Special Issue in honor of Alan C. Lazer Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 135–147. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu LIMIT FOR THE p-LAPLACIAN EQUATION WITH DYNAMICAL BOUNDARY CONDITIONS EYLEM ÖZTÜRK, JULIO D. ROSSI In memory of Alan C. Lazer, a great mathematician Abstract. In this article we study the limit as p→∞ in the evolution prob- lem driven by the p-Laplacian with dynamical boundary conditions. We prove that the natural energy functional associated with this problem converges to a limit in the sense of Mosco convergence and as a consequence we obtain con- vergence of the solutions to the evolution problems. For the limit problem we show an interpretation in terms of optimal mass transportation and provide examples of explicit solutions for some particular data. 1. Introduction Our main purpose in this article is to study a nonlinear diffusion equation ob- tained as the limit as p → ∞ to the p-Laplacian with dynamical boundary con- ditions. More precisely, we look for the limit as p → ∞ of the solutions to the problem 0 = ∆pu(x, t), x ∈ Ω, t > 0, ∂u ∂t (x, t) + |∇u|p−2 ∂u ∂η (x, t) = f(x, t), x ∈ ∂Ω, t > 0, u(x, 0) = u0(x), x ∈ ∂Ω. (1.1) Here Ω ⊂ RN is a bounded smooth domain, ∂u∂η denotes the outer normal derivative of u and f is a nonnegative function that represents a given source term localized on ∂Ω, which is interpreted physically as adding material to an evolving system, within which mass particles are continually rearranged by diffusion. In this model it is assumed that diffusion is much faster inside the domain than on the boundary, hence the time derivative appears only in the boundary condition. Associated with this evolution problem we have the functional Ep : L2(∂Ω) 7→ R ∪ {+∞}, Ep(u) =  min v∈W 1,p(Ω),trace(v)=u 1 p ∫ Ω |∇v|p u ∈ trace(W 1,p(Ω)), +∞ u 6∈ trace(W 1,p(Ω)). (1.2) 2010 Mathematics Subject Classification. 35K20, 35K55, 35K92, 47J35. Key words and phrases. p-Laplacian; dynamical boundary conditions; Mosco convergence. c©2021 This work is licensed under a CC BY 4.0 license. Published October 6, 2021. 135 136 E. ÖZTÜRK, J. D. ROSSI EJDE/SI/01 As we mentioned before our aim is to look for the limit as p → ∞ of the so- lutions up to (1.1). To this end we use a general result by Mosco [23, 24]: if the associated functionals converge to a limit functional (in an adequate sense, that roughly speaking, means convergence of the epigraphs, see Section 2 for the precise definition) then the corresponding solutions to the associated evolution problems converge to the solution associated with the limit functional. The limit of the functional Ep as p→∞ is E∞ : L2(∂Ω) 7→ R∪{+∞}, given by E∞(u) = { 0 u ∈ A∞, +∞ u 6∈ A∞, (1.3) where A∞ = { u ∈ C(∂Ω) : ∃v : Ω 7→ R with |∇v| ≤ 1a.e Ω, v|∂Ω = u } . Our first result reads as follows. Theorem 1.1. The functionals Ep converge to E∞ as p→∞ in the Mosco sense. As a consequence we have the convergence of the solutions to our evolution problem (1.1). Theorem 1.2. Let up(x, t) be the solution of problem (1.1) with a fixed initial condition u0 ∈ A∞ L2(∂Ω) and fixed right hand side f ∈ L1(0, T : L2(∂Ω)). Then up → u∞ (1.4) as p→∞ in C([0, T ] : L2(∂Ω)), that is, lim p→∞ max t∈[0,T ] ‖up(·, t)− u∞(·, t)‖L2(∂Ω) = 0. Moreover, the limit u∞ is characterized as the solution to f(x, t)− ∂u ∂t (x, t) ∈ ∂E∞(u(x, t)) x ∈ ∂Ω, t > 0, u(x, 0) = u0(x) x ∈ ∂Ω. (1.5) If we assume that u0 ∈ L1(∂Ω) and f is such that sup t∈[0,T ] ∫ ∂Ω |f(x, t)|dσ(x) + ∫ ∂Ω ∣∣∣∂f ∂t (x, t) ∣∣∣dσ(x) < +∞. Then there exists a subsequence pi →∞ such that upi → u∞ a.e. and strongly in L2(∂Ω× [0, T ]), ∇upi ⇀ ∇u∞ weakly in L2(∂Ω× [0, T ]), ∂upi ∂t ⇀ ∂u∞ ∂t weakly in L2(∂Ω× [0, T ]). (1.6) Finally, we relate the limit problem with an optimal mass transport problem in Theorem 1.3. the optimal mass transport problem is defined with a cost given by the distance between points on ∂Ω considering paths inside Ω, that is defined as the minimum of the lengths of the paths inside Ω that join the two points. We call this distance dΩ. It turns out that the limit of the solution to the limit problem, u∞(·, t), is a Kantorovich potential for the optimal mass transport problem between f(·, t) and ∂u∞ ∂t (·, t). EJDE-2021/SI/01 p-LAPLACIAN WITH DYNAMICAL BOUNDARY CONDITIONS 137 Theorem 1.3. The solution to the limit problem (1.5) satisfies∫ ∂Ω u∞(x, t) (∂u∞ ∂t (x, t)− f(x, t) ) dσ(x) = max v:|v(x)−v(y)|≤dΩ(x,y) ∫ ∂Ω v(x) (∂u∞ ∂t (x, t)− f(x, t) ) dσ(x), that is, u∞ is a Kantorovich potential for the dual formulation of the Monge- Kantorovich mass transport problem between f(·, t)dσ and ∂u∞ ∂t (·, t)dσ. As was pointed out in [1], the limit problem (1.5) can be interpreted as a model for the formation and growth of a sandpile where particles of sand are distributed on ∂Ω (here u∞(x, t) describes the amount of the sand at the point x at time t). The main assumption being that the sandpile is stable when the slope is less than or equal to one and unstable if not. We also include some explicit examples of solutions to the limit problem. In these examples one can appreciate the mass transport interpretation of the limit problem. Also we illustrate a curious phenomenon, the support of the solution on ∂Ω may be disconnected even if the domain is strictly convex, the initial condition is zero and the reaction has connected support. Dynamical boundary conditions appear in modeling physical phenomena when there is a thin layer around the boundary in which reaction takes place. We refer to [11, 12, 13, 16, 20, 21, 25] for general references concerning evolution problems with this kind of boundary conditions. As a precedent concerning limits as p→∞, we mention that problem (1.1) in the elliptic (time independent) case was studied in [17] (see also [18] for the associated eigenvalue problem). Here one needs to assume that ∫ ∂Ω f = 0 (otherwise, there is no solution) and to have uniqueness of solutions one normalizes according to∫ ∂Ω u = 0. Concerning evolution problems with the p-Laplacian, the counterpart of our results for the Cauchy problem was obtained in [1, 14]. In these references it was studied the limiting behavior as p → ∞ of solutions to the quasilinear parabolic problem ∂v ∂t (x, t)−∆pv(x, t) = f(x, t), in (0, T )× RN , v(x, 0) = u0(x), in RN . In [1], assuming that u0 is a Lipschitz function with compact support, satisfying |∇u0| ≤ 1, it is proved that vp → v∞ and the limit function v∞ satisfies f(x, t)− ∂v∞ ∂t (x, t) ∈ ∂F∞(v∞(x, t)), where F∞(v) = { 0, if |∇v| ≤ 1, +∞, otherwise. Other related papers that deal with limits as p → ∞ in p-Laplacian problems are [4, 5, 6, 19]. The relation between a limit as p → ∞ in a p-Laplacian problem and optimal mass transport was first found in [15] (see also [5]). The rest of the paper is organized as follows: in Section 2 we gather some preliminary results concerning Mosco convergence of functionals; in Section 3 we 138 E. ÖZTÜRK, J. D. ROSSI EJDE/SI/01 prove the convergence of the functionals Ep to E∞ stated Theorem 1.1 and we deduce the convergence of the solutions to the evolution problems in Theorem 1.2. In Section 4 we deal with the Mass transport interpretation of the limit problem. Finally, in Section 5 we include some explicit examples of solutions to the limit problem. 2. Preliminaries Now, we recall the definition of Mosco-convergence. If X is a metric space, and {An} is a sequence of subsets of X, we define lim inf n→∞ An := { x ∈ X : ∃xn ∈ An, xn → x } , lim sup n→∞ An := { x ∈ X : ∃xnk ∈ Ank , xnk → x } . If X is a normed space, we denote by s-lim and w-lim the above limits associated, respectively, to the strong and to the weak topology of X. Definition 2.1. Let H be a Hilbert space. Given Ψn,Ψ : H → (−∞,+∞] convex, lower-semicontinuous functionals, we say that Ψn converges to Ψ in the sense of Mosco if w-lim supn→∞ Epi(Ψn) ⊂ Epi(Ψ) ⊂ s-lim inf Epi(Ψn), (2.1) where Epi(Ψn) and Epi(Ψ) denote the epigraphs of the functionals Ψn and Ψ, defined by Epi(Ψn) := { (u, λ) ∈ L2(RN )× R : λ ≥ Ψn(u) } , Epi(Ψ) := { (u, λ) ∈ L2(RN )× R : λ ≥ Ψ(u) } . Remark 2.2. We note that (2.1) is equivalent to the requirement that the following two conditions are simultaneously satisfied: ∀u ∈ D(Ψ)∃un ∈ D(Ψn) : un → u and Ψ(u) ≥ lim sup n→∞ Ψn(un), (2.2) for every subsequence {nk}, Ψ(u) ≤ lim inf k Ψnk (uk) whenever uk ⇀ u. (2.3) Here D(Ψ) := {u ∈ H : Ψ(u) < ∞} and D(Ψn) := {u ∈ H : Ψn(u) < ∞} denote the domains of Ψ and Ψn, respectively. To identify the limit of the solutions un to problem (1.1) (see the Introduction), we use methods from Convex Analysis, and so we must first recall some terminology [10, 8, 2]. If H is a real Hilbert space with inner product (·, ·) and Ψ : H → (−∞,+∞] is convex, then the subdifferential of Ψ is defined as the multivalued operator ∂Ψ given by v ∈ ∂Ψ(u) ⇐⇒ Ψ(w)−Ψ(u) ≥ (v, w − u) ∀w ∈ H. Recall that the epigraph of Ψ is defined by Epi(Ψ) = { (u, λ) ∈ H × R : λ ≥ Ψ(u) } . Given K, a closed convex subset of H, we define the indicator function of K by IK(u) = { 0 if u ∈ K, +∞ if u 6∈ K. EJDE-2021/SI/01 p-LAPLACIAN WITH DYNAMICAL BOUNDARY CONDITIONS 139 Then the subdifferential is characterized by v ∈ ∂IK(u) ⇐⇒ u ∈ K and (v, w − u) ≤ 0 ∀w ∈ K. When the convex functional Ψ : H → (−∞,+∞] is proper, lower-semicontinuous, and such that min Ψ = 0, it is well known (see [8]) that the abstract Cauchy problem ut + ∂Ψ(u) 3 f, a.e. t ∈ (0, T ), u(0) = u0, has a unique solution for any f ∈ L1(0, T ;H) and u0 ∈ D(∂Ψ). The Mosco convergence is a very useful tool to study convergence of solutions of parabolic problems. The following theorem is a consequence of the results in [9, 2]. Theorem 2.3. Let Ψn,Ψ : H → (−∞,+∞] be convex and lower semicontinuous functionals. Then the following two statements are equivalent: (i) Ψn converges to Ψ in the sense of Mosco. (ii) (I + λ∂Ψn)−1u→ (I + λ∂Ψ)−1u for all λ > 0, u ∈ H. Moreover, either one of the above conditions, (i) or (ii), imply (iii) for every u0 ∈ D(∂Ψ) and u0,n ∈ D(∂Ψn) such that u0,n → u0, and for every fn, f ∈ L1(0, T ;H) with fn → f , if un(t), u(t) are solutions of the abstract Cauchy problems (un)t + ∂Ψn(un) 3 fn a.e. t ∈ (0, T ) un(0) = u0,n, and ut + ∂Ψ(u) 3 f a.e. t ∈ (0, T ) u(0) = u0, respectively, then un → u in C([0, T ] : H). 3. Mosco convergence of the functionals and convergence of the solutions First, we show some uniform bounds (independent of p) for the solutions up to (1.1). Lemma 3.1. Fix T > 0. Assume that u0 ∈ L1(∂Ω) and f is such that C(f) := sup t∈[0,T ] ∫ ∂Ω |f(x, t)|dσ(x) + ∫ ∂Ω ∣∣∣∂f ∂t (x, t) ∣∣∣dσ(x) < +∞. (3.1) Then, there exists a constant C such that sup ∂Ω×[0,T ] |up| ≤ C, ∫ T 0 ∫ ∂Ω ∣∣∣∂up ∂t ∣∣∣2 ≤ C, (∫ T 0 ∫ ∂Ω |∇up|p )1/p ≤ C1/p, (3.2) for every N + 1 ≤ p <∞. The constant C depends on u0, C(f) and T . 140 E. ÖZTÜRK, J. D. ROSSI EJDE/SI/01 Proof. In this proof we denote by C a generic constant that depends only on u0, C(f) and T and may change from one line to another. Now, we use the weak form of (1.1). It holds that∫ t 0 ∫ ∂Ω ∂up ∂t v + ∫ t 0 ∫ Ω |∇up|p−2∇up∇v = ∫ t 0 ∫ ∂Ω fv. Choose a smooth, nondecreasing function β : R 7→ R such that β(x) = sgn(x) for |x| ≥ δ > 0. By approximation we set v = β(up) as the test function in the weak form of (1.1), to obtain∫ t 0 ∫ ∂Ω ∂up ∂t β(up) ≤ ∫ t 0 ∫ ∂Ω fβ(up). Hence, ∫ ∂Ω B(up)(t)− ∫ ∂Ω B(u0) = ∫ t 0 ∫ ∂Ω ∂B(up) ∂t ≤ ∫ t 0 ∫ ∂Ω fβ(up), here B satisfies B′(s) = β(s). Letting δ → 0 we obtain sup t∈[0,T ] ∫ ∂Ω |up|(t) ≤ ∫ ∂Ω |u0|+ ∫ T 0 ∫ ∂Ω |f | ≤ ‖u0‖L1(∂Ω) + C(f)T, where C(f) is the constant that depends on f given in (3.1). Now, if we take v = up as a test function we obtain∫ t 0 ∫ ∂Ω ∂up ∂t up + ∫ t 0 ∫ Ω |∇up|p = ∫ t 0 ∫ ∂Ω fup. Since ∫ t 0 ∫ ∂Ω fup ≤ C(f) ∫ T 0 ‖up‖L∞(∂Ω), we obtain 1 2 ∫ ∂Ω |up|2(t) + ∫ t 0 ∫ Ω |∇up|p ≤ 1 2 ∫ ∂Ω |u0|2 + C(f) ∫ T 0 ‖up‖L∞(∂Ω). (3.3) Hence, sup t∈[0,T ] 1 2 ∫ ∂Ω |up|2(t) + ∫ T 0 ∫ Ω |∇up|p ≤ 1 2 ∫ ∂Ω |u0|2 + C(f) ∫ T 0 ‖up‖L∞(∂Ω). (3.4) Since up belongs to W 1,p(Ω), for p ≥ N + 1 we have ‖up(t)‖L∞(∂Ω) ≤ C { ‖∇up(t)‖LN+1(Ω) + ‖up(t)‖L1(∂Ω) } ≤ C { ‖∇up(t)‖Lp(Ω) + ‖up(t)‖L1(∂Ω) } . The constant C in this inequality in independent of p, (p ≥ N + 1), then we have ‖up(t)‖pL∞(∂Ω) ≤ C p { ‖∇up(t)‖pLp(Ω) + ‖up(t)‖pL1(∂Ω) } ≤ Cp { ‖∇up(t)‖pLp(Ω) + Cp } . (3.5) Therefore, ∫ T 0 ‖up(s)‖pL∞(∂Ω) ≤ C p {∫ T 0 ‖∇up(s)‖pLp(Ω) + CpT } . EJDE-2021/SI/01 p-LAPLACIAN WITH DYNAMICAL BOUNDARY CONDITIONS 141 Using (3.4) we obtain∫ T 0 ‖up(s)‖pL∞(∂Ω) ≤ Cp {1 2 ∫ ∂Ω |u0|2 + C(f) ∫ T 0 ‖up(s)‖L∞(∂Ω) + CpT } ≤ Cp‖u0‖2L2(∂Ω) + CpC(f) (∫ T 0 ‖up(s)‖pL∞(∂Ω) )1/p T 1−1/p + CpT ≤ Cp 2/(p−1) + 1 2 ∫ T 0 ‖up(s)‖pL∞(∂Ω). Hence, (∫ T 0 ‖up(s)‖pL∞(∂Ω) )1/p ≤ C. Here the constant C is independent of p. Then, (3.4) implies(∫ T 0 ∫ Ω |∇up|p )1/p ≤ C1/p. By an approximation procedure we can use v = ∂u ∂t as test function to obtain∫ T 0 ∫ ∂Ω ∣∣∣∂up ∂t ∣∣∣2 + ∫ T 0 ∫ Ω ∂ ∂t 1 p |∇up|p = ∫ T 0 ∫ ∂Ω f ∂up ∂t . Integrating by parts in time in the last integral, we obtain∫ T 0 ∫ ∂Ω ∣∣∣∂up ∂t ∣∣∣2 + ∫ Ω 1 p |∇up|p(T ) = ∫ Ω 1 p |∇u0|p − ∫ T 0 ∫ ∂Ω ∂f ∂t up + ∫ ∂Ω f(T )up(T )− ∫ ∂Ω f(0)u0. (3.6) Hence, ∫ Ω |∇up|p(T ) ≤ ∫ Ω |∇u0|p + pC(f) ∫ T 0 ‖up(s)‖L∞(∂Ω) + pC(f)‖up(T )‖L∞(∂Ω) + pC(f)‖u0‖L∞(∂Ω) ≤ ∫ Ω |∇u0|p + pC + pC‖up(T )‖L∞(∂Ω). Now, using (3.5) we obtain ‖up(T )‖pL∞(∂Ω) ≤ C p (∫ Ω |∇u0|p + pC + pC‖up(T )‖L∞(∂Ω) ) + Cp ≤ 1 2 ‖up(T )‖pL∞(∂Ω) + (CppC)p/(p−1) + Cp and then we conclude that ‖up(T )‖L∞(∂Ω) ≤ C. As T is any time we obtain sup t∈[0,T ] ‖up(t)‖L∞(∂Ω) ≤ C. Finally, since |∇u0| ≤ 1, from (3.6) we conclude that∫ T 0 ∫ ∂Ω ∣∣∣∂up ∂t ∣∣∣2 ≤ C 142 E. ÖZTÜRK, J. D. ROSSI EJDE/SI/01 This completes the proof. � Now, we prove that the functionals Ep converge in the sense of Mosco to the limit functional E∞. Proof of Theorem 1.1. First, we want to show that (2.2) holds, that is, ∀u ∈ D(E∞) ∃up ∈ D(Ep) : up → u and E(u) ≥ lim sup n→∞ Ep(up). (3.7) Given u ∈ D(E∞), that is, u ∈ A∞, we just take up ≡ u as the desired sequence. We clearly have up → u strongly in L2(∂Ω). Now, from the fact that u ∈ A∞, there exists v∗ : Ω 7→ R with |∇v∗| ≤ 1 a.e Ω and v∗|∂Ω = u. Hence, we obtain that u ∈ D(Ep), that is, u ∈ trace(W 1,p(Ω) and Ep(up) = min v∈W 1,p(Ω),trace(v)=u 1 p ∫ Ω |∇v|p ≤ 1 p ∫ Ω |∇v∗|p ≤ 1 p |Ω| → 0 as p→∞. Then we have 0 = E(u) ≥ lim sup n→∞ Ep(up) = 0 as we wanted to show. Now, we have to prove that (2.3) also holds, namely, for every subsequence {pk}, E∞(u) ≤ lim inf k Epk(uk) whenever uk ⇀ u. (3.8) To see this, we first observe that when u ∈ A∞ = D(E∞) we have E∞(u) = 0 and we trivially obtain E∞(u) ≤ lim infk Epk(uk) since Epk(uk) ≥ 0. Also, we can assume that lim infk Epk(uk) < +∞ (otherwise the desired inequal- ity holds trivially). Hence, for a subsequence we have that there is a constant C such that 1 pk min v∈W 1,pk (Ω),trace(v)=uk ∫ Ω |∇v|pk ≤ C. Call vk a function in W 1,pk(Ω) that attains the minimum. For this vk we have(∫ Ω |∇vk|pk )1/pk ≤ (pkC)1/pk . Now, for 2 < q <∞, we obtain(∫ Ω |∇vk|q )1/q ≤ |Ω|(pk−q)/pkq (∫ Ω |∇vk|pk )1/pk ≤ |Ω|(pk−q)/pkq(pkC)1/pk . The right-hand side is bounded and hence we can take the limit as pk → ∞ to obtain that vk ⇀ v∗ weakly in W 1,q(Ω). This limit v∗ satisfies(∫ Ω |∇v∗|q )1/q ≤ |Ω|1/q. Hence, taking q →∞ we conclude that v∗ ∈W 1,∞(Ω) and |∇v∗| ≤ 1 a.e. in Ω. Now, from the weak convergence of vk to v∗ in W 1,q(Ω) using the Sobolev trace embedding we obtain that uk = trace(vk)→ u = trace(v∗) strongly in L2(∂Ω) and hence we have that u ∈ A∞ = D(E∞). Then, we have 0 = E∞(u) ≤ lim inf k Epk(uk) since Epk(uk) ≥ 0, as we wanted to show. � EJDE-2021/SI/01 p-LAPLACIAN WITH DYNAMICAL BOUNDARY CONDITIONS 143 As a consequence we obtain the convergence of the corresponding solutions to the associated evolution problems. Proof of Theorem 1.2. We can apply Theorem 2.3 to obtain the first part of the result, namely, up → u∞ as p → ∞ in C([0, T ] : L2(∂Ω)) and the limit u∞ is characterized as the solution to the limit problem (1.5). To complete the proof we observe that, from the uniform bounds obtained in Lemma 3.1, we obtain the existence of a subsequence pi → ∞ such that the con- vergences stated in (1.6) hold. � 4. Mass transport interpretation of the limit problem We relate the limit problem with an optimal mass transport problem with a cost given by the distance between points inside Ω that is defined as the infimum of the lengths of curves going from x to y, that is, dΩ(x, y) = inf γ(0)=x,γ(1)=y length(γ(t)). When the domain Ω is convex the distance dΩ coincides with the Euclidean distance, we have dΩ(x, y) = |x− y|. Given two measures µ, ν on ∂Ω with the same total mass we consider the trans- port cost (Monge-Kantorovich mass transport problem) C(µ, ν) = min θ(x,y):θ|x=µ,θ|y=ν ∫ ∂Ω×∂Ω dΩ(x, y)dθ(x, y). Here by θ|x we denote the first marginal of θ, that is, θ|x(E) = θ(E × ∂Ω) (and similarly with θ|y we denote the second marginal of θ). Associated with an optimal mass transport problem we have its dual formulation that is given by C(µ, ν) = max v:|v(x)−v(y)|≤dΩ(x,y) ∫ ∂Ω v(x)(dµ(x)− dν(x)). Maximizers of the dual problem are called Kantorovich potentials for the optimal mass transport problem. It turns out that the limit of the solutions, u∞(·, t), is a Kantorovich potential for the optimal mass transport problem between f(·, t) and ∂u∞ ∂t (·, t). Proof of Theorem 1.3. First, let us prove that the limit function u∞ is admissible for the dual problem. Given two points x, y ∈ ∂Ω, using that |∇u∞(·, t)| ≤ 1 a.e. in Ω, we have |u∞(x, t)− u∞(y, t)| = ∣∣ ∫ 1 0 ∂u∞(γ(s), t) ∂s (s)ds ∣∣ = ∣∣ ∫ 1 0 〈∇u∞(γ(s), t), γ′(s)〉ds ∣∣ ≤ length(γ(s)) and hence we obtain |u∞(x)− u∞(y)| ≤ dΩ(x, y). 144 E. ÖZTÜRK, J. D. ROSSI EJDE/SI/01 Now, we show that in fact u∞(·, t) is a solution to the dual problem. We have that u∞(x, t) solves the limit equation f(x, t)− ∂u∞ ∂t (x, t) ∈ ∂E∞(u(x, t)), that is, E∞(v(x)) ≥ E∞(u∞(x, t)) + ∫ ∂Ω ( f(x, t)− ∂u∞ ∂t (x, t) ) (v(x)− u∞(x, t)) Take v ∈ A∞. Since u∞(·, t) ∈ A∞ we have 0 ≥ ∫ ∂Ω ( f(x, t)− ∂u∞ ∂t (x, t) ) (v(x)− u∞(x, t)) and therefore,∫ ∂Ω u∞(x, t) (∂u∞ ∂t (x, t)− f(x, t) ) ≥ ∫ ∂Ω v(x) (∂u∞ ∂t (x, t)− f(x, t) ) , for every v such that |v(x)− v(y)| ≤ dΩ(x, y). We have obtained that u∞(·, t) is a Kantorovich potential for the optimal mass transport problem between f(·, t)dσ and ∂u∞ ∂t (·, t)dσ. � 5. Examples In this final section we include some simple examples in which one can find the solution to the limit evolution problem f(x, t)− ∂u∞ ∂t (x, t) ∈ ∂E∞(u∞(x, t)) x ∈ ∂Ω, t > 0, u(x, 0) = u0(x) x ∈ ∂Ω. (5.1) Example 5.1. We consider the 1-dimensional case with Ω = (0, 1), and take f(x, t) = { 0, x = 0, t > 0, 1, x = 1, t > 0, (notice that f is defined on ∂Ω× (0, T )) and u0 ≡ 0. Then we have u∞(x, t) = { 0, x = 0, 0 ≤ t ≤ 1, t, x = 1, 0 ≤ t ≤ 1, and u∞(x, t) = { 1 2 (t− 1), x = 0, 1 ≤ t, 1 2 (t− 1) + 1, x = 1, 1 ≤ t. Notice that |u∞(1, t)− u∞(0, t)| ≤ 1 = dΩ(0, 1) = 1, for every t ≥ 0. Also we remark that the solution starts to grow at x = 1 with ∂u∞ ∂t (1, t) = 1 until it reaches u∞(1, t0) = 1 (this happens at t0 = 1) and next it grows at the slower rate ∂u∞ ∂t (1, t) = 1/2 (but also grows at x = 0 with ∂u∞ ∂t (0, t) = 1/2). This is due to the fact that the unit mass added at x = 1 is divided between two locations x = 0 and x = 1 in order to keep the constraint |u∞(1, t)− u∞(0, t)| ≤ 1 for times t ≥ 1. EJDE-2021/SI/01 p-LAPLACIAN WITH DYNAMICAL BOUNDARY CONDITIONS 145 Example 5.2. We consider a nontrivial initial condition for the setting of Ω, and f(x, t). Note that f is defined on ∂Ω × (0, T )) and fix a nonnegative C1 initial condition u0 with |u′0(x)| ≤ 1 for x ∈ [0, 1]. Then we have u∞(x, t) = { u0(0), x = 0, 0 ≤ t ≤ t0, u0(1) + t, x = 1, 0 ≤ t ≤ t0, with t0 the first time at which u0(1) + t0 − u0(0) = 1, that is t0 = u0(0)− u0(1) + 1. Note that t0 ≥ 0 because u0(0) − u0(1) + 1 = u′0(ξ) + 1 ≥ 0. Also note that u∞(x, t) ∈ A∞, since there exists a function v with |v′| ≤ 1 in [0, 1] such that v(0) = u0(0), v(1) = u0(1) + t (in addition, this function v can be chosen satisfying v ≥ u0 in [0, 1]). For times larger than t0 we have u∞(x, t) = { u0(0) + 1 2 (t− t0), x = 0, t0 ≤ t, u0(1) + 1 2 (t− t0) + t0, x = 1, t0 ≤ t. Example 5.3. Now, we extend above exmaples to several dimensions. Take a fixed domain Ω ⊂ RN , fix a subdomain of its boundary Γ ⊂ ∂Ω and consider f : ∂Ω× (0, T ) 7→ RN , f(x, t) = χΓ(x) and, as before, u0 ≡ 0. We remark that Example 5.1 is a particular case of this more general setting. In this case the solution u∞(x, t) to the limit problem is u∞(x, t) = (a(t)− dΩ(x,Γ))+, with a(t) the solution to the ODE a′(t) ∣∣∣{x ∈ ∂Ω : dΩ(x,Γ) < a(t)} ∣∣∣ HN−1 = |Γ|HN−1 , a(0) = 0. Here we denoted by |E|HN−1 the N−1-dimensional surface measure of a measurable set E ⊂ ∂Ω. Notice that the support of u∞(·, t) in ∂Ω can be disconnected even if the domain is strictly convex and the set where the source is localized Γ is connected. In fact, this is the case the set { x ∈ ∂Ω : dΩ(x,Γ) < k } is disconnected for some k > 0. Also notice that, since Ω is bounded and ∂Ω is smooth (it has finite HN−1−measure), there exists a finite time t0 such that the support of u∞(·, t) is the whole ∂Ω for times t ≥ t0. At this time t0 we have a(t0) = max x∈∂Ω dΩ(x,Γ) and then we have u∞(x, t) = ( max x∈∂Ω dΩ(x,Γ)− dist(x,Γ) ) , After this time the solution is u∞(x, t) = ( |Γ|HN−1 |∂Ω|HN−1 t+ max x∈∂Ω dΩ(x,Γ)− dist(x,Γ) ) , 146 E. ÖZTÜRK, J. D. ROSSI EJDE/SI/01 That is, after t0 the solution grows uniformly in the whole ∂Ω with speed |Γ|HN−1 |∂Ω|HN−1 . Acknowledgements. J. D. Rossi. is partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), PICT-2018-03183 (Argentina) and UBA- CyT grant 20020160100155BA (Argentina). References [1] G. Aronsson, L. C. 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Rossi Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria (C1428BCW), Buenos Aires, Argentina Email address: jrossi@dm.uba.ar 1. Introduction 2. Preliminaries 3. Mosco convergence of the functionals and convergence of the solutions 4. Mass transport interpretation of the limit problem 5. Examples Acknowledgements References