APPROXIMATION OF OPTIMAL CONTROL PROBLEMS IN THE COEFFICIENT FOR THE p-LAPLACE EQUATION . I . CONVERGENCE RESULT

We study a Dirichlet optimal control problem for a quasi-linear monotone elliptic equation, the so-called weighted p-Laplace problem. The coefficient of the p-Laplacian, the weight u, we take as a control in BV (Ω) ∩ L∞(Ω). In this article, we use box-type constraints for the control such that there is a strictly positive lower and some upper bound. In order to handle the inherent degeneracy of the p-Laplacian, we use a regularization, sometimes referred to as the ε-p-Laplacian. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted ε-pLaplacian, where we approximate the nonlinearity by a bounded monotone function, parametrized by k. Further, we discuss the asymptotic behavior of the solutions to the regularized problem on each (ε, k)-level as the parameters tend to zero and infinity, respectively.


Introduction.
Control in the coefficients of elliptic problems has a long history of its own, starting with the work of Murat [10,11] and Tartar [14].The constrained optimal control problem (OCP) in the coefficients of the leading order differential expressions was first discussed in detail by Casas [2] in the case of the classical Laplace equation, where the scalar coefficient u in the div(u∇•) formulation was taken as control satisfying box constraints with a strictly positive lower and some upper bound together with a slope constraint.The problem of existence and uniqueness of the underlying boundary value problem and the corresponding OCP was treated, and an optimality system has been derived and analyzed.Analogous results for the case of general quasi-linear elliptic equations of the type div(a(u, ∇•)) remained open.In this article we treat the case of the weighted p-Laplacian, where a(u, ∇y) = u|∇y| p−2 ∇y.The corresponding quasi-linear differential operator, − div(u|∇y| p−2 ∇y), in principle, has degeneracies as ∇y tends to zero and also if u approaches zero.Moreover, when the term u|∇y| p−2 is regarded as the coefficient of the Laplace operator, we also have the case of unbounded coefficients.In order to avoid degeneracy with respect to the control u, we assume that u is bounded away from zero.For the precise statements, see the next section.We leave the case of potentially degenerating controls to a future contribution.Instead, in this article, we focus on the degeneracies related to the nonlinearity.A number of regularizations have been suggested in the literature.
Finally, we have to deal with a two-parameter family of OCPs in the coefficients for monotone nonlinear differential equations.We consequently provide the wellposedness analysis for the underlying partial differential equations as well as for the OCPs.After that we pass to the limits as k → ∞ and ε → 0. The approximations and regularizations are considered to be useful not only for the mathematical analysis, but also for the purpose of numerical simulations.
An important point, arising after the solvability of the optimization problem, is the question of optimality conditions.The classical approach to deriving such conditions is based on the Lagrange principle.However, in the case when the control is considered as a scalar coefficient of the weighted p-Laplacian, the classical adjoint system often cannot be directly constructed due to the lack of differential properties of the solution to the boundary value problem with respect to control variables.To overcome this difficulty, in the forthcoming second part of this paper, we derived the optimality conditions passing to the limit in optimality conditions for a two-parameter family of approximating control problems.

Setting of the OCP.
Let Ω be a bounded open subset of R N (N ≥ 1) with a Lipschitz boundary.Let p be a real number such that 2 ≤ p < ∞.By BV (Ω) we denote the space of all functions in L 1 (Ω) for which the norm where α is a given positive value.Let z d ∈ L 2 (Ω) and f ∈ L 2 (Ω) be given distributions.The OCP we consider in this paper is to minimize the discrepancy between the distribution z d ∈ L 2 (Ω) and the solutions of the boundary value problem and the class of admissible BV -controls A ad is defined as follows: (2.4) It is clear that A ad is a nonempty convex subset of L 1 (Ω) with an empty topological interior.
More precisely, we are concerned with the following OCP: The existence of a unique solution to the boundary value problem (2.2)-(2.3)follows from an abstract theorem on monotone operators; see, for instance, [9] or [13, section II.2].
Theorem 2.1.Let V be a reflexive separable Banach space.Let V * be the dual space, and let A : V → V * be a bounded, semicontinuous, coercive, and strictly monotone operator.Then the equation Ay = f has a unique solution for each f ∈ V * .Moreover, Here, the above mentioned properties of the strict monotonicity, semicontinuity, and coercivity of the operator A have, respectively, the following meaning: In our case, we can define the operator A as a mapping W 1,p 0 (Ω) → W −1,q (Ω) by Then it is easy to show that Ay = −Δ p (u, y) + y and A satisfies all assumptions of Theorem 2.1 (for the details, we refer the reader to [9,12]).As a consequence of this theorem, we also know that y ∈ W  In this section we focus on the solvability of OCP (P).Hereinafter, we suppose that the space BV (Ω) × W 1,p 0 (Ω) is endowed with the norm (u, y) BV (Ω)×W 1,p 0 (Ω) := u BV (Ω) + y W 1,p 0 (Ω) .Moreover, q will denote the conjugate of p: q = p p−1 .Remark 3.1.We recall that a sequence {f k } ∞ k=1 converges weakly * to f in BV (Ω) if and only if the following two conditions hold (see [1]): f k → f strongly in L 1 (Ω) and Df k * Df weakly * in the space of Radon measures M(Ω; R N ).Moreover, if converges strongly to some f in L 1 (Ω) and satisfies sup k∈N Ω |Df k | < +∞, then (see, for instance, [1] and [6]) As an obvious consequence of these notions, we have the following property.
and y k y in W 1,p 0 (Ω).Then we have . Hence, it is immediate to pass to the limit and to deduce (3.2).
As a consequence, we have the following property.
Our next step concerns the study of topological properties of the set of admissible solutions Ξ to problem (2.2)-(2.5).
The following result is crucial for our further analysis.
Theorem 3.4.Let {(u k , y k )} k∈N ⊂ Ξ be a bounded sequence.Then, there is a pair (u, y) ∈ Ξ such that, up to a subsequence, u k * u in BV (Ω) and y k y in W 1,p 0 (Ω).Proof.By Remark 3.1 and compactness properties of BV (Ω) × W 1,p 0 (Ω), there exist a subsequence of {(u k , y k )} k∈N , still denoted by the same indices, and functions u ∈ BV (Ω) and y ∈ W 1,p 0 (Ω) such that Then by Lemma 3.2, we have Now, we show that the limit pair (u, y) is related by inequality (2.11).With that in mind we write down the Minty relation for (u k , y k ): In view of (3.3) and Lemma 3.2, we can pass to the limit in relation (3.4) as k → ∞ and arrive at the inequality (2.11) for every ϕ ∈ C ∞ 0 (Ω).Finally, from the density of (2.11) holds for every ϕ ∈ W 1,p 0 (Ω), and hence y ∈ W 1,p 0 (Ω) is a solution to the boundary value problem (2.2)- (2.3).This fact together with u ∈ A ad leads us to the conclusion: (u, y) ∈ Ξ.
To conclude this section, we give the existence result for optimal pairs to problem (2.2)-(2.5).
Proof.Since the set Ξ is nonempty and the cost functional is bounded from below on Ξ, it follows that there exists a minimizing sequence From Theorem 3.4 we deduce the existence of a subsequence, which we denote in the same way, and a pair (u * , y * ) ∈ Ξ such that u k * u in BV (Ω) and y k y * in W 1,p 0 (Ω).From these convergences we infer that lim Thus I(u * , y * ) ≤ inf (u,y)∈Ξ I(u, y) and, consequently, (u * , y * ) is a solution of (P).

Regularization of OCP (P).
As was pointed out in [12], the p-Laplacian Δ p (u, y) provides an example of a quasi-linear operator in divergence form with a socalled degenerate nonlinearity for p > 2. In this context we have nondifferentiability of the state y with respect to the control u.As follows from Theorem 3.5, this fact is not an obstacle to proving existence of optimal controls in the coefficients, but it causes certain difficulties when deriving the optimality conditions for the considered problem.To overcome this difficulty, we introduce the following family of approximating control problems (see, for comparison, the approach of Casas and Fernández [3] for quasilinear elliptic equations with a distributed control in the right-hand side): Here, k ∈ N, ε is a small parameter, which varies within a strictly decreasing sequence of positive numbers converging to 0, and where As for the function F k : R + → R + , it can, e.g., be defined by A direct calculation shows that in this case δ = 4/27.It is clear that the effect of such perturbations of Δ p (u, y) is its regularization around critical points where |∇y(x)| vanishes or becomes unbounded.In particular, if y ∈ W 1,p 0 (Ω) and Ω k (y shows that the Lebesgue measure of the set Ω k (y) satisfies the estimate i.e., the approximation F k (|∇y| 2 ) is essential on sets with small Lebesgue measure.
The main goal of this section is to show that for each ε > 0 and k ∈ N, the perturbed OCP (4.1)-(4.4) is well posed and its solutions can be considered as a reasonable approximation of optimal pairs to the original problem (2.2)-(2.5).To begin with, we establish a few auxiliary results concerning monotonicity and growth conditions for the regularized p-Laplacian Δ ε,k,p .
For our further analysis, we make use of the following notation: Remark 4.1.For an arbitrary element y * ∈ H 1 0 (Ω) let us consider the level set Downloaded 09/06/16 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Hence, the Lebesgue measure of the set Ω k (y * ) satisfies the estimate Now, we establish the following results.
Proposition 4.2.For every u ∈ A ad , k ∈ N, and ε > 0, the operator From the assumptions on F k and the boundedness of u, we obtain which concludes the proof.
Proposition 4.3.For every u ∈ A ad , k ∈ N, and ε > 0, the operator A ε,k,u is strictly monotone.Proof.To begin with, we make use of the following algebraic inequality: (4.9) In order to prove it, we note that the left-hand side of (4.9) can be rewritten as Since p ≥ 2 and 2 and we arrive at the inequality (4.9).With this we obtain Since the relation Ω) = 0 implies that y = v almost everywhere in Ω, it follows that the strict monotonicity property (2.8) holds in this case.Proof.To check this property it is enough to observe that for any y ∈ H 1 0 (Ω), k ∈ N, ε > 0, and u ∈ A ad , we have We are now in a position to apply the abstract theorem on monotone operators (see Theorem 2.1) to the equation A ε,k,u y = f with f ∈ L 2 (Ω).Closely following the arguments of section 2, we arrive at the following assertion.

I(u, y).
Analogously to problem (P), we can prove the following theorem.
Theorem 4.6.For every positive value ε > 0 and integer k ∈ N, the OCP (P ε,k ) has at least one solution.
The proof follows the steps of the proof of Theorem 3.5.Indeed, it is immediate to check that Ξ ε,k is not empty.Then, we can take a minimizing sequence {(u i , y i )} i∈N ⊂ Ξ ε,k .The lower boundedness of I implies the boundedness of {(u i , y i )} i∈N in BV (Ω)× H 1 0 (Ω).Then, arguing as in the proof of Theorem 3.4, we deduce the existence of a subsequence, denoted in the same way, and a pair (u * , y * ) ∈ Ξ ε,k such that u i * u * in BV (Ω) and y i y * in H 1 0 (Ω).Hence, I(u * , y * ) ≤ lim inf i→∞ I(u i , y i ).For our further analysis, we need to obtain some appropriate a priori estimates for the weak solutions to problem (4.2)-(4.3).With that in mind, we make use of the following auxiliary results.Proposition 4.7.Let u ∈ A ad , k ∈ N, and ε > 0 be given.Then, for arbitrary g ∈ L 2 (Ω) and y ∈ H 1 0 (Ω), we have Proof.Let us fix an arbitrary element y of H 1 0 (Ω).We associate with this element the set Ω k (y), where Ω k (y) := {x ∈ Ω : |∇y(x)| > k}.Then, by Friedrich's inequality, Using the fact that To conclude this section, let us show that for every u ∈ A ad and f ∈ L 2 (Ω), the sequence of weak solutions to the boundary value problem (4.2)-( 4 , is bounded with respect to the • ε,k,u -quasi-seminorm in the sense of Definition 4.8. Indeed, the integral identity (4.10) together with estimate (4.13) (for g = f ) immediately leads us to the relation As a result, it follows from (4.17) that where . Moreover, taking g = y = y ε,k in (4.13) and using (4.18), we also have (4.19)

Asymptotic analysis of the approximate OCP (P ε,k
).Our main intention in this section is to show that optimal solutions to the original OCP (P) can be attained (in some sense) by optimal solutions to the approximated problems (P ε,k ).With that in mind, we make use of the concept of variational convergence of constrained minimization problems (see [8]) and study the asymptotic behavior of a family of OCPs (P ε,k ) as ε → 0 and k → ∞.We begin with some auxiliary results concerning the weak compactness in H where u ∈ A ad is an arbitrary admissible control and Ω k (y ε,k To establish the second part of the theorem, let us take a subsequence {y εi,ki } i∈N of {y ε,k } ε>0 k∈N (here, ε i → 0 and k i → ∞ as i → ∞) and a function y ∈ H 1 0 (Ω) such that y εi,ki y in H 1 0 (Ω) as i → ∞.Further, we fix an index i ∈ N and associate it with the following set: (5.1) Due to estimates (4.8) and (4.18), we see that and, therefore,
Proof.The proof is divided into five steps.
Step 1: y i y in H 1 0 (Ω).From Lemma 5.1 we deduce the existence of a subsequence, still denoted by {y i } i∈N ⊂ H 1 0 (Ω), and an element y ∈ W 1,p 0 (Ω) such that y i y in H 1 0 (Ω).Let us prove that y is the solution of (2.2)-(2.3).Let us fix an arbitrary test function ϕ ∈ C ∞ 0 (Ω) and pass to the limit in the Minty inequality ( as i → ∞.Take into account that In view of the convergences ∇y i ∇y in L 2 (Ω) N and u i → u strongly in L r (Ω), for all r < ∞, and the boundedness of Thus, passing to the limit in relation (5.8) as n → ∞, we arrive at the inequality (2.11) for every ϕ ∈ C ∞ 0 (Ω).Finally, from the density of C ∞ 0 (Ω) in W 1,p 0 (Ω), we infer that (2.11) holds for every ϕ ∈ W 1,p 0 (Ω), and hence y ∈ W 1,p 0 (Ω) is the solution to the boundary value problem (2.2)-(2.3).Since the solution of (2.2)-(2.3) is unique, the whole sequence {y i } i∈N converges weakly to y = y(u) in H 1 0 (Ω).
Step 2: χ Ω\Ω k (yi) ∇y i ∇y in L p (Ω) N .Following the definition of the sets Ω ki (y i ) and using (4.18), we obtain On the other hand, in view of the weak convergence ∇y i ∇y in εi,ki,ui by (4.8),(4.18) it follows from (5.9) and (5.10) that Hence, g = ∇y almost everywhere in Ω, and χ Ω\Ω k (yi) ∇y i ∇y in L p (Ω) N holds.
Step 3: χ Ω\Ω k (yi) ∇y i → ∇y in L p (Ω) N .For each i ∈ N, we have the energy equalities (5.11) From (5.11) and the fact that y i y in H 1 0 (Ω), we deduce (5.12) lim , and u i (x) ≥ α for almost all x ∈ Ω, it is easy to check that χ Ω\Ω k i (yi) ∇y i u 1/p i ∇yu 1/p in L p (Ω) N .Using this convergence and (5.13), we get The weak convergence χ Ω\Ω k i (yi) ∇y i u Step 4: Proof of (5.7).From (5.6) and (5.13) we obtain (5.14) lim Let us prove that (5.15) lim This is established as follows.From (4.6) we deduce From (5.6) we know that the last term converges in L Now, combining this estimate and (5.6) we conclude that This completes the proof.
We are now in a position to show that optimal pairs to the approximated OCP (P ε,k ) lead in the limit to optimal solutions of the original OCP (P).
Since Theorem 5.3 does not give an answer to whether the entire set of solutions Ξ opt to problem (2.2)-(2.5)can be attained in such a way, the following result sheds some light on this matter.
Proof.By the strict local optimality of (u 0 , y 0 ), we have that it is the unique solution of (Q) min (u,y)∈Ξ,u∈U (u 0 )

I(u, y).
For every ε and k let us consider the control problems