,r, i =1, 2, (u 1,r u 2,r ) + + (β r (u 1,r ) β r (u 2,r )) + (3.20) (ψ 1 ψ 2 ) + + (φ 1 φ 2 ) +. Taking limits in (3.20) as r goes to + we can get (ii).

11 [11] Quasi-linear elliptic problems in L 1 with etc. 301 Proof of Theorem 3.2. Existence. Let us approximate φ in L 1 () by φ m,n = sup{inf{m, φ}, n}, which is bounded, non decreasing in m and non increasing in n, and ψ in L 1 ( ) by ψ m,n = sup{inf{m, ψ}, n}. Then, if m n, bytheorem 3.3 and Theorem 3.4, there exist u m,n W 1,p () L () and w m,n L 1 ( ), w m,n (x) β(u m,n (x)) a.e. on, such that (3.21) a(., Du m,n ) D(u m,n v)+ + 1 m ψ m,n (u m,n v)+ u + m,n(u m,n v) 1 n u m,n (u m,n v)+ φ m,n (u m,n v), u m,n(u m,n v) for any v W 1,p () L (), v(x) D(β) a.e. on. Moreover (3.22) w m,n (u m,n v)+ u m,n + w m,n ψ m,n + φ m,n ψ + φ. Fixed m IN, by Theorem 3.3 (ii) and Theorem 3.4 (ii), {u m,n } n=m and {w m,n } n=m are monotone non increasing. Then, by (3.22) and the Monotone convergence theorem, there exists û m L 1 (), ŵ m L 1 ( ) and a subsequence n(m), such that u m,n(m) û m 1 1 m and w m,n(m) ŵ m 1 1 m. Thanks to Theorem 3.3 (ii) and Theorem 3.4 (ii), û m and ŵ m are non decreasing in m. Now, by(3.22), we have that û m and ŵ m are bounded. Using again the Monotone convergence theorem, there exist u L 1 () and w L 1 ( ) such that û m converges a.e. and in L 1 () to u and Consequently, ŵ m converges a.e. and in L 1 ( ) to w. u m := u m,n(m) converges a.e. and in L 1 () to u and (3.23) w m := w m,n(m) converges a.e. and in L 1 ( ) to w.

12 302 K. AMMAR F. ANDREU J. TOLEDO [12] Taking v = u m T k (u m )in(3.21) with n = n(m), (3.24) λ DT k (u m ) p k ( φ 1 + ψ 1 ), k IN. From (3.24), we deduce that T k (u m )isbounded in W 1,p (). suppose that Then, we can and T k (u m ) converges weakly in W 1,p () to T k (u), T k (u m ) converges in L p () and a.e. on to T k (u) T k (u m ) converges in L p ( ) and a.e. on tot k (u). Taking G = { u m u n k 2, u m A, u n A, C(x, A, t) k, Du m A, Du n A, Du m Du n >t}, and arguing as in Theorem 3.3, it is not difficult to see that Du m is a Cauchy sequence in measure. Similarly we can prove that DT k (u m ) converges in measure to DT k (u). Then, up to extraction of a subsequence, Du m converges to Du a.e. in. From here, (3.25) a(., DT k (u m )) converges weakly in L p () N and a.e. in to a(., DT k (u)). Let us see now that u T 1,p tr (). Obviously, u m u a.e. in. On the other hand, since DT k (u m )isbounded in L p () and DT k (u m ) DT k (u) inmeasure, it follows from [4, Lemma 6.1] that DT k (u m ) DT k (u) inl 1 (). Finally, let us see that γ(u m ) converges a.e. in. For every k>0, let A k := {x : T k (u)(x) <k} and C := k>0 A k. Then, by (3.22), (3.24) and the Trace theorem, there exists positive constants M 1,M 2 such that λ N 1 ({x : T k (u)(x) = k}) 1 k p T k (u) p (3.26) M 1 k p ( T k (u) T k (u) p 1 + DT k (u) p ) Hence, λ N 1 (C) =0. Thus, if we define in the function v by it is easy to see that v(x) =T k (u)(x) if x A k, (3.27) u n v =: τ(u) a.e. in. M 2 k p (kp 1 + k).

13 [13] Quasi-linear elliptic problems in L 1 with etc. 303 Therefore, u T 1,p tr () and moreover, by (3.26), u M p0 ( ), p 0 = inf{p 1, 1}, where M p0 ( ) is the Marcinkiewicz space of exponent p 0 (see, for instance, [5]). Since w m (x) β(u m (x)) a.e. on, from (3.23), (3.27) and from the maximal monotonicity of β, wededuce that w(x) β(u(x)) a.e. on. Finally let us pass to the limit in (3.21) to prove that u is an entropy solution of (S). For this step, we introduce the class F of functions S C 2 (IR) L (IR) satisfying S(0) = 0, 0 S 1, S (s) =0fors large enough, S( s) = S(s), and S (s) 0 for s 0. Let v W 1,p () L (), v(x) D(β) a.e. if D(β), and S F. u m S(u m v) astest function in (3.21) we get (3.28) a(x, Du m ) DS(u m v)+ + 1 m ψ m S(u m v)+ u + ms(u m v) 1 n(m) We can write the first term of (3.28) as Taking u m S(u m v)+ w m S(u m v)+ u n(m) S(u m v) φ m S(u m v). (3.29) a(x, Du m ) Du m S (u m v) a(x, Du m ) DvS (u m v). Since u m u and Du m Du a.e., Fatou s lemma yields a(x, Du) DuS (u v) lim inf a(x, Du m ) Du m S (u m v). m The second term of (3.29) is estimated as follows. Let r := v + S. By (3.25) (3.30) a(x, DT r u m ) a(x, DT r u) weakly in L p (). On the other hand, DvS (u m v) Dv L p (). Then, by the Dominated Convergence theorem, we have (3.31) DvS (u m v) DvS (u v) in L p () N.

14 304 K. AMMAR F. ANDREU J. TOLEDO [14] Hence, by (3.30) and (3.31), it follows that lim a(x, Du m ) DvS (u m v) = a(x, Du) DvS (u v). m Therefore, applying again the Dominated Convergence theorem in the other terms of (3.28), we obtain a(x, Du) DS(u v)+ ψs(u v)+ us(u v)+ ws(u v) φs(u v). From here, to conclude, we only need to apply the technique used in the proof of [4, Lemma 3.2]. Uniqueness. Let u be an entropy solution of problem (S), taking T h (u) as test function in (3.1), h>0, we have a(x, Du) DT k (u T h (u)) + ψt k (u T h (u)) + ut k (u T h (u)) + wt k (u T h (u)) φt k (u T h (u)). Now, using (H 1 ) and the positivity of the second and third terms, it follows that (3.32) λ Du p k ψ + k φ. {h< u <h+k} { u h} { u h} Let now u 1 and u 2 be entropy solutions of problem (S), following the lines of [4], we shall see that u 1 = u 2. Let w 1,w 2 L 1 ( ) with w 1 (x) β(u 1 (x)) and w 2 (x) β(u 2 (x)) a.e. on such that for every h>0, a(x, Du 1 ) DT k (u 1 T h (u 2 )) + u 1 T k (u 1 T h (u 2 ))+ + w 1 T k (u 1 T h (u 2 )) ψt k (u 1 T h (u 2 )) + φt k (u 1 T h (u 2 )) and a(x, Du 2 ) DT k (u 2 T h (u 1 )) + u 2 T k (u 2 T h (u 1 ))+ + w 2 T k (u 2 T h (u 1 )) ψt k (u 2 T h (u 1 )) + φt k (u 2 T h (u 1 )).

15 [15] Quasi-linear elliptic problems in L 1 with etc. 305 Adding both inequalities and taking limits when h goes to, onaccount of the monotonicity of β, weget (u 1 u 2 )T k (u 1 u 2 ) lim inf I h,k, h where I h,k := a(x, Du 1 ) DT k (u 1 T h (u 2 )) + a(x, Du 2 ) DT k (u 2 T h (u 1 )). Then, in order to prove that u 1 = u 2,itisenough to prove that (3.33) lim inf h I h,k 0 for any k. To prove this, we split where Ih,k 1 := Ih,k 2 := + Ih,k 3 := + Ih,k 4 := + { u 1 <h, u 2 <h} { u 1 <h, u 2 h} { u 1 <h, u 2 h} { u 1 <h, u 2 h} { u 1 h, u 2 <h} { u 1 h, u 2 <h} { u 1 h, u 2 <h} { u 1 h, u 2 h} { u 1 h, u 2 h} I h,k = I 1 h,k + I 2 h,k + I 3 h,k + I 4 h,k, (a(x, Du 1 ) a(x, Du 2 )) DT k (u 1 u 2 ) 0, a(x, Du 1 ) DT k (u 1 h sgn(u 2 ))+ a(x, Du 2 ) DT k (u 2 u 1 ) a(x, Du 2 ) DT k (u 2 u 1 ), a(x, Du 1 ) DT k (u 1 u 2 )+ a(x, Du 2 ) DT k (u 2 h sgn(u 1 )) a(x, Du 1 ) DT k (u 1 u 2 ), a(x, Du 1 ) DT k (u 1 h sgn(u 2 ))+ a(x, Du 2 ) DT k (u 2 h sgn(u 1 )) 0.

16 306 K. AMMAR F. ANDREU J. TOLEDO [16] Combining the above estimates we get where and Now, if we put we have L 2 h,k L 1 h,k := L 2 h,k := I h,k L 1 h,k + L 2 h,k, { u 1 <h, u 2 h} { u 1 h, u 2 <h} a(x, Du 2 ) DT k (u 2 u 1 ) a(x, Du 1 ) DT k (u 1 u 2 ). C(h, k) :={h < u 1 <k+ h} {h k< u 2 <h}, { u 1 u 2 <k, u 1 h, u 2 <h} C(h,k) a(x, Du 1 ) Du 1 + Then, by Hölder s inequality, we get Now, by (H 2 ), ( L 2 h,k ( + ) 1/p a(x, Du 1 ) p C(h,k) ) 1/p ) Du 2 p. C(h,k) ( ) 1/p ( a(x, Du 1 ) p C(h,k) c2 1 p a(x, Du 1 ) (Du 1 Du 2 ) C(h,k) (( a(x, Du 1 ) Du 2. ) 1/p Du 1 p + C(h,k) ( ) p c p g(x)+ Du 1 p 1 C(h,k) ( g p p + On the other hand, applying (3.32), we obtain {h< u 1 <k+h} {h< u 1 <k+h} ( Du 1 p k ψ + λ { u 1 h} { u 1 h} ) 1/p ) 1/p Du 1 p. φ )

17 [17] Quasi-linear elliptic problems in L 1 with etc. 307 and ( Du 2 p k ) ψ + φ. {h k< u 2 <h} λ { u 2 h k} { u 2 h k} Then, since u 1,u 2,φ,ψ L 1 () and u 1,u 2 M p0 ( ), we have that lim h L2 h,k =0. Similarly, lim h L 1 h,k =0. Therefore (3.33) holds. Finally, let u 1 be the entropy solution of problem (S) corresponding to φ 1 L 1 () and ψ 1 L 1 ( ) and let u 2 be the entropy solution of problem (S) corresponding to φ 2 L 1 () and ψ 2 L 1 ( ). As a consequence of uniqueness we can construct u 1 and u 2 following the proof of (i), then, taking into account Theorem 3.3 (ii) and Theorem 3.4 (ii), we prove (ii). Definition 3.5. Let us suppose that D(β) isclosed or a is smooth. For ψ L 1 ( ), let us define the operator A in L 1 () L 1 () by (u, φ) Aif u L 1 () T 1,p tr (), φ L 1 () and there exists w L 1 ( ), w(x) β(u(x)) a.e. on, such that a(., Du) DT k (u v)+ wt k (u v) ψt k (u v)+ φt k (u v) for all v W 1,p () L (), v(x) D(β) a.e. in, and all k>0. By Theorem 3.2 we have that A is an m-accretive operator. Moreover, it is not difficult to see that D(A) =L 1 (). Then by the Nonlinear Semigroup Theory it is possible to solve in the mild sense the evolution problem in L 1 () { ut + Au 0 in ]0, + [, u(0) = u 0 L 1 (). The mild solution of the above problem in the case ψ =0is characterized in [3] in the entropy sense for particular graphs β. 4 Existence and uniqueness of solutions of problem (P) Let us now study problem (P) { div a(., Du) =0 in a(., Du) η + u = ψ on, for any a satisfying (H 1 ), (H 2 ) and (H 3 ) and any ψ L 1 ( ).

18 308 K. AMMAR F. ANDREU J. TOLEDO [18] Using classical variational methods ([9], [10]), for every data ψ L ( ) this problem can be solved in W 1,p (). In fact, let us define the following capacity operator C : W 1 p,p ( ) W 1 p,p ( ) by < Cf,g >= a(., Du) Dv where u W 1,p () is the solution of the Dirichlet problem (D) { div a(., Du) =0 in u = f on, and v W 1,p () is such that γ(v) =g. Function u is called the A-harmonic lifting of f, where A is the operator associated to the formal differential expression diva(x, Du). It is easy to see that the operator C is bounded from W 1 p,p ( ) to its dual W 1 p,p ( ), hemicontinuous and strictly monotone. Therefore, (4.34) Cf + f = ψ has a unique solution f W 1 p,p ( ) L ( ). In the general case where ψ L 1 ( ), the variational methods are not available. For this reason we introduce a new concept of solution, named entropy solution, and we will give an existence and uniqueness result of solutions in this sense. Definition 4.1. A measurable function u : IR is an entropy solution of (P) if u T 1,p tr (), τ(u) L 1 ( ) and a(., Du) DT k (u v)+ ut k (u v) ψt k (u v) for all v L () W 1,p () and all k>0. Theorem 4.2. For any ψ L 1 ( ), there exists a unique entropy solution of problem (P). Moreover, if u 1 is an entropy solution of problem (P) corresponding to ψ 1 L 1 ( ) and u 2 is an entropy solution of problem (P) corresponding to ψ 2 L 1 ( ) then u 1 u 2 ψ 1 ψ 2.

19 [19] Quasi-linear elliptic problems in L 1 with etc. 309 Proof. Let n IN, using Theorem 3.2 with β(r) =r for all r IR and φ =0,wehave that, given ψ L 1 ( ), there exists u n L 1 () T 1,p tr (), τ(u n ) L 1 ( ), such that a(., Du n ) DT k (u n v)+ 1 u n T k (u n v)+ u n T k (u n v) n (4.35) ψt k (u n v) for all v L () W 1,p () and all k>0. Taking v =0as test function in (4.35), and using (H 1 ), it is easy to see that (4.36) 1 DT k (u n ) p M k λ n IN and k >0, (4.37) u n M n IN and 1 (4.38) n u n M n IN, where M = ψ L1 ( ). Then, by (4.36), we can suppose that T k (u n ) converges weakly in W 1,p () to σ k W 1,p (), T k (u n ) converges in L p () and a.e. to σ k and T k (u n ) converges in L p ( ) and a.e. to σ k. Since there exists C 1 > 0 such that, for all n IN and for all k>0, ( ) 1/p ( ( ) ) 1/p T k (u n ) p C 1 T k (u n ) + DT k (u n ) p, where p = Np N p,wededuce, thanks to (4.36) and (4.37), that there exists C 2 > 0 such that T k (u n ) L p () C 1 (M + ( Mk λ ) 1 ) p C 2 k 1 p k 1. Now, λ N {x : σ k (x) = k} T k (u n ) p lim inf n k p σ k p k p C p 1 2 for all k 1. k N(p 1)/(N p)

20 310 K. AMMAR F. ANDREU J. TOLEDO [20] Hence, there exists C 3 > 0 such that λ N {x : σ k (x) = k} C 3 1 k N(p 1)/(N p) for all k>0. Let u be defined on by u(x) =σ k (x) on{x : σ k (x) <k}. Then and we can suppose that and u n converges to u a.e. in, T k (u n ) converges weakly in W 1,p () to T k (u) W 1,p (), T k (u n ) converges in L p () and a.e. to T k (u), T k (u n ) converges in L p ( ) and a.e. to T k (u). Consequently, u T 1,p (). On the other hand, thanks to (4.37) λ N 1 {x : T k (u)(x) = k} 1 k 1 lim inf T k (u n ) M k n k. T k (u) Therefore, if we define v(x) =T k (u)(x) on{x : T k (u)(x) <k}, u n v a.e. in. Consequently, u T 1,p tr () and, by (4.37), u L 1 ( ). Taking G = { u m u n k 2, u m A, u n A, C(x, A, t) k, Du m A, Du n A, Du m Du n >t}, and arguing as in Theorem 3.3, it is not difficult to see that Du m is a Cauchy sequence in measure. Similarly, DT k (u m ) converges in measure to DT k (u). Then, up to extraction of a subsequence, Du m converges to Du a.e. in. From here, a(., DT k (u m )) converges weakly in L p () N and a.e. in to a(., DT k (u)). Let us see finally that (4.39) (4.40) u n converges to u in L 1 ( ), 1 n u n converges to 0 in L 1 ().

21 [21] Quasi-linear elliptic problems in L 1 with etc. 311 In fact, taking v = T h (u n )astest function in (4.35), dividing by k and letting k 0, we get 1 (4.41) u n + u n ψ. n {x : u n(x) h} {x : u n(x) h} {x : u n(x) h} Now, by (4.37), λ N 1 {x : u n (x) h} 0ash +. Then, by (4.41), it is easy to see that the sequence { 1 n u n} is equiintegrable in L 1 () and that the sequence {u n } is equiintegrable in L 1 ( ). Since 1 n u n 0 a.e. in and u n u a.e. in, applying Vitali s convergence theorem we get (4.39) and (4.40). We can then pass to the limit in (4.35) (as in the proof of Theorem 3.2) to conclude that u is an entropy solution of (P ). Let us prove now the uniqueness. Let u 1 be an entropy solution of problem (P) corresponding to ψ 1 L 1 ( ) and u 2 be an entropy solution of problem (P) corresponding to ψ 2 L 1 ( ). Working as in the proof of the uniqueness of Theorem 3.2, we get (ψ 1 ψ 2 )T k (u 1 u 2 ) (u 1 u 2 )T k (u 1 u 2 ) ( lim inf a(x, Du 1 ) DT k (u 1 T h (u 2 ))+ h + ) + a(x, Du 2 ) DT k (u 2 T h (u 1 )) ( (4.42) and lim inf h lim h + (a(x, Du 1 ) a(x, Du 2 )) DT k (u 1 u 2 )+ { u 1 <h, u 2 <h} { u 1 <h, u 2 h} { u 1 h, u 2 <h} ( a(x, Du 2 ) DT k (u 2 u 1 )+ ) a(x, Du 1 ) DT k (u 1 u 2 ), a(x, Du 2 ) DT k (u 2 u 1 )+ { u 1 <h, u 2 h} ) + a(x, Du 1 ) DT k (u 1 u 2 ) { u 1 h, u 2 <h} =0. Since { u 1 <h, u (a(x, Du 2 <h} 1) a(x, Du 2 )) DT k (u 1 u 2 ) 0, dividing by k and letting k 0, we get that u 1 u 2 ψ 1 ψ 2.

22 312 K. AMMAR F. ANDREU J. TOLEDO [22] In order to prove that u 1 = u 2 in if ψ 1 = ψ 2,itisenough to observe that the inequalities (4.42) become equalities. Consequently lim inf (a(x, Du 1 ) a(x, Du 2 )) DT k (u 1 u 2 )=0. h + { u 1 <h, u 2 <h} From here, since { u 1 <h, u (a(x, Du 2 <h} 1) a(x, Du 2 )) DT k (u 1 u 2 )ispositive and non decreasing in h, itfollows that DT h (u 1 )=DT h (u 2 ) a.e. in for all h, but since u 1 = u 2 a.e. in, we get u 1 = u 2 a.e. in. Definition 4.3. We define the following operator B in L 1 ( ) L 1 ( ) by (f,ψ) Bif f,ψ L 1 ( ) and there exists u T 1,p tr () with τ(u) =f such that a(., Du) DT k (u v) ψt k (u v), for all v L () W 1,p () and all k>0. By Theorem 4.2, B is an m-accretive operator in L 1 ( ). Now, on the one hand, operator C considered as an operator on L 1 ( ) L 1 ( ), denoted again C, iscompletely accretive (see [6]). In fact, let ρ C (IR), 0 ρ 1, supp(ρ ) compact and 0 / supp(ρ). If (f 1,ψ 1 ), (f 2,ψ 2 ) C, then, (ψ 1 ψ 2 )ρ(f 1 f 2 )= (a(., Du 1 ) a(., Du 2 )) Dρ(u 1 u 2 )= = (a(., Du 1 ) a(., Du 2 )) D(u 1 u 2 )p (u 1 u 2 ) 0, where u i is the A-harmonic lifting of f i, i = 1, 2. Consequently, by (4.34), C L1 ( ) L 1 ( ) is m-accretive in L 1 ( ). On the other hand, if (f,ψ) Cthen <ψ,t k (û v) >= a(., Dû) DT k (û v), for any v W 1,p () L (), where û W 1,p () is the solution of the Dirichlet problem { div a(., Dû) =0 in û = f on. Therefore (f,ψ) B, and consequently, since B is m-accretive, C L1 ( ) L 1 ( ) = B.

23 [23] Quasi-linear elliptic problems in L 1 with etc. 313 Remark 4.4. In [1], the operator B is also characterized as follows, (f,ψ) B if f,ψ L 1 ( ), T k (f) W 1 p,p ( ) for all k>0 and < C(g + T k (f g)),t k (f g) > ψt k (f g), for all g L ( ) W 1 p,p ( ) and for all k>0. Remark 4.5. It is not difficult to see that D(B) isdense in L 1 ( ). Then, by the Nonlinear Semigroup Theory, it is possible to solve in the mild sense the evolution problem in L 1 ( ) { ut + Bu =0 in ]0, + [, u(0) = u 0 L 1 ( ), which rewrites, from the point of view of Nonlinear Semigroup Theory, the following problem div a(x, Du) =0 in ]0, + [, u (t)+a(x, Du) η =0 on ]0, + [, u(0) = u 0 L 1 ( ). In a forthcoming paper the mild solutions of the above problem will be characterized in the entropy sense. Acknowledgements We want to thank J. M. Mazón and S. Segura de León for many suggestions and interesting remarks during the preparation of this paper. REFERENCES [1] K. Ammar: Solutions entropiques et renormalisées de quelques E.D.P. non linéaires dans L 1, Thesis, Université Louis Pasteur, Strasbourg, [2] F. Andreu J. M. Mazón S. Segura de León J. Toledo: Quasi-linear elliptic and parabolic equations in L 1 with nonlinear boundary conditions, Adv. Math. Sci. Appl., 7 (1) (1997), [3] F. Andreu J. M. Mazón S. Segura de León J. Toledo: Existence and uniqueness for a degenerate parabolic equation with L 1 -data, Trans. Amer. Math. Soc., 351 (1) (1999), [4] Ph. Bénilan L. Boccardo Th. Gallouët R. Gariepy M. Pierre J. L. Vázquez: An L 1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 22 (2) (1995),

24 314 K. AMMAR F. ANDREU J. TOLEDO [24] [5] Ph. Benilan H. Brezis M. G. Crandall: A semilinear equation in L 1 (R N ), Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 2 (4) (1975), [6] Ph. Bénilan M. G. Crandall: Completely accretive operators, InSemigroup theory and evolution equations (Delft, 1989), Lecture Notes in Pure and Appl. Math., vol. 135, pp , Dekker, New York, [7] Ph. Bénilan M. G. Crandall P. Sacks: Some L 1 existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions, Appl. Math. Optim., 17 (3) (1988), [8] L. Boccardo Th. Gallouët: Nonlinear elliptic equations with right-hand side measures, Comm. in Partial Diff. Equations, 17 (1992), [9] D. Kinderlehrer G. Stampacchia: An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, [10] J. Leray J. L. Lions: Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), Lavoro pervenuto alla redazione il 5 aprile 2004 ed accettato per la pubblicazione il 1 febbraio Bozze licenziate il 26 settembre 2006 INDIRIZZO DEGLI AUTORI: K. Ammar U.L.P U.F.R de Mathématiques et Informatique 7 rue René Descartes Strasbourg (France) F. Andreu J. Toledo Departamento de Análisis Matemático Universitat de València Dr. Moliner Burjassot (Spain) The second and third authors have been partially supported by PNPGC project, reference BFM