ASYMPTOTICS FOR QUASILINEAR OBSTACLE PROBLEMS IN BAD DOMAINS

. We study two obstacle problems involving the p-Laplace operator in domains with n-th pre-fractal and fractal boundary. We perform asymptotic analysis for p → ∞ and n → ∞ .


1.
Introduction. In this paper we consider obstacle problems involving p-Laplacian in bad domains in R 2 . With "bad domains" we refer to domains with pre-fractal and fractal boundary (see Figures 1 and 2).
On the one side, the study of fractals provides models for problems connected to various phenomena in different fields: in Biology, in Medicine, in Engineering applications and in other Applied Sciences (see, for instance, [10] and the references quoted there). On the other side, the p-Laplace operator allows to give an answer for many applied problems: the non-Newtonian fluid mechanics, reaction-diffusion problems, flows through porous media (see [8] and references therein).
The motivation of the present paper is the study of an optimal mass transport problem as investigated in [13]. More precisely, in [13] the authors study a double obstacle problem for p-Laplacian in smooth domains and by passing to the limit as p → ∞ they obtain a complete answer to an optimal mass transport problem for the Euclidean distance. We recall that this connection was the key to the first complete proof of the existence of an optimal transport map for the classical Monge problem (see [9]). In particular, in [13] the authors relate this optimization problem either with an optimal mass transport problem with taxes or an optimal mass transport problem with courier (for notation and general results on mass transport theory we refer to [20]).
In the present paper, given f ∈ L 1 (Ω α ), we consider two obstacle problems involving p-Laplacian (p > 2) on domains with pre-fractal boundary Ω n α : where K n = {v ∈ W 1,p (Ω n α ) : ϕ 1,n v ϕ 2,n in Ω n α } with obstacles ϕ 1,n and ϕ 2,n . We remark that to give sense to the previous problem and to prove the results of the present paper it is sufficient to take f ∈ L 1 (Ω α ) whereas in [13] it is required that f ∈ L ∞ (Ω).
We also consider two obstacle problem on domain with fractal boundary Ω α : Here, under suitable assumptions about the obstacles (see (5)), we prove that, up to pass to a subsequence, the solutions of the problems P p converge uniformly in C(Ω α ), as p → ∞, to a solution of the following problem In a similar way, we prove that, up to pass to a subsequence, the solutions of the problems P p,n converge uniformly in C(Ω n α ), as p → ∞, to a solution of the following problem As for n → ∞ the n-th pre-fractal curves converge to the fractal curve in the Hausdorff metric, the pre-fractal domains Ω n α converge to the fractal domain Ω α . Then we study the asymptotic behavior of P p,n and P ∞,n for n → ∞ (we refer to [15] for the connections between convergence of convex sets and solutions of variational inequalities).
More precisely, under suitable assumptions about the obstacles, we obtain the strong convergence in W 1,p (Ω α ) of suitable extensions of solutions of P p,n as n → ∞ to a solution to problem P p (see Theorem 4.1). Similarly, we obtain the -weakly convergence in W 1,∞ (Ω α ) of suitable extensions of solutions of P ∞,n as n → ∞ to a solution to problem P ∞ (see Theorem 4.2).

Figure 2. A bad domain
We note that, passing to the limit first for p → ∞ and then for n → ∞, the sequence of suitable extensions of solutions of P p,n converges in C(Ω α ) to a solution of problem (10). Otherwise, passing to the limit first for n → ∞ and then for p → ∞, the sequence of solutions of P p,n converges in C(Ω α ) to a solution of the same problem (10). As we do not have uniqueness result for the problem (10), we cannot deduce that the limit solutions are equal. We remark that it would be interesting to find suitable assumptions that guarantee uniqueness results.
The organization of the paper is the following. In Section 2 we describe the bad domains. In Section 3 we state and prove the results concerning the limit for p → ∞ in the obstacle problems for the p-Laplacian in fractal and pre-fractal domains. In Section 4 we perform asymptotic analysis for n → ∞ for fixed p > 2 and p = ∞. Finally in Section 5 we discuss about uniqueness. Moreover, we conclude by a scheme that underlines how by exchanging the order of limits (p → ∞ and n → ∞) we obtain (possibly different) solutions of the same problem P ∞ .
2. Bad domains. In the following, writing "bad domains" we refer to domains (in R 2 ) Ω n α with pre-fractal boundary and to domains Ω α with fractal boundary. More precisely, we construct the domain Ω α with fractal boundary starting by any regular polygon in R 2 (triangle, square, pentagon, etc.) and replacing each side by a Koch curve type fractal K α . We recall the K α is the unique closed bounded set in R 2 which is invariant with respect to a family of 4 contractive similarities Ψ α , that is, . . , ψ 4,α }, with ψ i,α : C → C, i = 1, . . . , 4, 2 < α < 4 : (see [11]). Furthermore, there exists a unique Borel regular measure ν α with supp ν α = K α , invariant with respect to Ψ α , which coincides with the normalized d f -dimensional Hausdorff measure on K α , where Hausdorff dimension d f = ln α 4 (see [11]).   Let K 0 be the line segment of unit length that has as endpoints A = (0, 0) and B = (1, 0). For each n ∈ N, we set where, for each integer n > 0, ψ i|n,α = ψ i1,α • ψ i2,α • · · · • ψ in,α is the map associated with arbitrary n−tuple of indices i|n = (i 1 , i 2 , . . . , i n ) ∈ {1, . . . , 4} n and it is the identity map in R 2 for n = 0. K n α is the so-called n-th pre-fractal Koch curve type fractal. We recall that for n → ∞ the n-th pre-fractal curves K n α converge to the fractal curve K α in the Hausdorff metric (see [11]).
The pre-fractal domains Ω n α are polygonal domains having as sides pre-fractal Koch curve type fractals. More precisely, we obtain the pre-fractal domains Ω n α starting by the regular polygon used to construct Ω α and replacing each side by a pre-fractal Koch curve. In particular, for all n ≥ 1, Ω n α are polygonal, non convex and with an increasing number of sides which develop at the limit a fractal geometry. To have an idea of pre-fractal curves we can look at the Figure 3; there, we can see the iterations for n = 2 obtained by choosing different contraction factors (α = 2.2, α = 3 and α = 3.8); instead, in Figures 4 and 5 we can see the iterations for n = 3 and n = 4, respectively.
A particular example is the pre-fractal snowflakes: in Figure 1 we have chosen outward curves starting from a triangle and α = 3. Instead, in Figure 2 we have an example of domain with fractal boundary obtained by choosing inward curves starting from a pentagon and α = 3+ 3. Asymptotic analysis for p −→ ∞. Let Ω α be the domain introduced before. Let f belongs to L 1 (Ω α ).
We consider two obstacle problem (3) in the domain Ω α : where and We note that the condition (5) implies that K = ∅.
In fact, let us consider the function In fact, from the properties of modulus, we have Adding ϕ 1 (z) in both sides and passing to the max, we have From this, exchanging the role of x and y, we obtain and then (6).
Obviously choosing y = x, we obtain Finally, from (5), we have From this, taking the maximum of the left-hand side, we obtain that As K = ∅, since K is a closed convex subset of W 1,p (Ω α ) and the functional is convex, weakly lower semicontinuous and coercive in K, then the variational problem min has a minimizer u p in K, that is Also, it is known that u p is a minimizer iff it is a solution of the variational inequality (3) (see [19]).
Let us observe that in general this minimizer is not unique: in Section 5 we briefly discuss about uniqueness. Now we perform the asymptotic analysis for p −→ ∞ as in [13] (see also [3]).
. Assume that ϕ 1 and ϕ 2 verify (5). Then, for any p ∈ (2, ∞), a minimizer u p of the problem (8) exists. Moreover, there exists a subsequence, as p −→ ∞, such that u p −→ u ∞ weakly in W 1,m (Ω α ), ∀m > 2, u ∞ being a maximizer of the following variational problem where Proof. Assuming that ϕ 1 and ϕ 2 verify (5), we have that K = ∅ and a minimizer u p exists. In a similar matter, it is possible to prove that (5) Furthermore, since u p ∈ K, we have, naturally, Moreover, from (11), using Hölder's inequality and having in mind (12), we get with C independent from p. From (12) and (13), we obtain that {u p } p>2 is bounded in W 1,p (Ω α ). By (13), we have that ||∇u p || L p (Ωα) ≤ (pC) By using (14), for Morrey-Sobolev's embedding we have By Ascoli-Arzelà compactness criterion, by using (12) and (15) we can extract a subsequence of the previous one, that we indicate again with {u p }, such that for Moreover, thanks to (12) and (14) we have, for all m > 2, Then, there exists a subsequence denoted by u p k , such that, for k −→ ∞, From (16) so we obtain that u ∞ ∈ K ∞ . Finally, passing to the limit in (11), we have Hence, we obtain as we wanted to prove.
As in the fractal case, K n = ∅. Since K n is a closed convex subset of W 1,p (Ω n α ) and the functional is convex, weakly lower semicontinuous and coercive in K n , then the variational problem min has a minimizer u p,n in K n , that is J p,n (u p,n ) = min v∈Kn J p,n (v).
As in the case of fractal domain, the following theorem holds.
Theorem 3.2. Let f ∈ L 1 (Ω n α ). Assume that ϕ 1,n and ϕ 2,n verify (18). Then, a minimizer u p,n of the problem (20) exists. Moreover, there exists a subsequence, as p −→ ∞, such that u p,n −→ u ∞,n weakly in W 1,m (Ω n α ), ∀m > 2, u ∞,n being a maximizer of the following variational problem where 4. Asymptotic analysis for n −→ ∞. In this section, we perform asymptotic analysis for n → ∞ and we consider p fixed. By Theorem 5.7 in [5], there exists a bounded linear extension operator Ext J : W 1,p (Ω n α ) → W 1,p (R 2 ), whose norm is independent of n, that is, with C J independent of n.
We putû p,n = (Ext J u p,n )| Ωα , where u p,n is a solution of the problem (17).

ASYMPTOTICS FOR QUASILINEAR OBSTACLE PROBLEMS IN BAD DOMAINS 9
Theorem 4.1. Let f ∈ L 1 (Ω α ), and Then, there exists a subsequence of functionsû p,n defined in (24) such thatû p,n strongly converges as n → ∞ in W 1,p (Ω α ) to a solution to problem (3).
Proof. First, we note that problem (17) admits a solution as condition (26) guarantees that the convex K n is non-empty. In a similar way, problem (3) admits a solution as condition (25) implies that the convex K is non-empty. Proceeding as in Theorem 3.1, we obtain that and with C independent from n. From (28) and (29), we obtain that {u p,n } p>2 is bounded in W 1,p (Ω n α ). Sinceû p,n = (Ext J u p,n )| Ωα , where u p,n solve problem (17), we have ||û p,n || W 1,p (R 2 ) C J ||u p,n || W 1,p (Ω n α ) (30) with C J independent of n. Then there existsû ∈ W 1,p (Ω α ) and a subsequence of u p,n , denoted byû p,n again, weakly converging toû in W 1,p (Ω α ).
We recall that solutions u p,n to problems (17) realize the minimum on K n of the functional J p,n (·) (see (19)).
We prove that where J p (·) is the functional defined in (7). In fact, asû p,n weakly converges toû in W 1,p (Ω α ), we have that, for all fixed m ∈ N, Then, passing to the limit for m → ∞, we obtain Moreover, given u p solution of the problem (3), we can construct a sequence of functions w n ∈ K n that strongly converges to u p in W 1,p (Ω α ) by setting w n = ϕ 2,n ∧ (u p ∨ ϕ 1,n ) (where we denote by u ∧ v = inf(u, v), u ∨ v = sup(u, v), u + = u ∨ 0). We have that (see Theorem 1.56 in [19]). From (33) and (34), we obtain (31). Now, we will prove that the convergence is strong in W 1,p (Ω α ). From the estimate (see, for instance, Lemma 2.1 in [2]) and with the choice of v n = (û ∨ ϕ 1,n ) ∧ ϕ 2,n (v n ∈ K n and v n →û in W 1,p (Ω α )), ∀m ∈ N, n ≥ m, we have that Then, considering the first and the last member of this chain of equalities and inequalities and passing to lim sup as n → ∞, we obtain In fact, the third term goes to zero by the weak convergence ofû p,n toû in W 1,p (Ω α ). The first one goes to zero, since because, as n → ∞,û p,n and v n (strongly) converge toû in L p (Ω α ) and |Ω α \Ω n α | → 0. The second one tends to zero because v n strongly converges toû in W 1,p (Ω α ). In conclusion, passing to the limit for m → ∞, we obtain the thesis.

Concluding remarks.
After the analysis of the asymptotic behavior, we discuss the issue of the uniqueness. In the case p = ∞, Example 3.6 (case 0 < k < 1) in [13] shows that in general there is not uniqueness of the solution of problem (10).
In the case of two obstacle problem in fractal and pre-fractal domain (p ∈ (2, ∞)) in order to obtain uniqueness, we can make the following assumption: In following Theorem 5.1, we show how condition (46) implies uniqueness for the solution of two obstacle problem in fractal domain (in the pre-fractal case, the proof is similar).
Theorem 5.1. Let a p (u, v) defined as in (4). Let us assume that (5)  Proof. We have already proved the existence of solutions. Now, let us prove the uniqueness.
Let us assume condition (46) holds: for example, we suppose that Ωα f dx < 0. We show the uniqueness by contradiction. Let u 1 and u 2 be two solutions of (3). Choosing in (3) first u 1 and then u 2 as test function, we have Then, again thanks to (35), we obtain We have also uniqueness in the case of homogeneous Dirichlet boundary condition. We want to remark that two obstacle problem with Dirichlet boundary condition on pre-fractal and fractal domain have been studied in [7]. In particular, an analogous theorem to Theorem 4.1 is stated under the assumptions (25), (26), (27), ϕ 1 0 ϕ 2 in ∂Ω α , ϕ 1,n 0 ϕ 2,n in ∂Ω n α . Moreover in [7] sequences of obstacles ϕ i,n , i = 1, 2, that satisfy the previous assumptions have been constructed by using suitable arrays of fibers Σ n around the boundary of the domain Ω n α as in [16] (see also [6]).
As consequences of uniqueness results either in the pre-fractal case or in the fractal one, we can deduce that all the sequenceû p,n of Theorem 4.1 converges to the solution to problem (3) as n → ∞.
We conclude by the following scheme that provides a summary of all the results we have obtained here.

p → ∞(subseq)
We note that, passing to the limit first for n → ∞ (see Theorem 4.1) and then for p → ∞ (see Theorem 3.1) the sequence ofû p,n converges in C(Ω α ) to a solution u of problem (10) (where "subseq" indicates the convergence along subsequences). Otherwise, passing to the limit first for p → ∞ (see Theorem 3.2 ) and then, after suitable extensions, for n → ∞ (see Theorem 4.2), the sequence of u p,n converges in C(Ω α ) to a solutionũ of the same problem (10).
As we do not have uniqueness result for this problem, we cannot deduce thatû = u. It would be interesting to find suitable assumptions that guarantee uniqueness results (see [1], [12], [17] and the references therein).
Finally, in the framework of fractal sets, we want to recall the recent paper [4] where it is studied the infinity Laplace operator and the corresponding Absolutely Minimizing Lipschitz Extension problem on the Sierpinski gasket by introducing a notion of infinity harmonic functions on pre-fractal sets (see Section 5 for the relation between infinity and p-harmonic function).