On obstacle problems for non coercive linear operators

In this paper we prove the existence and uniqueness of the solution to the one and the two obstacles problems associated with a linear elliptic operator, which is non coercive due to the presence of a convection term. We show that the operator is weakly T-monotone and, as a consequence, we establish the Lewy–Stampacchia dual estimates and we study the comparison and the continuous dependence of the solutions as the obstacles vary. As an application, we prove also the existence of solutions for a class of non coercive implicit obstacle problems.


Introduction and main results
We consider various obstacle problems associated with the partial differential operator, from H 1 0 (Ω) into its dual (H 1 0 (Ω)) � = H −1 (Ω) , defined by Av = Lv + D ⋅ (vE) = −D ⋅ (M Dv − vE) 1 3 where Ω is a bounded, open subset of ℝ N , N > 2 , M = M(x) is an elliptic matrix with bounded and measurable coefficients satisfying with positive constants and and E = E(x) is a convection vector field such that Here, given a function v and a vector field G, we denote Dv = grad v and D ⋅ G = div G.
Let be a non empty convex set of H 1 0 (Ω) and We consider the following variational inequality, briefly denoted by (A, , F) where ⟨⋅, ⋅⟩ denotes the duality between H 1 0 (Ω) and its dual, so that Here we shall consider the cases where is the convex set related to the lower or the upper or the two obstacles problem, by setting with when the lower obstacle problem is considered, or with when the constraint is the upper obstacle, while for the two obstacles problem we set with (1) | | 2 ≤ M(x) ⋅ , |M(x)| ≤ , a.e. x ∈ Ω, ∀ ∈ ℝ N (3) F ∈ H −1 (Ω).

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On obstacle problems for non coercive linear operators The Dirichlet problem associated with general elliptic operators of second order with discontinuous coefficients has been studied in the well known paper [18] by G. Stampacchia who proved the existence, uniqueness and regularity of the weak solution in the coercive case assuming more summability on E than (2) or a smallness condition of ‖E‖ L N with respect to the ellipticity constant in the bounded domain Ω . We recall that for dimensions greater than two the power N is optimal, by the Sobolev inequality ‖v‖ L 2 * ≤ C * ‖Dv‖ L 2 , for v ∈ H 1 0 (Ω) where 2 * = 2N∕(N − 2) . In the framework of variational inequalities associated with the non coercive operator A with other lower order terms, existence and comparison results were proven in [4] and in Section 4.7 of [15], under the assumption D ⋅ E ≥ 0 in addition to (2).
Without additional assumptions on E, only with the general integrability condition (2) the operator A is not monotone and the coerciveness fails; nevertheless, the existence and uniqueness results of G. Stampacchia have been retrieved in [1] for the Dirichlet problem using a nonlinear approach.
Here assuming only (2) and developing the nonlinear approximation of [1], which produces the key a priori estimate (21) below, we prove first the existence of a solution of the problem (4) with one or two obstacles.
Moreover, proving an important property of the operator A, the so-called weak T-monotonicity (see below for the definition), we derive some comparison principles and the dual estimates of Lewy-Stampacchia, extending [10] (see also [14,15,17] and their references).
Finally, we extend the continuous dependence of the solutions with respect to the Mosco-convergence of the convex sets to these non coercive obstacle problems. This allows us to consider, as an application motivated by a semiconductors problem that can be modelled as an implicit obstacle problem, the existence of solutions to certain quasi-variational inequalities of obstacle type, when the convex = [u] depends on the solution u through appropriate nonlinear mappings Ψ ∶ u ↦ = Ψ(u) and Φ ∶ u ↦ = Φ(u).
We observe that, by Sobolev embeddings, all ours results are still valid in dimension N = 2 with E ∈ [L p (Ω)] 2 , with any p > 1 , and in dimension N = 1 with E ∈ L 1 (Ω).

Existence and comparison theorems
The first goal of the paper is to prove the following existence result Theorem 1 Assume that hypotheses (1), (2) and (3) hold. If assumption (5), or (6) or (7) is satisfied then there exists a unique solution u to the lower obstacle, or to the upper obstacle or to the two obstacles problem (4), respectively.
Next, we will highlight a property of the pseudomonotone operator A, which in general is not monotone due to the presence of a general convection term. To this aim we need to specify some notations.
Given h > 0 , T h (s) denotes the standard truncation function defined by and, as usual, we set We will prove the following

Theorem 2 Assume that hypothesis (1) holds and let
Then the operator A is weakly T-monotone.
We point out that the new notion of weakly T-monotonicity given above plays, in our framework, the role of the T-monotonicity property introduced by Brézis and Stampacchia in [6] and allows us to obtain some new comparison results and duality inequalities already known for the coercive obstacle problems.
First of all we derive the following comparison principles, which, in turn, imply the uniqueness of the solution of the problem (A, , F) in the cases of one (lower or upper) and two obstacles. (1), (2) hold and let such that and or or, for the case of the two obstacles, for i = 1, 2, let also

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On obstacle problems for non coercive linear operators Let u i denote the solutions of one of the following obstacle problems (A, corresponding respectively, to the lower obstacle, to the upper obstacle or to the two obstacles case. Then

Lewy-Stampacchia inequalities
As already pointed out, the notion of weakly T-monotonicity of the operator A introduced above also allows the extension to our framework of some dual estimates already known for solutions of variational inequalities related to coercive and T-monotone operators (see [6,14,15,17]). For this purpose we recall some definitions.
Let X be an Hilbert space, which is a vector lattice with respect to a partial order relation ≤ (that is, we denote by V * the subspace of the dual space V ′ generated by the positive cone that is, V * = P � − P � , where ⟨ ⋅, ⋅⟩ denotes the duality pairing between V and V ′ . The space V * is called the order dual of V. It is well known that the space H 1 (Ω) is a vector lattice under the ordering Moreover, H 1 0 (Ω) is a sublattice and F ∈ H −1 (Ω) is a positive element for the dual ordering if and only if F is a positive distribution, hence a positive measure belonging to H −1 (Ω) . Thus, the order dual [H 1 0 (Ω)] * is the space of all distributions in H −1 (Ω) which can be written as the difference of two positive measures belonging to H −1 (Ω).
The following result follows from the comparison corollary stated above and extends a property already known for coercive and strictly T-monotone operator (see Theorem 3.2 of [14]) to the non coercive and weakly T-monotone operator A. (1) and (2) hold and let u, v ∈ H 1 0 (Ω) such that u 2 ≥ u 1 , a.e. in Ω.

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Then, In particular, since A0 = 0, we also have We say that w ∈ H 1 (Ω) is a supersolution to the lower obstacle problem (A, , F) and w ≥ 0 on Ω , in the sense that w − ∈ H 1 0 (Ω) . We can also extend to non coercive operators the Stampacchia's result which establishes that the solution to (A, , F) is the lowest of its supersolutions (see [15] and its references). Defining similarly subsolutions, using instead ≤ , to the upper obstacle problem (A, , F) we leave to the reader the formulation of the symmetrical case of the following interesting properties for the one obstacle problem.
Next we state the following Lewy-Stampacchia inequalities for the one and the two obstacle problems associated to weakly T-monotone operators.

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On obstacle problems for non coercive linear operators Remark 2 The Lewy-Stampacchia inequalities for the two obstacles problem in abstract form for T-monotone coercive operators have been proved in [17].

Mosco convergence
Next, we consider the continuous dependence of the solution u with respect to the Mosco convergence of the convex sets, which definition we recall for convenience.
The first result we will state concerns the stability of the solutions of lower (or upper) obstacle problems, as the obstacles vary. if suffices that one of the following assertions holds (see [2,3,12,13]): 1. the sequence { n } is weakly convergent in H 1 0 (Ω) to 0 , and n ≤ 0 , ∀ n; 2. the sequence { n } is weakly convergent in L p (Ω) to 0 and decreasing; 3. the sequence { n } strongly converges in H 1 0 (Ω) to 0 ; Remark 4 The convergence of the solutions of the upper obstacles problems, as the obstacles vary, reads exactly in the same manner, just by replacing n by n and n by n . Also the sufficient conditions for the Mosco convergence of the convex sets hold with the obvious adaptations to upper obstacles.
To conclude, we state the similar result on the convergence of the solutions of the two obstacles problem. Indeed, for any sequence v n ∈ n n such that v n ⇀ v 0 weakly in H 1 0 (Ω) , for a subsequence, we have v n → v 0 in L 2 (Ω) and a.e. in Ω , so it is clear that v 0 ∈ 0 0 . On the other hand, for every v 0 ∈ 0 0 we have v n = ( n ∨ v 0 ) ∧ n ∈ n n and the strong convergence of the obstacles implies v n → v 0 in H 1 0 (Ω).

Implicit obstacle problems
Suppose that the obstacles depend, through some functional relation, from the solution u, by assuming given the mappings and therefore, in each one of the three cases (5), (6) and (7), we also have a func- n ≤ n , n → 0 and n → 0 strongly in H 1 0 (Ω).

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On obstacle problems for non coercive linear operators Then each implicit obstacle problem may be formulated as a quasivariational inequality, denoted by (A, [u], F) Theorem 7 Assume that the assumptions (1), (2) and (3) hold and the obstacle mappings Ψ ∶ L 2 (Ω) → H 1 0 (Ω) and Φ ∶ L 2 (Ω) → H 1 0 (Ω) are continuous and have bounded ranges. Then, there exists at least a solution to the implicit lower (resp. upper) obstacle problem (18) , then there exists at least a solution to the implicit non coercive two obstacles problem (18)

Remark 6
Other examples of implicit obstacle problems are discussed for instance in [14]. In general the uniqueness of the solution cannot be expected. However, in special situations, the uniqueness of certain coercive implicit obstacle problems can be obtained under smallness conditions on the data as, for instance, in the case of a semiconductor model [16]. In Sect. 5 we also give an application of Theorem 7.

Proof of Theorems 1 and 2
For each n ∈ ℕ let u n be a solution of the following approximated obstacle problem The existence of u n follows by the well known results by [8,11], since, for every n ∈ ℕ , the nonlinear composition is bounded with respect to u n .

Proof of Theorem 1
We begin with the proof in the case = .
Step 1: We claim that there exists a positive constant C 0 = C , ,E, ,F,N,Ω , independent of n, such that ∀k > 0 (19). Note that v ≥ . Then Then the use of the Young inequality in the right-hand side yields to and by the Poincaré inequality ( Recall that 0 ≤ u n − ∈ H 1 0 (Ω) implies also log(1 + u n − ) ∈ H 1 0 (Ω). Since we deduce that and (20) follows.
Step 2: We prove that there exists a positive constant C 1 , independent of n such that Let k > 0 and G k (s) be the function defined by On obstacle problems for non coercive linear operators Using the ellipticity condition in the left hand side and the Young inequality in the right one, we obtain Note that the function v = u n − G k (u n − ) is an admissible test function in (19) and that

This choice implies
Then the use of the Sobolev inequality with constant C * yields Now, by virtue of (20) we can choose k > 0 such that and we prove an estimate on |DG k (u n − ) in L 2 (Ω) , since the left hand side grows quadratically, while the right hand side grows linearly, with respect to ‖DG k (u n − )‖ L 2 . At last, taking into account the estimate (22) we obtain (21).
Step 3: As a consequence of (21) there exists a subsequence, still denoted by u n , and a function u such that Note that u ∈ . We prove that u is a solution of the problem (4). Given w ∈ we take in (19) v = u n − T k (u n − w) and we get Here we have used the property u n = T k (u n − w) + w if |u n − w| < k and DT k (u n − w) = 0 if |u n − w| ≥ k . Thanks to the weak lower semicontinuity of the quadratic form H 1 0 (Ω) ∋ ↦ ⟨L , ⟩ we pass to the limit as n → +∞ in the left hand side; moreover, since the sequence T k (u n −w) 1+ 1 n |u n | is bounded we use the Lebesgue theorem in the right hand side and we have that is Taking the limit as k → ∞ and observing that T k (u − w) → u − w in H 1 0 (Ω) , we obtain that is, u is a solution of the obstacle problem (A, , F) . Finally, the uniqueness will be a consequence of the next Corollary 1.
In order to prove the existence of solution for the upper obstacle problem, we just observe that u is solution of (A, , F) iff −u solves the lower obstacle problem (A, − , −F).

On obstacle problems for non coercive linear operators
Finally, the existence of solution of the two obstacles problem (A, , F) can be achieved essentially by repeating the proof made for the lower obstacle problem. Indeed, first we remark that u is the solution of the two obstacles problem (A, , F) iff u * = u − is the solution of (A, 0 * , F * ) , with * = − ∈ H 1 0 (Ω) and F * = F − L + D ⋅ ( E) ∈ H −1 (Ω) . Secondly, we repeat the three steps of the proof noting that in the corresponding approximating problem for u * n ∈ 0 * the test func- , with any w ∈ 0 * are negative and therefore are admissible, since they belong also to 0 * . ◻

Proof of Theorem 2
Let v ∈ H 1 0 (Ω) such that Then and Observe that the inequality and the Poincaré inequality (with constant P > 0 ) yield Now, we fix > 0 and let 0 < h < . We note that Since the last integral goes to zero as h → 0 we obtain

Remark 7
The main point of the above proof is the inequality (24). It is worth noting that, it is only needed the L 2 -summability of E, instead of the L N -summability required in the proof of the existence result.
Proof of Corollary 1 Let u 1 , u 2 be solutions of the obstacle problems (A, 1 , F 1 ) and (A, 2 , F 2 ) , respectively. Given h > 0 , we may choose v = u 1 − T h ((u 1 − u 2 ) + ) as test function in the formulation of the problem (A, 1 , F 1 ) and v = u 2 + T h ((u 1 − u 2 ) + ) in the formulation of the problem (A, 2 , F 2 ) . Thus, we have Adding the two above inequalities and using the assumption F 1 ≤ F 2 in H −1 (Ω) we deduce and the thesis of Corollary 1 for the lower obstacle problem easily follows applying the weakly T-monotonicity property of the operator A.
For the two obstacles problem the proof is similar. ◻

Proof of the Lewy-Stampacchia inequalities
In this section we will prove the Lewy-Stampacchia inequalities (11) and (13).

Proof of Corollary 2
The proof is similar to the case of coercive T-monotone operators of [14]. Let z ∈ H 1 0 (Ω) be the unique solution of (A, u∧v , Au ∧ Av) , that is the lower obstacle problem The existence and uniqueness of z follows by Theorem 1 and Corollary 1. Let ∈ H 1 0 (Ω) such that ≥ 0 . The function w = z + belongs to u∧v and choosing w as test function in (25) we obtain which, in turn, implies Since u and v are the solutions of the obstacle problems (A, u∧v , Au) and (A, u∧v , Av) , respectively, and Au ∧ Av ≤ Au and Au ∧ Av ≤ Av , applying twice Corollary (1) we deduce and then Thus z = u ∧ v and the conclusion follows by (26).
The proof of the second inequality in (9) can be performed in a similar way, using the existence, uniqueness and comparison results for the upper obstacle problem.

Now, taking w = z + T h ((u − z) + ) as test function in (27) we have and adding the last two inequalities we get
Again, by virtue of the weakly T-monotonicity of the operator A we have u ≤ z . Thus, z = u and (28) yields the conclusion. ◻

Proof of Theorem 4
The proof is an extension to non coercive weakly T-monotone operators of the one in [17]. Let u ∈ be the solution of the problem (A, , F) . Thus u ∈ and it satisfies the following inequality The upper bound in the inequality (13) can be achieved as in the proof of Theorem 3. In order to prove the lower bound in (13), let z ∈ H 1 0 (Ω) be the unique solution of the one obstacle problem (A, u , F ∧ A ) , that is For any ∈ H 1 0 (Ω) , ≥ 0 we test problem (31) with the function w = z + (note that w ≥ u ) and we obtain Our goal is to prove that z = u and then it is enough to show that z ≤ u . First of all, we claim that z ≤ . As a matter of the fact, for any h > 0 the choice w = z − T h ((z − ) + ) (note that w ≥ u ) in (31) gives which in turn implies By virtue of the weakly T-monotonicity of the operator A we conclude that z ≤ and, since z ≥ u ≥ , we get z ∈ .
Adding the last two inequalities we have

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On obstacle problems for non coercive linear operators which implies z ≤ u . ◻ Remark 8 It is well known, since [9] for the one obstacle problem and [19] for the two obstacles problem, that some regularity results on the solution of the obstacle problems (A, , F) can be obtained as a consequence of the Lewy-Stampacchia inequalities as they are a direct consequence of the regularity results already known for the solutions of the Dirichlet problem. Thus, by the linear theory, we may conclude (see [1]): 2 and E ∈ (L (Ω)) N , > N , then u ∈ L ∞ (Ω).

Remark 9
Under the assumption (32) the Lewy-Stampacchia inequality (33) can be obtained through an alternative proof based on an approximation based in the bounded penalization with a family of monotone increasing Lipschitz functions n introduced in [9], for instance with n (s) = 0 for s ≤ 0 and n (s) = 1 for s ≥ 1∕n (see [15], for instance) or with a homographic function n (s) = s |s|+1∕n (see [5]). This method, both of theoretical and numerical interest, represents an alternative to the classical unbounded penalty method.
We set For each n ∈ ℕ , let u n be a weak solution of the following approximating Dirichlet problem: u n ∈ H 1 0 (Ω), The existence of u n follows by the well known results by [8,11], since, for every n, the nonlinear composition is bounded with respect to u n . Given h > 0 let us take T h ((u n − ) − ) as test function in the weak formulation of (35) and we have Since and we obtain that is Now, working as in the proof of Theorem 2 we get (u n − ) − = 0 a.e. in Ω. Consequently, u n being a weak solution of the Dirichlet problem, the positivity of g yields, in the sense of distributions, the following inequalities The proof of the boundedness of the sequence {u n } in H 1 0 (Ω) can be carried on as in the proof of Theorem 1 (see step 2). Thus, there exists a subsequence, still denoted by u n and a function u such that Note that u ∈ ; moreover, letting n → +∞ in the distributional sense in (35) and in the inequalities (36), we deduce that u is the solution of the problem (4) and satisfies the Lewy-Stampacchia estimate (37) u n ⇀ u weakly inH 1 0 (Ω) u n → u strongly inL 2 (Ω)and a.e. inΩ.

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On obstacle problems for non coercive linear operators

Proof of the Mosco convergence
This section is devoted only to the proof of Theorem 5, since the proof of Theorem 6 can be performed exactly in the same way.
Let u n and u 0 be the solutions of the obstacle problems (A, n , F) and (A, 0 , F) , respectively, that is Working as in the proof of Theorem 1 (step 1 and step 2) we deduce that the sequence {u n } is bounded in H 1 0 (Ω) ; thus, there exists a subsequence, which we denote also by {u n } , which weakly converges to some u * in H 1 0 (Ω) . Since u n ∈ n , the condition (ii) of the definition of Mosco convergence implies that u * ∈ 0 , that is u * ≥ 0 , a.e. in Ω.
Now, we claim that We note that, the sequence {u 2 n } is bounded in W 1,q 0 (Ω) . Thus, up another subsequence, we can say that {u 2 n } weakly converges in W 1,q 0 (Ω) to some w ∈ W 1,q 0 (Ω) . Let Φ ∈ (L (Ω)) N , > q ′ . We have Since u n strongly converges to u in L (Ω) , < 2 * , and Du n weakly converges to Du in (L 2 (Ω)) N , we may to pass to the limit and we deduce that and then w = u 2 * . Moreover Now we need to prove that u * = u 0 . By (i) of the definition of Mosco-convergence, for every w 0 ∈ 0 , there exists a sequence {w n } , with w n ∈ n and ||w n − w 0 || H 1 0 (Ω) → 0 . We take v = w n as test function in (38) and we have ⟨Lu n , u n − w n ⟩ ≤ � Ω u n E ⋅ D(u n − w n ) + ⟨F, u n − w n ⟩.

Then
We pass to the limit thanks to the weak lower semicontinuity of the quadratic form ⟨Lv, v⟩ , (41) and ||w n − w 0 || H 1 0 (Ω) → 0 . Thus we obtain Then the uniqueness of the solution of the unilateral problem on the convex K 0 implies that u * = u 0 and the full sequence {u n } weakly converges to u 0 . Finally, we prove that the sequence {u n } strongly converges to u 0 . Since u 0 ∈ 0 , due to the condition (i) of the definition of Mosco convergence, there exists a sequence {z n } , with z n ∈ n , ||z n − u 0 || H 1 0 (Ω) → 0 , . We can take v = z n in (38) and we have where the right hand side converges to zero also thanks (41). Thus we have which says that the sequence {u n } converges strongly in H 1 0 (Ω) to the same limit of the sequence {z n } ; that is {u n } converges strongly in H 1 0 (Ω) to u 0 . ◻

Application to quasi-variational inequalities of obstacle type
In this section we prove Theorem 7 and we provide an example of application inspired in a semiconductor model (see [7,16] for references of the physical free boundary problem).

Proof of Theorem 7
We limit ourselves to the two obstacles problem, since the other two cases are similar. For any w ∈ L 2 (Ω) , denote by u w = S(w) the unique solution to (A, , F) . By the Theorem 1, it is clear that the solution map S ∶ L 2 (Ω) ↦ Φ(w) Ψ(w) ⊂ H 1 0 (Ω) is well defined and its range is bounded, since the ranges of Ψ and of Φ are also bounded in H 1 0 (Ω) by hypothesis.
Since Ψ and of Φ are also continuous and compatible, i.e. Ψ(w) ≤ Φ(w) for all w ∈ L 2 (Ω) , if we take a sequence w n → w in L 2 (Ω) , by Theorem 6 we have Ψ(w) and consequently S(w n ) → S(w) in H 1 0 (Ω).
⟨L(u n − w n ), u n − w n ⟩ + ⟨Lw n , u n − w n ⟩ ≤ � Ω u n E ⋅ Du n − � Ω u n E ⋅ Dw n + ⟨F, u n − w n ⟩.

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On obstacle problems for non coercive linear operators Therefore the solution map S is also continuous from L 2 (Ω) into H 1 0 (Ω) . Finally, since the embedding of H 1 0 (Ω) into L 2 (Ω) is compact, the image of S, S(L 2 (Ω)) , is not only bounded but in fact also compact in L 2 (Ω) . Then by the Schauder fixed point theorem, there is a u = S(u) ∈ Φ(u) Ψ(u) solving (18) for the implicit two obstacles problem. ◻ A simplified variant of a model with a free boundary for the determination of the depletion zone in certain semiconductor diodes, which is the non coincidence set of a two obstacles problem, corresponds to an asymptotic coupled system (44)-(45)-(46) below for the electrostatic potential u = u(x) , subjected to a drift by the vector field E = E(x) , x ∈ Ω , and lying in between two Fermi quasi-potentials = (x) and = (x) , which are both functions depending implicitly of the potential u through two Dirichlet problems depending on a function v ∈ L 2 (Ω) of the following type: Here the coefficients B(v)(x) = B(x, v(x)) are given by a Carathéodory matrix B(x, s) ∶ Ω × ℝ → ℝ N 2 , i.e., it is measurable in x for each s ∈ ℝ and continuous in s for a.e. x ∈ Ω , satisfying for positive constants * and * : It is clear that for G ∈ H −1 (Ω) and any v ∈ L 2 (Ω) the problem (42) has a unique solution w = T(v, G) ∈ H 1 0 (Ω) . Fixing G, we have the continuity of the map T G ∶ L 2 (Ω) ∋ v ↦ w = T(v, G) ∈ H 1 0 (Ω) for the strong topologies. Indeed v n → v in L 2 (Ω) , then by (43) and Lebesgue theorem, B n = B(v n ) → B(v) = B in L p (Ω) N 2 , for all 1 ≤ p < ∞ , and for a subsequence also a.e. in Ω . Since by (43) we have * ‖Dw n ‖ L 2 ≤ ‖G‖ H −1 , we may assume that w n ⇀ w * in H 1 0 (Ω)-weak and the strong convergence of B n → B clearly implies w * = w.
Also by Lebesgue theorem, ‖(B n − B)Dw‖ L 2 → 0 , and comparing the equations (42) in variational form for w n = T(v n , G) and for w = T(v, G) , by addition, we obtain which by (43) yields and therefore w n → w in H 1 0 (Ω)-strong. So the continuity of T G holds. By the comparison principle, for any fixed v ∈ L 2 (Ω) , and given G, H ∈ H −1 (Ω) , such that