Regularity of the obstacle problem for the parabolic biharmonic equation
aa r X i v : . [ m a t h . A P ] M a y Regularity of the obstacle problemfor the parabolic biharmonic equation
M. Novaga and
S. Okabe
Abstract
We study the regularity of solutions to the obstacle problem for the parabolic biharmonicequation. We analyze the problem via an implicit time discretization, and we prove someregularity properties of the solution.
The purpose of this paper is to investigate the regularity properties of solutions to the obstacleproblem for the parabolic biharmonic equation.The parabolic biharmonic equation is a prototype of higher order parabolic equations, andhas been intensively studied in the mathematical literature. We refer for instance to [5, 11, 13,16, 17, 18, 19, 20, 26] and references therein, for a nonexhaustive list of works on this equation,and for a discussion of possible applications.The obstacle problem for elliptic and parabolic PDE’s is a topics which attracted a greatinterest in the past years. However, even if many studies are available on second order ellipticand parabolic equations (see for instance [8, 12] and references therein), there are relatively fewresults for higher order obstacle problems, even in the linear fourth order case. In particular,while the elliptic obstacle problem for the biharmonic operator has been considered in [7, 9,10, 15, 24], to the best of our knowledge no result is available for the corresponding parabolicobstacle problem.We let Ω ⊂ R N be a bounded domain, with boundary of class C , and we let f : Ω → R bethe obstacle function, satisfying f ∈ C (Ω) , f < ∂ Ω . (1.1)We consider an initial datum u : Ω → R such that u ∈ H (Ω) , u ≥ f a.e. in Ω . (1.2)We recall that u ∈ H (Ω) implies u = 0 and ∇ u · ν Ω = 0 (weakly) on ∂ Ω, that is, u satisfiesthe so-called Dirichlet boundary conditions on ∂ Ω (see [2, 18]), where ν Ω denotes the unit outernormal of ∂ Ω.We shall consider the following fourth order parabolic obstacle problem: u t ( x, t ) + ∆ u ( x, t ) ≥ × R + ,u t ( x, t ) + ∆ u ( x, t ) = 0 in { ( x, t ) ∈ Ω × R + : u ( x, t ) > f ( x ) } ,u ( x, t ) = 0 on ∂ Ω × R + , ∇ u ( x, t ) · ν Ω ( x ) = 0 on ∂ Ω × R + ,u ( x, t ) ≥ f ( x ) in Ω × R + ,u ( x,
0) = u ( x ) in Ω . (P) 1n order to state the main result of this paper precisely, we define a weak solution of (P). Letus set K := { u ∈ L (0 , T ; H (Ω)) | u t ∈ L (Ω × (0 , T )) , u ≥ f a.e. in Ω × (0 , T ) , (1.3) u ( x,
0) = u ( x ) a.e. in Ω } Then a weak solution of (P) is defined as follows:
Definition 1.1. u is a weak solution of (P) if (i) u ∈ K , (ii) For any w ∈ K , it holds that Z T Z Ω [ u t ( w − u ) + ∆ u ∆( w − u )] dxdt ≥ . (1.4)We now state the main result of this paper. Theorem 1.1.
Let N ≥ . Let f be a function satisfying (1.1) . Then, for any initial data u satisfying (1.2) , the problem (P) has a unique weak solution u ∈ L ∞ ( R + ; H (Ω)) ∩ H loc ( R + ; L (Ω)) , with u t ∈ L ( R + × Ω) . (1.5) Furthermore, for a.e. t ∈ R + the quantity (1.6) µ t := u t ( · , t ) + ∆ u ( · , t ) defines a Radon measure in Ω , and for any T > there exists a constant C > such that Z T µ t (Ω) dt < C. (1.7) Moreover, when N ≤ , the following regularity properties hold: (i) u ∈ L (0 , T ; W , ∞ (Ω)) for any T < + ∞ . In particular, if N = 1 , u ∈ C ,β ([0 , T ]; C ,γ (Ω) with < γ < and < β < − γ , (1.8) if N ∈ { , } , u ∈ C ,β ([0 , T ]; C ,γ (Ω)) with < γ < − N and < β < − N − γ . (1.9)(ii) For any < T < + ∞ , it holds that supp µ t ⊂ { ( x, t ) ∈ Ω × (0 , T ) | u ( x, t ) = f ( x ) } (1.10) and u satisfies (P) in the sense of distribution.
2e need to impose the restriction on the dimension N ≤ W , ∞ estimate on the solution u ( · , t ) (see Remark 2.1 for further comments on this). However, inanalogy with the regularity results in the stationary case [15, 9], one may expect that the W , ∞ estimate holds in any dimension.Let us point out that problem (P) corresponds to the gradient flow of a convex functionaldefined on the Hilbert space L (Ω), hence we can apply the general theory of maximal monotoneoperators developed in [6]. Indeed, given f as above, we can define the functional E f ( u ) : L (Ω) → [0 , + ∞ ] as E f ( u ) = Z Ω | ∆ u | if u ∈ H (Ω) and u ≥ f, + ∞ otherwise . Notice that E f ( u ) is convex and lower semicontinuous on L (Ω), and the problem (P) corre-sponds to the gradient flow(1.11) u t + ∂E f ( u ) ∋ , u (0) = u , where ∂E f denotes the subdifferential of E f in L (Ω). In particular, given an initial datum u ∈ H (Ω) with u ≥ f , by the results in [6] it follows that the evolution problem (1.11) has aunique solution u satisfying (1.5).In this paper we characterize the solution u by means of an implicit variational scheme, cor-responding to the minimizing movements introduced by De Giorgi (see e.g. [3]). This approachwill allow us to extend some of the arguments in [9], concerning the regularity of the ellipticobstacle problem for the biharmonic operator. We point out that the method does not rely onthe linear structure of the problem and can be applied to more general fourth order parabolicequations. Indeed, one motivation for this work comes from the motion of planar closed curvesby the elastic flow, in presence of obstacles. The elastic flow is the L gradient flow of the elasticenergy E ( γ ) = Z γ κ ds, where γ is a planar closed curve and κ denotes the curvature of γ . Among other applications, thisflow models the evolution of lipid bilayer membranes (see for instance [14]), where the presenceof obstacles is a natural features.Although this flow is governed by a fourth order quasilinear parabolic equation, we expectthat the method of this paper can be adapted, and this will be subject of future investigation.The paper is organized as follows: in Section 2 we introduce the implicit scheme correspond-ing to problem (P), by means of an appropriate variational problem; in Section 3 we studythe regularity of solutions to the variational problem; in Section 4 we pass to the limit in theapproximating scheme and prove Theorem 1.1. The equation in (P) is the L gradient flow for the functional E ( u ) = 12 Z Ω | ∆ u ( x ) | dx. T > n ∈ N , and set τ n = Tn .
Let us set u ,n = u . For i = 1 , · · · , n , we define inductively u i,n as a solution of the minimumproblem min { G i,n ( u ) : u ∈ K } , ( M i,n )where G i,n ( u ) := E ( u ) + P i,n ( u )(1.12)with P i,n ( u ) := 12 τ n Z Ω ( u − u i − ,n ) dx, (1.13)and K is a convex set given by K := { u ∈ H (Ω) : u ( x ) ≥ f ( x ) a.e. in Ω } . In the following, we let V i,n ( x ) := u i,n ( x ) − u i − ,n ( x ) τ n . (1.14) Definition 1.2. (Piecewise linear interpolation)
Define u n : Ω × [0 , T ] → R as u n ( x, t ) := u i − ,n ( x ) + ( t − ( i − τ n ) V i,n ( x )(1.15) if ( x, t ) ∈ Ω × [( i − τ n , iτ n ] for i = 1 , · · · , n . Definition 1.3. (Piecewise constant interpolation)
Define ˜ u n : Ω × [0 , T ] → R as ˜ u n ( x, t ) := u i,n ( x ) , (1.16) V n ( x, t ) := V i,n ( x ) , (1.17) if ( x, t ) ∈ Ω × [( i − τ n , iτ n ) for i = 1 , · · · , n . ( M i,n ) We first mention a well-known compactness result in H (Ω) [1, 2]. Proposition 2.1.
The following embedding is compact : H (Ω) ֒ → C ,γ (Ω) for < γ < if N = 1 ,C ,γ (Ω) for < γ < − N if N = 2 , ,L q (Ω) for ≤ ∀ q < + ∞ if N = 4 ,L q (Ω) for ≤ ∀ q < NN − if N ≥ . (2.1) 4e now show the existence of minimizers of ( M i,n ). Theorem 2.1. (Existence of minimizers)
Let f be a function satisfying (1.1) . Let u satisfy (1.2) . Then the problem ( M i,n ) possesses a unique solution u i,n ∈ H (Ω) with u i,n ( x ) ≥ f ( x ) a.e. in Ω for each i = 1 , · · · , n .Proof. Fix n ∈ N , T >
0, and i = 1 , · · · , n , arbitrarily. From (1.12)-(1.13) and the minimalityof a solution u to ( M i,n ), we obtain that E ( u ) ≤ G i,n ( u ) ≤ G i,n ( u i − ,n ) = E ( u i − ,n ) , and then 0 ≤ inf H (Ω) G i,n ( u ) ≤ G i,n ( u i − ,n ) = E ( u i − ,n ) ≤ · · · ≤ E ( u ) . Thus we can take a minimizing sequence { u j } ⊂ H (Ω) for ( M i,n ) such that u j ( x ) ≥ f ( x ) a.e.in Ω for each j ∈ N and sup j G i,n ( u j ) < ∞ .Observing that the norm k ∆ u k L (Ω) is equivalent to k u k H (Ω) (see [23]), it follows from k ∆ u j k L (Ω) = q E ( u j ) ≤ p E ( u ) = k ∆ u k L (Ω) that { u j } is uniformly bounded in H (Ω). Thus there exists u ∈ H (Ω) such that u j ⇀ u in H (Ω) , (2.2)in particular, ∆ u j ⇀ ∆ u in L (Ω) , (2.3)up to a subsequence. Thanks to Proposition 2.1, we obtain that u j → u in C ,γ ( ¯Ω) for 0 < γ <
12 if N = 1 ,C ,γ (Ω) for 0 < γ < − N N = 2 , ,L q (Ω) for 1 ≤ ∀ q < + ∞ if N = 4 ,L q (Ω) for 1 ≤ ∀ q < NN − N ≥ . In particular u j → u a.e. in Ω up to a subsequence.(2.4)Recalling u j ≥ f a.e. in Ω for each j ∈ N , (2.4) yields that u ≥ f a.e. in Ω. Making use ofFatou’s Lemma, we conclude that P i,n ( u ) ≤ lim inf j →∞ P i,n ( u j ) . (2.5)Furthermore (2.3) implies E ( u ) = 12 k ∆ u k L (Ω) ≤
12 lim inf j →∞ k ∆ u j k L (Ω) = lim inf j →∞ E ( u j ) . (2.6)Combining (2.5) with (2.6), we see that u ∈ H (Ω) is the minimizer of ( M i,n ) with u ≥ f a.e.in Ω. The uniqueness follows from the fact that the functional G i,n ( · ) is strictly convex.5egarding the regularity of the minimizer u i,n obtained in Theorem 2.1, we start with thefollowing: Theorem 2.2.
Let u i,n be the solution of ( M i,n ) obtained by Theorem . Then, for any n ∈ N ,it holds that Z T Z Ω | V n ( x, t ) | dxdt ≤ E ( u ) , (2.7) sup i k ∆ u i,n k L (Ω) ≤ p E ( u ) . (2.8) Proof.
Fix
T > n ∈ N . For each i = 1 , · · · , n , it follows from (1.12)-(1.13) and theminimality of u i,n that G i,n ( u i,n ) ≤ G i,n ( u i − ,n ) = E ( u i − ,n ) . (2.9)Hence we get P i,n ( u i,n ) ≤ E ( u i − ,n ) − E ( u i,n ) , i.e., 12 τ n Z Ω ( u i,n − u i − ,n ) dx ≤ E ( u i − ,n ) − E ( u i,n ) . (2.10)Combining (2.10) with definitions (1.14) and (1.17), we obtain12 Z T Z Ω | V n ( x, t ) | dxdt = 12 n X i =1 Z iτ n ( i − τ n Z Ω | V i,n ( x ) | dxdt ≤ n X i =1 ( E ( u i − ,n ) − E ( u i,n )) = E ( u ) − E ( u n,n ) ≤ E ( u ) , i.e., (2.7).By (2.9), we obtain that E ( u i,n ) ≤ E ( u i − ,n ) for each i = 1 , · · · , n , and then12 Z Ω (∆ u i,n ) dx = E ( u i,n ) ≤ E ( u ) . (2.11)It is clear that (2.11) is equivalent to (2.8).By the definition of u i,n , we see that Z Ω | ∆( u i,n + εζ ) | dx + 12 τ n Z Ω ( u i,n − u i − ,n + εζ ) dx ≥ Z Ω | ∆ u i,n | dx + 12 τ n Z Ω ( u i,n − u i − ,n ) dx for any ε > ζ ∈ H (Ω) with ζ ≥
0. This implies Z Ω ∆ u i,n ∆ ζ dx + 1 τ n Z Ω ( u i,n − u i − ,n ) ζ dx ≥ , so that µ i,n := ∆ u i,n + V i,n ≥ µ i,n is a measure in Ω (e.g., see [25]).Regarding the finiteness of µ i,n , we have the following:6 heorem 2.3. Let u i,n be the solution of ( M i,n ) obtained by Theorem . Then µ i,n definedin (2.12) is a measure in Ω for each i = 1 , · · · , n . Moreover there exists a positive constant C being independent of n such that τ n n X i =1 µ i,n (Ω) < C. (2.13) Proof.
Fix
T > n ∈ N and i = 1 , · · · , n arbitrarily. For any ε >
0, we define γ ε ( λ ) = λ ε if λ < , λ > , (2.14) β ε ( λ ) = γ ′ ε ( λ ) . (2.15)Let us consider the minimization problem: min v ∈ H (Ω) G εi,n ( v ), where G εi,n ( v ) := Z Ω (cid:20)
12 (∆ v ) + 12 τ n ( v − u i − ,n ) + γ ε ( v − f ) (cid:21) dx. (2.16)A standard argument implies that the problem has a unique solution w ε . Since the variationalprinciple yields that for any ϕ ∈ H (Ω) Z Ω (cid:20) ∆ w ε ∆ ϕ + 1 τ n ( w ε − u i − ,n ) ϕ + β ε ( w ε − f ) ϕ (cid:21) dx = 0 , we have ∆ w ε + 1 τ n ( w ε − u i − ,n ) + β ε ( w ε − f ) = 0 in Ω . (2.17)The standard elliptic regularity theory implies that w ε is a classical solution of (2.17).For any ϕ ∈ H (Ω) with ϕ ≥ f a.e. on Ω, the minimality of w ε asserts that G εi,n ( w ε ) ≤ G εi,n ( ϕ ) = Z Ω (cid:20)
12 (∆ ϕ ) + 12 τ n ( ϕ − u i − ,n ) (cid:21) dx. (2.18)Since Theorem 2.1 allows us to take u i − ,n as ϕ in (2.18), we have G εi,n ( w ε ) ≤ Z Ω (∆ u i − ,n ) dx ≤ E ( u ) , (2.19)i.e., 12 Z Ω (∆ w ε ) dx ≤ E ( u ) , (2.20) 12 τ n Z Ω ( w ε − u i − ,n ) dx ≤ E ( u ) , (2.21)and Z Ω γ ε ( w ε − f ) dx ≤ E ( u ) . (2.22) 7he inequality (2.20) implies that there exist a sequence { ε ′ } and a function ¯ u ∈ H (Ω) suchthat, as ε ′ → w ε ′ ⇀ ¯ u in H (Ω) , (2.23) w ε ′ → ¯ u a.e. in Ω . (2.24)By (2.14) and (2.22), we obtain Z Ω (cid:12)(cid:12) ( w ε − f ) − (cid:12)(cid:12) dx ≤ Cε.
Combining (2.24) with Chebychev’s inequality, we deduce that (¯ u − f ) − = 0 a.e. in Ω, i.e.,¯ u ≥ f a.e. in Ω. Thus it holds that ¯ u ∈ K . In the following we shall prove that ¯ u is a minimizerof ( M i,n ), i.e., min v ∈ V Z Ω (cid:20)
12 (∆ v ) + 12 τ n ( v − u i − ,n ) (cid:21) dx. To prove the assertion, fix v ∈ K arbitrarily. Then we observe that Z Ω (cid:20)
12 (∆ v ) + 12 τ n ( v − u i − ,n ) (cid:21) dx = E ( v ) + P i,n ( v ) + Z Ω γ ε ( v − f ) dx ≥ E ( w ε ) + P i,n ( w ε ) + Z Ω γ ε ( w ε − f ) dx ≥ Z Ω (cid:20)
12 (∆ w ε ) + 12 τ n ( w ε − u i − ,n ) (cid:21) dx. Making use of (2.23)-(2.24), we have Z Ω (cid:20)
12 (∆ v ) + 12 τ n ( v − u i − ,n ) (cid:21) dx ≥ lim inf ε ′ → Z Ω (cid:20)
12 (∆ w ε ′ ) + 12 τ n ( w ε ′ − u i − ,n ) (cid:21) dx ≥ Z Ω (cid:20)
12 (∆¯ u ) + 12 τ n (¯ u − u i − ,n ) (cid:21) dx. This implies that ¯ u is a minimizer of ( M i,n ). Then the uniqueness of minimizer yields ¯ u = u i,n .Recalling β ε ≤
0, we find∆ w ε + 1 τ n ( w ε − u i − ,n ) = − β ε ( w ε − f ) ≥ , i.e., µ εi,n := ∆ w ε + 1 τ n ( w ε − u i − ,n )is a measure in Ω. To begin with, we shall prove that µ εi,n converges to a measure as ε → i and n , { µ εi,n ( U ) } is uniformly boundedwith respect to ε for any compact subset U of Ω. Indeed, for each i , n and fixed ψ ∈ C ∞ (Ω)with ψ ≡ U and 0 ≤ ψ < µ εi,n ( U ) = Z U ψdµ εi,n ≤ Z Ω ψdµ εi,n (2.25) 8 Z Ω (cid:20) ∆ w ε ∆ ψ + 1 τ n ( w ε − u i − ,n ) ψ (cid:21) dx ≤ (cid:18)Z Ω (∆ w ε ) dx (cid:19) (cid:18)Z Ω (∆ ψ ) dx (cid:19) + 1 √ τ n (cid:18) τ n Z Ω ( w ε − u i − ,n ) dx (cid:19) (cid:18)Z Ω ψ dx (cid:19) . Since (2.19) yields that12 τ n Z Ω ( w ε − u i − ,n ) dx ≤ E ( u i − ,n ) − E ( w ε ) − Z Ω γ ε ( w ε − f ) dx (2.26) ≤ E ( u i − ,n ) − E ( w ε ) , and ψ is fixed, combining (2.25) with (2.20) and (2.26), we obtain µ εi,n ( U ) ≤ C ( U ) " (2 E ( u )) + (cid:18) E ( u i − ,n ) − E ( w ε ) τ n (cid:19) . (2.27)Then, for each i and n , there exist a sequence { ε ′′ } ⊂ { ε ′ } and a measure ¯ µ in Ω such that, as ε ′′ → µ ε ′′ i,n ⇀ ¯ µ, (2.28)where (2.28) means that for any function ζ ∈ C (Ω) Z Ω ζdµ ε ′′ i,n → Z Ω ζd ¯ µ. (2.29)Furthermore, taking ζ ∈ C (Ω) in (2.29), we find Z Ω ζd ¯ µ = lim ε ′′ → Z Ω (cid:20) ∆ ζ ∆ w ε ′′ + 1 τ n ζ ( w ε ′′ − u i − ,n ) (cid:21) dx = Z Ω (cid:20) ∆ ζ ∆¯ u + 1 τ n ζ (¯ u − u i − ,n ) (cid:21) dx, so that ¯ µ = µ i,n .Next we shall prove that τ n P ni =1 µ i,n ( U ) is uniformly bounded with respect to n for anycompact set U ⊂ Ω. Combining (2.27) with (2.23) and (2.28), we see that µ i,n ( U ) ≤ C ( U ) (2 E ( u )) + C ( U ) lim inf ε → (cid:18) E ( u i − ,n ) − E ( w ε ) τ n (cid:19) ≤ C ( U )(2 E ( u )) + C ( U ) (cid:18) E ( u i − ,n ) − E ( u i,n ) τ n (cid:19) . Multiplying τ n and summing over i = 1 , · · · , n , we obtain τ n n X i =1 µ i,n ( U ) ≤ C ( U ) ′ E ( u ) T + C ( U ) ′ [ E ( u ) − E ( u n,n )] ≤ C ( U ) ′ E ( u )( T + 1) . τ n P ni =1 µ i,n (Ω) is uniformly bounded with respect to n . Multiplyingthe equation (2.17) by w ε − f , we find Z Ω (cid:20) ∆ w ε + 1 τ n ( w ε − u i − ,n ) (cid:21) ( w ε − f ) dx = − Z Ω β ε ( w ε − f )( w ε − f ) dx ≤ . (2.30)Let Ω δ denote the intersection of Ω and δ -neighborhood of ∂ Ω. Since f < ∂ Ω, there existsa positive constant c such that f ( x ) < − c in Ω δ (2.31)for δ > Z Ω (cid:20) ∆ w ε + 1 τ n ( w ε − u i − ,n ) (cid:21) f dx (2.32) ≤ − c Z Ω δ (cid:20) ∆ w ε + 1 τ n ( w ε − u i − ,n ) (cid:21) dx + Z Ω \ Ω δ (cid:20) ∆ w ε + 1 τ n ( w ε − u i − ,n ) (cid:21) f dx. On the other hand, it follows from (2.26) and R Ω ∆ w ε w ε dx ≥ Z Ω (cid:20) ∆ w ε + 1 τ n ( w ε − u i − ,n ) (cid:21) w ε dx ≥ − k w ε k L (Ω) (cid:18) τ n Z Ω ( w ε − u i − ,n ) dx (cid:19) (2.33) ≥ − (2 E ( u )) (cid:18) E ( u i − ,n ) − E ( w ε ) τ n (cid:19) . Then (2.30), (2.32), and (2.33) imply that c Z Ω δ dµ εi,n ≤ k f k L ∞ (Ω \ Ω δ ) Z Ω \ Ω δ dµ εi,n + (2 E ( u )) (cid:18) E ( u i − ,n ) − E ( w ε ) τ n (cid:19) , so that µ εi,n (Ω δ ) ≤ c − k f k L ∞ (Ω \ Ω δ ) µ εi,n (Ω \ Ω δ ) + c − (2 E ( u )) (cid:18) E ( u i − ,n ) − E ( w ε ) τ n (cid:19) . Thus we get µ εi,n (Ω) ≤ C µ εi,n (Ω \ Ω δ ) + c − (2 E ( u )) (cid:18) E ( u i − ,n ) − E ( w ε ) τ n (cid:19) , where C = 1 + c − k f k L ∞ (Ω \ Ω δ ) . Then, by (2.23) and (2.28) we obtain µ i,n (Ω) ≤ C µ i,n (Ω \ Ω δ ) + c − (2 E ( u )) lim inf ε → (cid:18) E ( u i − ,n ) − E ( w ε ) τ n (cid:19) ≤ C µ i,n (Ω \ Ω δ ) + c − (2 E ( u )) (cid:18) E ( u i − ,n ) − E ( u i,n ) τ n (cid:19) . Since Ω \ Ω δ is a compact subset of Ω, multiplying τ n and summing over i = 1 , · · · , n , we observethat τ n n X i =1 µ i,n (Ω) ≤ C C δ + 2 c − E ( u )( E ( u ) − E ( u n,n )) ≤ C C δ + 2 c − E ( u ) , where C δ := τ n P ni =1 µ i,n (Ω \ Ω δ ) is independent of n . This completes the proof.10n the rest of this section, we shall prove that u i,n ∈ W , ∞ (Ω) if N ≤
3. In what follows, wedenote the mollifier as follows: J ε ( h )( x ) := Z R N j ε ( x − y ) h ( y ) dy, where j ε ( x − y ) = 1 ε n j (cid:18) x − yε (cid:19) and the function j ( x ) = j ( | x | ) satisfies j ∈ C ∞ ( R ) , j ( t ) = 0 if | t | > , j ( t ) ≥ , Z R N j ( | x | ) dx = 1 . Here we show a property of the support of µ i,n . Lemma 2.1.
Let x ∈ Ω . Assume that there exist a neighborhood W of x and a constant δ > such that J ε ( u i,n )( x ) − f ( x ) > δ in W. (2.34) Then µ i,n = 0 in W .Proof. We extend u i,n ∈ H (Ω) to become a function in H ( R n ). By the assumption (2.34), itholds that u i,n ± ζ ∈ K for any ζ ∈ C ∞ c ( W ) with | ζ | < δ . Since u i,n is the unique minimizer of( M i,n ), one can verify that for any ζ ∈ C ∞ c ( W ) with | ζ | < δ Z Ω | ∆ J ε ( u i,n ) | dx + 12 τ n Z Ω ( J ε ( u i,n ) − u i − ,n ) dx (2.35) ≤ Z Ω | ∆ J ε ( u i,n ) ± ∆ ζ | dx + 12 τ n Z Ω ( J ε ( u i,n ) ± ζ − u i − ,n ) dx. Letting ε ↓ Z Ω | ∆ u i,n | dx + 12 τ n Z Ω ( u i,n − u i − ,n ) dx ≤ Z Ω | ∆ u i,n ± ∆ ζ | dx + 12 τ n Z Ω ( u i,n ± ζ − u i − ,n ) dx, so that 0 ≤ ± (cid:18)Z Ω { ∆ u i,n ∆ ζ + V i,n ζ } dx (cid:19) + 12 Z Ω | ∆ ζ | dx + 12 τ n Z Ω ζ dx (2.36)for any ζ ∈ C ∞ c ( W ) with | ζ | < δ . Fix ζ ∈ C ∞ c ( W ) with | ζ | < δ arbitrarily. Then we assertsfrom (2.36) that0 ≤ ± ε (cid:18)Z Ω { ∆ u i,n ∆ ζ + V i,n ζ } dx (cid:19) + ε Z Ω | ∆ ζ | dx + ε τ n Z Ω ζ dx. (2.37)Since µ i,n ≥
0, it follows from (2.37) that0 ≤ R Ω { ∆ u i,n ∆ ζ + V i,n ζ } dx R Ω | ∆ ζ | dx + τ n R Ω ζ dx ≤ ε. ε > Z Ω { ∆ u i,n ∆ ζ + V i,n ζ } dx = 0 . This completes the proof.We denote the inverse operator of the Laplacian by ∆ − , i.e., if w satisfies ( − ∆ w = g in Ω ,w = 0 on ∂ Ω , then we write ∆ − g = w . We note that the estimate (cid:13)(cid:13) ∆ − g (cid:13)(cid:13) H (Ω) ≤ C k g k L (Ω) (2.38)is followed from the elliptic regularity (e.g., see [21]).We start with the following lemma: Lemma 2.2.
For each n ∈ N and i ∈ { , · · · , n } , there exists a function v i,n satisfying thefollowing properties :(a) v i,n = ∆ u i,n + ∆ − V i,n a.e. in Ω;(b) v i,n is upper semicontinuous in Ω;(c)
For any x ∈ Ω and for any sequence of balls B ρ ( x ) with center x and radius ρ , it holdsthat | B ρ ( x ) | Z B ρ ( x ) v i,n dx ↓ v i,n ( x ) as ρ ↓ . (2.39) Proof.
Let us define v ρi,n ( x ) = 1 | B ρ ( x ) | Z B ρ ( x ) (cid:2) ∆ u i,n ( y ) + ∆ − V i,n ( y ) (cid:3) dy. We claim that, for any x ∈ Ω, v ρi,n ( x ) is decreasing in ρ . Indeed, if u i,n ∈ C ∞ (Ω), we obtainfrom Green’s formula that∆ u i,n ( x ) + ∆ − V i,n ( x ) = 1 | ∂B ρ ( x ) | Z ∂B ρ ( x ) (cid:2) ∆ u i,n + ∆ − V i,n (cid:3) dS − Z B ρ ( x ) (cid:2) ∆ u i,n ( x ) + V i,n ( x ) (cid:3) G ρ ( x − x ) dx, where G ρ is Green’s function given by G ρ ( r ) =
12 ( r − ρ ) if N = 1 , π log ρr if N = 2 , N ( N − ω ( N ) ( r − N − ρ − N ) if N ≥ . (2.40) 12emark that ω ( N ) denotes the volume of unit ball in R N . From (2.12) and G ρ ′ > G ρ if ρ ′ > ρ ,we get 1 | ∂B ρ ( x ) | Z ∂B ρ ( x ) (cid:2) ∆ u i,n + ∆ − V i,n (cid:3) dS ≤ (cid:12)(cid:12) ∂B ρ ′ ( x ) (cid:12)(cid:12) Z ∂B ρ ′ ( x ) (cid:2) ∆ u i,n + ∆ − V i,n (cid:3) dS, and, by integration, 1 | B ρ ( x ) | Z B ρ ( x ) (cid:2) ∆ u i,n ( x ) + ∆ − V i,n ( x ) (cid:3) dx (2.41) ≤ (cid:12)(cid:12) B ρ ′ ( x ) (cid:12)(cid:12) Z B ρ ′ ( x ) (cid:2) ∆ u i,n ( x ) + ∆ − V i,n ( x ) (cid:3) dx. For general u i,n ∈ H (Ω) with (2.12), we introduce the C ∞ functions U m := J /m (∆ u i,n + ∆ − V i,n ) . Since ∆ U m ≥
0, we can deduce from (2.41) that1 | B ρ ( x ) | Z B ρ ( x ) U m dx ≤ (cid:12)(cid:12) B ρ ′ ( x ) (cid:12)(cid:12) Z B ρ ′ ( x ) U m dx. Letting m → + ∞ , we obtain (2.41) for general u i,n ∈ H (Ω). Thus we conclude that v ρi,n ( x ) ↓ v i,n ( x ) as ρ ↓ , (2.42)where v i,n is a some function.Since v ρi,n is continuous in x , we see that v i,n is upper semicontinuous. Recalling that ∆ u i,n +∆ − V i,n ∈ L (Ω), we also obtain that, as ρ ↓ v ρi,n → ∆ u i,n + ∆ − V i,n a.e. in Ω . Consequently we have v i,n = ∆ u i,n + ∆ − V i,n a.e. in Ω . This completes the proof.
Lemma 2.3.
Let ≤ N ≤ , then for any point x ∈ Ω that belongs to the support of µ i,n , itholds that v i,n ( x ) − ∆ − V i,n ( x ) ≥ ∆ f ( x )(2.43) for each n ∈ N and i = 1 , · · · , n .Proof. With the aid of Lemma 2.1, we asserts that supp µ i,n is contained in the set of pointswhere (2.34) is not satisfies. Thus, if x ∈ supp µ i,n , then there exist sequences x m → x and ε m ↓ J ε m u i,n )( x m ) − f ( x m ) → . (2.44)By Green’s formula, we have( J ε u i,n )( x m ) = 1 | S ρ,m | Z S ρ,m J ε u i,n dS − Z B ρ,m ∆( J ε u i,n )( y ) G ρ ( x m − y ) dy, (2.45) 13here B ρ,m := {| y − x m | < ρ } , S ρ,m := ∂B ρ,m . Similarly it holds that( J ε f )( x m ) = 1 | S ρ,m | Z S ρ,m J ε f dS − Z B ρ,m ∆( J ε f )( y ) G ρ ( x m − y ) dy. (2.46)Since it follows from u i,n ≥ f , also J ε u ≥ J ε f , that1 | S ρ,m | Z S ρ,m J ε u i,n dS ≥ | S ρ,m | Z S ρ,m J ε f dS, using the inequality and (2.44), we obtain, by comparing (2.45) with (2.46), thatlim inf m → + ∞ "Z B ρ,m ∆( J ε u i,n )( y ) G ρ ( x m − y ) dy − Z B ρ,m ∆( J ε f )( y ) G ρ ( x m − y ) dy ≥ . (2.47)Using a change of variables and integrating by parts, we can reduce the first term in (2.47) to Z B ρ,m ∆( J ε u i,n )( y ) · G ρ ( x m − y ) dy = Z B ρ,m ( J ε G ρ )( x m − y )∆ u i,n ( y ) dy + λ ε,m , (2.48)where λ ε,m := − Z B ρ + ε,m \ B ρ,m G ρ ( x m − y )∆( J ε u i,n )( y ) dy + Z B ρ + ε,m G ρ ( x m − y ) Z B ρ +2 ε,m \ B ρ,m j ε ( y − z )∆ u i,n ( z ) dy and λ ε,m → ε ↓ m . A similar relation holds for the second integral in (2.47).Therefore we obtainlim inf m → + ∞ Z B ρ,m ( J ε m G ρ )( x m − y )[ v i,n ( y ) − ∆ − V i,n ( y ) − ∆ f ( y )] dy ≥ . (2.49)Recalling that V i,n ∈ H (Ω) for each n ∈ N , we see that ∆ − V i,n ∈ H (Ω) by the ellipticregularity (see [21]). Then it follows from Sobolev’s embedding that ∆ − V i,n is continuious in Ωfor 1 ≤ N ≤
7. Furthermore since v i,n is upper semicontinuous, there exists a point x m,ρ ∈ B ρ,m such that the maximum of the function v i,n ( x ) − ∆ − V i,n ( x ) − ∆ f ( x ) in B ρ,m attains at x = x m,ρ .Then (2.49) implies that v i,n ( x m,ρ ) − ∆ − V i,n ( x m,ρ ) − ∆ f ( x m,ρ ) ≥ − δ m , δ m → m → + ∞ . We may assume that x m,ρ → x ρ for some x ρ ∈ { y ∈ R N : | y − x | ≤ ρ } , for the sequence { x m,ρ } is bounded. By the upper semicontinuity of v i,n , as m → + ∞ , it holds that v i,n ( x ρ ) − ∆ − V i,n ( x ρ ) − ∆ f ( x ρ ) ≥ . Letting ρ → v i,n , we see that x ρ → x and v i,n ( x ) − ∆ − V i,n ( x ) − ∆ f ( x ) ≥ . Making use of Lemmas 2.2 and 2.3, we can obtain a local bound of ∆ u i,n :14 emma 2.4. Let N ≤ . It holds that ∆ u i,n ∈ L ∞ loc (Ω)(2.50) for each n ∈ N and i = 1 , · · · , n . Moreover, for any R > with B R ⊂ Ω , there exist positiveconstants C , C , and C being independent of i and n such that k ∆ u i,n k L ∞ ( B R/ ) ≤ C E ( u ) + C k V i,n k L (Ω) + C µ i,n ( D R/ ) + k ∆ f k L ∞ ( B R/ ) , (2.51) where D R/ := B R \ B R/ .Proof. Set U i,n := u i,n + (∆ ) − V i,n , (2.52)where (∆ ) − V i,n denotes a unique solution of ( − ∆ w = ∆ − V i,n in Ω ,w = 0 on ∂ Ω . Let fix x ∈ Ω arbitrarily and denote by B ρ the ball with center x and radius ρ . Choose R > B R ⊂ Ω and ζ ∈ C ∞ ( B R ), ζ = 1 in B R/ , 0 ≤ ζ ≤ x ∈ B R/ ,we have ∆( J ε U i,n )( x ) = ∆( J ε U i,n )( x ) ζ ( x ) = − Z B R G R ( x − y )∆(∆( J ε U i,n ) ζ )( y ) dy, where G R is Green’s function defined in (2.40). Expanding the right-hand side, we obtain∆( J ε U i,n )( x ) = − Z B R/ G R ( x − y )∆ ( J ε U i,n )( y ) dy (2.53) − Z D R/ G R ( x − y )∆ ( J ε U i,n )( y ) ζ ( y ) dy + α ε ( x ) , where D R/ := B R \ B R/ and α ε ( x ) := − Z D R/ G R ( x − y ) ∇ (∆( J ε U i,n ))( y ) · ∇ ζ ( y ) dy − Z D R/ G R ( x − y )∆( J ε U i,n )( y )∆ ζ ( y ) dy := α ε, ( x ) + α ε, ( x ) . Noticing that supp ∇ ζ is contained in D R/ := B R \ B R/ , we get α ε, ( x ) = − Z D R/ ∆( J ε U i,n )( y ) ∇ · ( G R ( x − y ) ∇ ζ ( y )) dy. Since the fact that u i,n ∈ H (Ω) implies Z Ω | ∆( J ε U i,n )( y ) | dy ≤ k ∆ U i,n k L (Ω) , α ε, ( x ) and α ε, ( x ) are estimated for any x ∈ B R/ as follows: | α ε, ( x ) | ≤ C k ∆ U i,n k L (Ω) ( k∇ ζ k L ( D R/ ) + k ∆ ζ k L ( D R/ ) ); | α ε, ( x ) | ≤ C k ∆ U i,n k L (Ω) k ∆ ζ k L ( D R/ ) . Thus we deduce that | α ε ( x ) | ≤ C k ∆ U i,n k L (Ω) for all x ∈ B R/ , (2.54)where the constant C is independent of ε , i , and n .Along the same line as in (2.48), the first term in the right-hand side of (2.53) is reduced to Z B R/ G R ( x − y )∆ ( J ε U i,n )( y ) dy = Z B R/ ( J ε G R )( x − y )∆ U i,n ( y ) dy + β ε ( x ) , (2.55)where β ε ( x ) → ε ↓ x ∈ B R/ .Consider the integral ˜ G R ( x ) := Z B R/ G R ( x − y ) dµ i,n ( y ) . The integral is well defined in the sense of improper integrals, that is, aslim δ → Z { y ∈ B R/ : | x − y | >δ } G R ( x − y ) dµ i,n ( y ) for a.e. x. Indeed, this follows from Fubini’s theorem since for any k < + ∞ it holds that Z B R/ Z | x |
0. Thus we deduce from Lebesgue’s convergence theorem that for x ∈ B R/ , as ε ↓ Z B R \ B R/ G R ( x − y )∆ ( J ε U i,n )( y ) ζ ( y ) dy → Z B R \ B R/ G R ( x − y )∆ U i,n ( y ) ζ ( y ) dy. (2.57) 16e can write∆( J ε U i,n )( x ) = Z | z − x | <ε U i,n ( z )∆ j ε ( x − z ) dz = Z | z − x | <ε ∆ U i,n ( z ) j ε ( x − z ) dz = Z | z − x | <ε v i,n ( z ) j ε ( x − z ) dz = Z ε ε N j (cid:16) ρε (cid:17) Z ∂B ρ ( x ) v i,n ( ρ, θ ) dS θ dρ, where ( ρ, θ ) is the spherical coordinates about x and λ ε ( ρ ) is a smooth nonnegative function.Since it follows from the proof of Lemma 2.2 that1 | ∂B ρ ( x ) | Z ∂B ρ ( x ) v i,n ( ρ, θ ) dS θ ↓ v i,n ( x ) as ρ ↓ , the mean value theorem yields that∆( J ε U i,n )( x ) = 1 (cid:12)(cid:12) ∂B ρ ′ (cid:12)(cid:12) Z ∂B ρ ′ v i,n ( ρ ′ , θ ) dS θ Z ε ε N j (cid:16) ρε (cid:17) ω N ρ N − dρ = 1 (cid:12)(cid:12) ∂B ρ ′ (cid:12)(cid:12) Z ∂B ρ ′ v i,n ( ρ ′ , θ ) dS θ Z | y | < j ( | y | ) dy = 1 (cid:12)(cid:12) ∂B ρ ′ (cid:12)(cid:12) Z ∂B ρ ′ v i,n ( ρ ′ , θ ) dS θ ↓ v i,n ( x ) as ε ↓ , where ω N ρ N − denotes the area of surface ∂B ρ and ρ ′ ∈ (0 , ε ). Combining this with (2.55),(2.56), and (2.57), letting ε ↓ x ∈ B R/ there holds v i,n ( x ) = − ˜ G R ( x ) − Z D R/ ζ ( y ) G R ( x − y )∆ U i,n ( y ) dy + δ ( x ) . (2.58)Remark that (2.54) implies | δ ( x ) | ≤ C k ∆ U i,n k L (Ω) for all x ∈ B R/ , (2.59)where the constant C is independent of i and n . Recalling that ˜ G R is superharmonic, we shallapply a maximal principle for superharmonic functions to ˜ G R . It follows from Lemma 2.3 that v i,n ( x ) ≥ ∆ − V i,n ( x ) + ∆ f ( x ) on supp µ i,n ⌊ B R/ . Since the integral on the right-hand side of (2.58) is non-negative, we see that˜ G R ( x ) ≤ − v i,n ( x ) + δ ( x ) ≤ − ∆ − V i,n ( x ) − ∆ f ( x ) + δ ( x )(2.60) ≤ (cid:13)(cid:13) ∆ − V i,n (cid:13)(cid:13) C ( B R/ ) + k ∆ f k L ∞ ( B R/ ) + k δ k L ∞ ( B R/ ) on supp µ i,n ⌊ B R/ . Furthermore Proposition 2.1 and (2.38) assert that (cid:13)(cid:13) ∆ − V i,n (cid:13)(cid:13) C ( B R/ ) ≤ k ∆ − V i,n k C k,γ (Ω) ≤ C (cid:13)(cid:13) ∆ − V i,n (cid:13)(cid:13) H (Ω) ≤ C k V i,n k L (Ω) , (2.61)where k = 1 and 0 < γ < / N = 1, k = 0 and 0 < γ < − N/ N = 2, 3, and the constant C is independent of i and n . Thus, combining (2.60) with (2.59) and (2.61), we observe that˜ G R ( x ) ≤ C k ∆ U i,n k L ((Ω) + C k V i,n k L (Ω) + k ∆ f k L ∞ ( B R/ ) on supp µ i,n ⌊ B R/ , G R ( x ) ≤ C k ∆ U i,n k L ((Ω) + C k V i,n k L (Ω) + k ∆ f k L ∞ ( B R/ ) in R N . Observing that the integral in (2.58) is estimated as Z D R/ ζ ( y ) G R ( x − y )∆ U i,n ( y ) dy ≤ C µ i,n ( D R/ ) in B R/ , we deduce that, for any x ∈ B R/ , | v i,n ( x ) | ≤ C k ∆ U i,n k L ((Ω) + C k V i,n k L (Ω) + C µ i,n ( D R/ ) + k ∆ f k L ∞ ( B R/ ) , so that | ∆ u i,n ( x ) | ≤ C k ∆ U i,n k L ((Ω) + 2 C k V i,n k L (Ω) + C µ i,n ( D R/ ) + k ∆ f k L ∞ ( B R/ ) . (2.62)Since (2.8) yields that k ∆ U i,n k L (Ω) ≤ k ∆ u i,n k L (Ω) + (cid:13)(cid:13) ∆ − V i,n (cid:13)(cid:13) L (Ω) ≤ p E ( u ) + C k V i,n k L (Ω) , we obtain k ∆ u i,n k L ∞ ( B R/ ) ≤ C ′ p E ( u ) + C ′ k V i,n k L (Ω) + C µ i,n ( D R/ ) + k ∆ f k L ∞ ( B R/ ) . This completes the proof.
Remark 2.1.
We need to impose the restriction on the dimension N ≤ k ∆ − V i,n k L ∞ ( B R/ ) ≤ C k V i,n k L (Ω) in (2.61). Such an estimate will allow us to prove a uniform W , ∞ bound on u i,n with respectto n . Theorem 2.4.
Let N ≤ . It holds that u i,n ∈ W , ∞ (Ω)(2.63) for each n ∈ N and i = 1 , · · · , n . Moreover, for any R > with B R ⊂ Ω , there exist positiveconstants C and C being independent of n such that τ n n X i =1 (cid:13)(cid:13) D u i,n (cid:13)(cid:13) L ∞ (Ω) ≤ C + C k ∆ f k L ∞ (Ω) . (2.64) Proof.
Thanks to Theorem 2.2, we see that u i,n is uniformly bounded in H (Ω). Then, Propo-sition 2.1 asserts that u i,n is also uniformly bounded in C ,γ (Ω) with 0 < γ < / N = 1, andin C ,γ (Ω) with γ ∈ (0 , − N/
2) if N = 2, 3. Since u i,n = 0 on ∂ Ω, there exists a neighborhoodΩ δ of ∂ Ω such that u i,n > f in Ω δ . By the standard elliptic regularity theory, we observe that∆ u i,n ∈ H (Ω δ ) with k ∆ u i,n k H (Ω δ ) ≤ C ( k V i,n k L (Ω δ ) + k ∆ u i,n k L (Ω δ ) ) , (2.65) 18here the positive constant C depends only on Ω δ . Combining (2.65) with the interpolationinequality k ∆ u i,n k L ∞ (Ω δ ) ≤ K k ∆ u i,n k N/ H (Ω δ ) k ∆ u i,n k − N/ L (Ω δ ) , where K is a positive constant depending only on N , we deduce that k ∆ u i,n k L ∞ (Ω δ ) ≤ C ′ ( k V i,n k L (Ω) + k ∆ u i,n k L (Ω δ ) ) . (2.66)In the sequel, we let N = 2, 3. Let fix x ∈ Ω \ Ω δ arbitrarily and B ρ denote the ball withcenter x and radius ρ . Choose R > B R ⊂ Ω and ζ ∈ C ∞ ( B R ), ζ = 1 in B R/ ,0 ≤ ζ ≤ x ∈ B R/ , we can write( J ε U i,n )( x ) = Z B R W ( x − y )∆ ( ζJ ε U i,n )( y ) dy, where U i,n is the function defined by (2.52) and W is the fundamental solution of ∆ : W ( x ) = ( γ N | x | (log | x | −
1) if N = 2 , − γ N | x | if N = 3 , where γ N are constants chosen such that ∆ W = δ, where δ denotes the Dirac measure (e.g., see [15]). Expanding ∆ ( ζJ ε U i,n ) and performingintegrations by parts, we obtain( J ε U i,n )( x )(2.67) = Z B R/ W ( x − y )∆ ( ζJ ε U i,n )( y ) dy + Z D R/ W ( x − y )∆ ( ζJ ε U i,n )( y ) dy = Z B R/ W ( x − y ) ζ ( y )∆ ( J ε U i,n )( y ) dy + Z D R/ W ( x − y ) h ∆ ζ ( J ε U i,n ) + 4 ∇ (∆ ζ ) · ∇ ( J ε U i,n ) + 6∆ ζ ∆( J ε U i,n )+ 4 ∇ ζ · ∇ ∆( J ε U i,n ) + ζ ∆ ( J ε U i,n ) i ( y ) dy = Z B R W ( x − y ) ζ ( y )∆ ( J ε U i,n )( y ) dy + α ε ( x ) , where D R/ := R R \ B R/ and α ε ( x ) = Z D R/ W ( x − y ) h ∆ ζ ( J ε U i,n ) + 4 ∇ (∆ ζ ) · ∇ ( J ε U i,n ) + 2∆ ζ ∆( J ε U i,n ) i ( y ) dy − Z D R/ ∇ W ( x − y ) · ∇ ζ ( y )∆( J ε U i,n )( y ) dy. Since it follows from a direct calculation that (cid:18) ∂ ∂x j −
12 ∆ (cid:19) W ( x ) = ( γ N (cid:16) x j | x | − − (cid:17) if N = 2 ,γ N x j | x | − if N = 3 , (cid:18) ∂ ∂x j −
12 ∆ (cid:19) W ≥ − c, (2.68)where c is a positive constant. Applying ∂ /∂x j − ∆ / ζ ∆ ( J ε U i,n ) ≥
0, we obtain, if x ∈ B R/ , (cid:18) ∂ ∂x j −
12 ∆ (cid:19) J ε U i,n ( x ) ≥ − c Z B R ζ ( y )∆ ( J ε U i,n )( y ) dy + (cid:18) ∂ ∂x j −
12 ∆ (cid:19) α ε ( x ) . Since the integral in the right-hand side can be written as Z B R ( J ε ζ )( y )∆ U i,n ( y ) dy + β ε , where β ε → ε ↓
0, we conclude that ∂ J ε U i,n ∂x j ( x ) ≥ − k ∆ J ε U i,n k L ∞ ( B R/ ) − c Z B R ( J ε ζ )( y )∆ U i,n ( y ) dy (2.69) − cβ ε + (cid:18) ∂ ∂x j −
12 ∆ (cid:19) α ε ( x ) in B R/ . On the other hand, it also holds that ∂ J ε U i,n ∂x j = ∆( J ε U i,n ) − X k = j ∂ J ε U i,n ∂x k (2.70) ≤ N + 12 k ∆( J ε U i,n ) k L ∞ ( B R/ ) + c ( N − Z B R ( J ε ζ )( y )∆ U i,n ( y ) dy + c ( N − β ε − ( N − (cid:18) ∂ ∂x j −
12 ∆ (cid:19) α ε ( x ) in B R/ . Lemma 2.4 implies that k ∆( J ε U i,n ) k L ∞ ( B R/ ) ≤ k ∆ U i,n k L ∞ ( B R/ ) (2.71) ≤ C E ( u ) + ( C + 1) k V i,n k L (Ω) + C µ i,n ( D R/ ) + k ∆ f k L ∞ ( B R/ ) . Letting ε ↓
0, we find Z B R ( J ε ζ )( y )∆ U i,n ( y ) dy → Z B R ζ ( y )∆ U i,n ( y ) dy ≤ µ i,n ( B R ) . (2.72)Furthermore it follows from the Gagliardo-Nirenberg type interpolation inequality that k α ε k L ∞ ( B R/ ) ≤ C {k J ε U i,n k L (Ω) + k∇ ( J ε U i,n ) k L (Ω) + k ∆( J ε U i,n ) k L (Ω) }≤ C {k J ε U i,n k L (Ω) + k ∆( J ε U i,n ) k L (Ω) }≤ C {k U i,n k L (Ω) + k ∆ U i,n k L (Ω) } . Observing k U i,n k L (Ω) ≤ k u i,n k L (Ω) + (cid:13)(cid:13) (∆ ) − V i,n (cid:13)(cid:13) L (Ω) ≤ k u i,n k H (Ω) + C k V i,n k L (Ω) ,
20e obtain k α ε k L ∞ ( B R/ ) ≤ C ′ E ( u ) + C ′ k V i,n k L (Ω) + C ′ µ i,n ( D R/ ) + C k ∆ f k L ∞ ( B R/ ) (2.73)Recalling β ε → ε ↓ ε ↓ (cid:13)(cid:13)(cid:13)(cid:13) ∂ u i,n ∂x j (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( B R/ ) ≤ C E ( u ) + C k V i,n k L (Ω) + C µ i,n ( B R ) + C k ∆ f k L ∞ ( B R/ ) (2.74)Since x j can be in any direction, the inequality (2.74) implies that (cid:13)(cid:13) D u i,n (cid:13)(cid:13) L ∞ ( B R/ ) ≤ C ′ E ( u ) + C ′ k V i,n k L (Ω) + C ′ µ i,n ( B R ) + C ′ k ∆ f k L ∞ ( B R/ ) , (2.75)where the constants C ′ , C ′ , C ′ , and C ′ are independent of i and n . Recalling (2.66), along thesame line as above, one can verify that (cid:13)(cid:13) D u i,n (cid:13)(cid:13) L ∞ (Ω δ ) ≤ C ( k V i,n k L (Ω δ ) + E ( u ) ) , (2.76)where the constant C depends only on Ω δ . Since Ω \ Ω δ is compact, combining (2.75) with(2.76), we obtain the assertion u i,n ∈ W , ∞ (Ω) and (cid:13)(cid:13) D u i,n (cid:13)(cid:13) L ∞ (Ω) ≤ CE ( u ) + C k V i,n k L (Ω) + Cµ i,n (Ω) + C k ∆ f k L ∞ (Ω) (2.77)Finally multiplying (2.77) by τ n and summing over i = 1 , · · · , n , we conclude from (2.7) and(2.13) that τ n n X i =1 (cid:13)(cid:13) D u i,n (cid:13)(cid:13) L ∞ (Ω) ≤ CT E ( u ) + C Z T k V n ( t ) k L (Ω) dt + Cτ n n X i =1 µ i,n (Ω) + CT k ∆ f k L ∞ (Ω) ≤ CT E ( u ) + 2 CE ( u ) + C + CT k ∆ f k L ∞ (Ω) . This completes the proof.When we restrict to dimensions N ≤
3, Proposition 2.1 implies that u i,n is continuous.Under such restriction, we define C i,n := { x ∈ Ω : u i,n ( x ) = f ( x ) } , (2.78) N i,n := { x ∈ Ω : u i,n ( x ) > f ( x ) } . (2.79)It is clear that C i,n ∪ N i,n = Ω. We can show a relation between the support of µ i,n and the sets. Lemma 2.5.
Let N ≤ . If x ∈ N i,n , then there exists a neighborhood of x such that µ i,n ( N i,n ) = 0 . Furthermore we have supp µ i,n ⊆ C i,n . (2.80) 21 roof. Let N ≤ x ∈ N i,n arbitrarily. Since N i,n is an open set, there exist a constant δ > W of x such that u i,n ( x ) − f ( x ) > δ for all x ∈ W. Notice that u i,n satisfies Z Ω ∆ u i,n ∆( u i,n − ϕ ) dx ≤ − Z Ω V i,n ( u i,n − ϕ ) dx (2.81)for any ϕ ∈ K , for u i,n is a solution of ( M i,n ). Then for any ζ ∈ C ∞ ( W ) with 0 ≤ ζ ≤ δ/
2, thefunction ψ = u i,n − ζ belongs to K . Taking this ψ as ϕ in (2.81), we have Z Ω [∆ u i,n ∆ ζ + V i,n ζ ] dx ≤ , Since µ i,n ≥
0, this asserts that Z Ω [∆ u i,n ∆ ζ + V i,n ζ ] dx = 0 , i.e., µ i,n = 0 in W . (P) We first prove a convergence result which holds in any dimension N ≥ Theorem 3.1.
Let u n be the piecewise linear interpolation of { u i,n } . Then there exists a function u ∈ L ∞ ([0 , + ∞ ); H (Ω)) ∩ H loc (0 , + ∞ ; L (Ω)) such that u n ⇀ u in L (0 , T ; H (Ω)) ∩ H (0 , T ; L (Ω)) as n → + ∞ , (3.1) up to a subsequence, for any < T < + ∞ . Moreover Z T Z Ω u t dx dt ≤ E ( u ) ,u ( x, t ) ≥ f ( x ) for a.e. x ∈ Ω and for every t ∈ [0 , + ∞ ) , and for each α ∈ (0 , ) it holds u n → u in C ,α ([0 , T ]; L (Ω)) as n → + ∞ . (3.2) 22 roof. Recalling that u n ( x, · ) is absolutely continuous on [0 , T ], for all t , t ∈ [0 , T ] with t < t ,H¨older’s inequality and Fubini’s Theorem give us k u n ( · , t ) − u n ( · , t ) k L (Ω) = Z L (cid:18)Z t t ∂ u n ∂t ( x, t ) dt (cid:19) dx ! ≤ Z t t (cid:13)(cid:13)(cid:13)(cid:13) ∂ u n ∂t ( · , t ) (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) dt ! ( t − t ) . Then it follows from (2.7) that Z t t Z Ω u t dx dt ≤ E ( u )(3.3)and k u n ( · , t ) − u n ( · , t ) k L (Ω) ≤ p E ( u )( t − t ) . (3.4)Since (2.8) yields thatsup t ∈ [0 ,T ] k ∆ u n ( · , t ) k L (Ω) ≤ sup ≤ i ≤ n k ∆ u i,n k L (Ω) ≤ p E ( u ) , (3.5)there exists a function u ∈ L (0 , T ; H (Ω)) such that u n ⇀ u in L (0 , T ; H (Ω)) up to asubsequence. On the other hand, the estimate (2.7) implies that V n = ∂ u n ∂t ⇀ ∂ u∂t in L (0 , T ; L (Ω)) . (3.6)This means that ∂u/∂t ∈ L (0 , T ; L (Ω)), i.e., u ∈ H (0 , T ; L (Ω)). Combining (3.4) withAscoli-Arzel`a’s Theorem (see e.g. [4, Proposition 3.3.1]), we conclude (3.2).Since (3.5) means that { u n ( t ) } is uniformly bounded in H (Ω) with respect to t ∈ [0 , T ] and n ∈ N , we deduce from (3.2) that, for each t ∈ [0 , T ] u n ( t ) ⇀ u ( t ) in H (Ω)(3.7)up to a subsequence. This asserts that u ∈ L ∞ ([0 , T ]; H (Ω)). Moreover, Proposition 2.1 impliesthat for each t ∈ [0 , T ] u n ( t ) → u ( t ) in C ,γ (Ω) for 0 < γ < if N = 1 ,C ,γ (Ω) for 0 < γ < − N if N = 2 , ,L q (Ω) for 0 < q < + ∞ if N = 4 ,L q (Ω) for 0 < q < NN − if N ≥ . (3.8)In particular, if N ≥ u n ( t ) → u ( t ) a.e. in Ω(3.9)up to a subsequence. Since u n ( t ) ≥ f a.e. in Ω for each n ∈ N and t ∈ [0 , T ], the fact (3.8)-(3.9)yields that u ( t ) ≥ f a.e. in Ω for each t ∈ [0 , T ]. This completes the proof.When N = 1, we can improve the convergence result obtained in Theorem 3.1:23 heorem 3.2. Let N = 1 . Let u be the function obtained by Theorem . Then it holds that u ∈ L (0 , T ; W , ∞ (Ω)) ∩ C ,β ([0 , T ]; C ,α (Ω)) and u n → u weakly * in L (0 , T ; W , ∞ (Ω)) as n → ∞ , (3.10) u n → u in C ,β ([0 , T ]; C ,α (Ω)) as n → ∞ (3.11) for every α ∈ (0 , ) and β ∈ (0 , − α ) . Furthermore u ( · , t ) → u in C ,α (Ω) as t ↓ .Proof. Fix
T > n ∈ N . To begin with, we shall prove (3.10). By (2.64) we see that u n isuniformly bounded in L (0 , T ; W , ∞ (Ω)) with respect to n ∈ N . Since L (0 , T ; W , ∞ (Ω)) is thedual of L (0 , T ; W , (Ω)), Banach-Alaoglu’s Theorem asserts that u n subconverges to u weakly*in L (0 , T ; W , ∞ (Ω)). In particular, combining (2.64) with k u k L (0 ,T ; W , ∞ (Ω)) ≤ lim inf n → + ∞ k u n k L ([0 ,T ]; W , ∞ (Ω)) , we observe that u ∈ L (0 , T ; W , ∞ (Ω)).Next we prove (2.64). In the sequel we let Ω = (0 , L ). Let us define the function g := u n ( · , t ) − u n ( · , t ). Since g ∈ H (Ω) for each t , t ∈ [0 , T ] with t < t , we have Z Ω ( g ′ ( x )) dx = − Z Ω g ( x ) g ′′ ( x ) dx ≤ k g k L (Ω) (cid:13)(cid:13) g ′′ (cid:13)(cid:13) L (Ω) , (3.12)and ( g ′ ( x )) = Z x { ( g ′ ( x )) } ′ dx ≤ (cid:13)(cid:13) g ′ (cid:13)(cid:13) L (Ω) (cid:13)(cid:13) g ′′ (cid:13)(cid:13) L (Ω) . (3.13)Then (3.12) and (3.13) yield (cid:13)(cid:13) g ′ (cid:13)(cid:13) L ∞ (Ω) ≤ √ (cid:13)(cid:13) g ′′ (cid:13)(cid:13) L (Ω) k g k L (Ω) . (3.14)Since k g ′′ k L (Ω) ≤ i,n (cid:13)(cid:13)(cid:13) u ′′ i,n (cid:13)(cid:13)(cid:13) L (Ω) , we observe from (3.5) that (cid:13)(cid:13) g ′ (cid:13)(cid:13) L ∞ (Ω) ≤ √ p E ( u )) k g k L (Ω) . Then, by (3.4), we obtain (cid:13)(cid:13)(cid:13)(cid:13) ∂ u n ∂x ( · , t ) − ∂ u n ∂x ( · , t ) (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (Ω) ≤ p E ( u )( t − t ) . (3.15)Moreover, by the Mean Value Theorem, there exists ¯ x ∈ Ω such that g (¯ x ) = 1 L Z L g ( x ) dx, and then | g ( x ) | ≤ | g ( x ) − g (¯ x ) | + | g (¯ x ) | ≤ L (cid:13)(cid:13) g ′ (cid:13)(cid:13) L ∞ (Ω) + 1 √ L k g k L (Ω) for each x ∈ [0 , L ]. Thus, by (3.4) and (3.15), we find k u n ( · , t ) − u n ( · , t ) k L ∞ (Ω) ≤ L p E ( u )( t − t ) + r E ( u ) L ( t − t ) (3.16) 24 L p E ( u ) T √ L ! ( t − t ) . Furthermore, for each α ∈ (0 , ), we have (cid:12)(cid:12) g ′ (cid:12)(cid:12) α := sup (cid:26) | g ′ ( x ) − g ′ ( y ) || x − y | α (cid:12)(cid:12)(cid:12) x, y ∈ Ω , x = y (cid:27) ≤ (cid:12)(cid:12) g ′ (cid:12)(cid:12) α (2 (cid:13)(cid:13) g ′ (cid:13)(cid:13) L ∞ (Ω) ) − α . (3.17)Using Morrey’s inequality, it is followed from (3.5) that (cid:12)(cid:12)(cid:12)(cid:12) ∂ u n ∂x ( · , t ) − ∂ u n ∂x ( · , t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ K M (cid:13)(cid:13)(cid:13)(cid:13) ∂ u n ∂x ( · , t ) − ∂ u n ∂x ( · , t ) (cid:13)(cid:13)(cid:13)(cid:13) H (Ω) ≤ K M C p E ( u ) , where K M denotes the constant of Morrey’s inequality. Then, from (3.15) and (3.17), we deducethat (cid:12)(cid:12)(cid:12)(cid:12) ∂ u n ∂x ( · , t ) − ∂ u n ∂x ( · , t ) (cid:12)(cid:12)(cid:12)(cid:12) α ≤ p E ( u )( K M C ) α T √ L ! − α ( t − t ) − α . (3.18)Therefore it follows from (3.15), (3.16), and (3.18), that for every α ∈ (0 , ), u n is uniformlyequicontinuous with respect to the C ,α (Ω)-norm topology and that k u n ( · , t ) − u n ( · , t ) k C ,α (Ω) ≤ C ( t − t ) − α (3.19)for some C ( L, E ( u ) , α, T ) >
0. We then obtain (3.11) by applying the Ascoli-Arzel`a’s Theorem(see e.g. [4, Proposition 3.3.1]). Finally, since k u n ( · , t ) − u n ( · , t ) k C ,α (Ω) → t → t , we obtain the conclusion by selecting t = 0.When N = 2, 3, we can also improve the result obtained in Theorem 3.1: Theorem 3.3.
Let N = 2 , . Let u be the function obtained by Theorem . Then it holdsthat u ∈ L (0 , T ; W , ∞ (Ω)) ∩ C ,β ([0 , T ]; C ,γ (Ω)) and u n → u weakly * in L (0 , T ; W , ∞ (Ω)) as n → + ∞ , (3.20) u n → u in C ,β ([0 , T ]; C ,γ (Ω)) as n → + ∞ (3.21) for every < β < (cid:18) − N (cid:19) (cid:18) − γ − N/ (cid:19) , < γ < − N . Furthermore u ( · , t ) → u in C ,γ (Ω) as t ↓ .Proof. Let N = 2, 3. Fix T > n ∈ N . To begin with, the convergence (3.20) followsfrom the same line as in the proof of (3.10). In the sequel, we shall prove (3.21). For each t , t ∈ [0 , T ] with t < t , set g ( x ) := u n ( x, t ) − u n ( x, t ) .
25y (3.4), we have already known k g k L (Ω) ≤ (2 E ( u )) ( t − t ) . (3.22)Since (2.8) asserts that k g k H (Ω) ≤ E ( u )) , combining this with (3.22) and the interpolation inequality k g k L ∞ (Ω) ≤ C k g k − N L (Ω) k g k N H (Ω) , (3.23)we obtain k g k L ∞ (Ω) ≤ C k g k − N L (Ω) ≤ C ( t − t ) − N , (3.24)where the constant C is independent of n . For each γ ∈ (0 , − N/ | g | γ := sup (cid:26) | g ( x ) − g ( y ) || x − y | γ (cid:12)(cid:12)(cid:12)(cid:12) x, y ∈ Ω , x = y (cid:27) ≤ | g | γ/ (2 − N/ − N/ (2 k g k L ∞ (Ω) ) − γ − N/ . Since it follows from Sobolev’s embedding theorem that k g k C , − N/ (Ω) ≤ C k g k H (Ω) ≤ CE ( u ) , we get | g | γ ≤ C ( t − t )( − N ) (cid:16) − γ − N/ (cid:17) (3.25)Therefore we deduce from (3.24) and (3.25) that u n is uniformly equicontinuous with respect tothe C ,γ -norm topology for each γ ∈ (0 , − N/ k u n ( · , t ) − u n ( · , t ) k C ,γ (Ω) ≤ C ( t − t )( − N ) (cid:16) − γ − N/ (cid:17) (3.26)for some constant C = C (Ω , E ( u ) , γ, T ) >
0. By the Ascoli-Arzel`a’s Theorem (see e.g. [4,Proposition 3.3.1]), we get (3.21). Finally, since k u n ( · , t ) − u n ( · , t ) k C ,γ (Ω) → t → t , we obtain the conclusion by selecting t = 0.Regarding the piecewise constant interpolation ˜ u n for { u i,n } defined in Definition 1.3, wecan verify the following: Lemma 3.1.
Let ˜ u n be the piecewise constant interpolation of { u i,n } . If N = 1 , then ˜ u n → u in L ∞ ([0 , T ]; C ,γ (Ω)) as n → + ∞ (3.27) for every γ ∈ (0 , / , where u is the function obtained in Theorem . If N = 2 , , then ˜ u n → u in L ∞ ([0 , T ]; C ,γ (Ω)) as n → + ∞ (3.28) for every γ ∈ (0 , − N/ . Furthermore, for any N ≥ , it holds that ∆˜ u n ⇀ ∆ u in L (0 , T ; L (Ω)) as n → + ∞ . (3.29) 26 roof. By (2.8) we see that ˜ u n ∈ L ∞ ([0 , T ]; H (Ω)). Since N ≤
3, Proposition 2.1 implies that˜ u n ∈ ( L ∞ ([0 , T ]; C ,γ (Ω)) for 0 < γ < if N = 1 ,L ∞ ([0 , T ]; C ,γ (Ω)) for 0 < γ < − N if N = 2 , . Then, along the same line as in the proof of Theorem 3.1, we verify that ˜ u n ( t ) converges to afunction ˜ u ( t ), with ˜ u ( x, t ) ≥ f ( x ) in Ω, for each t ∈ [0 , T ] in C ,γ (Ω) if N = 1 and C ,γ (Ω) if N = 2, 3.We shall show that ˜ u coincides with u which is obtained as the limit of u n . Let us fix t ∈ [0 , T ] arbitrarily. Then there exists a sequence of intervals { [( i n − τ n , i n τ n ) } n ∈ N such that t ∈ [( i n − τ n , i n τ n ) for each n ∈ N . Recalling Definitions 1.2-1.3, if N = 1, we observe from(3.19) that k ˜ u n ( t ) − u n ( t ) k C ,γ (Ω) = k u i,n − u n ( t ) k C ,γ (Ω) = k u n ( i n τ n ) − u n ( t ) k C ,γ (Ω) ≤ C ( i n τ n − t ) − γ ≤ Cτ − γ n → n → + ∞ , and if N = 2, 3, we deduce from (3.26) that k ˜ u n ( t ) − u n ( t ) k C ,γ (Ω) = k u n ( i n τ n ) − u n ( t ) k C ,γ (Ω) ≤ Cτ ( − N ) (cid:16) − γ − N/ (cid:17) n → n → + ∞ . Hence we obtain (3.27) and (3.28).Finally we prove (3.29). It follows from Definitions 1.2 and 1.3 that u n ( x, t ) − ˜ u n ( x, t ) = 1 τ n ( t − iτ n )( u i,n ( x ) − u i − ,n ( x )) , so that, 12 sup t ∈ [0 ,T ] Z Ω | u n ( x, t ) − ˜ u n ( x, t ) | dx (3.30) ≤ n X i =1 sup t ∈ [( i − τ n ,iτ n ] ( t − iτ n ) τ n Z Ω τ n ( u i,n ( x ) − u i − ,n ( x )) dx ≤ τ n n X i =1 ( E ( u i − ,n ) − E ( u i,n ))= τ n ( E ( u ) − E ( u n,n )) ≤ τ n E ( u ) → n → + ∞ . Then we observe that for any ϕ ∈ C ∞ c (Ω) Z T Z Ω (∆ u n − ∆˜ u n ) ϕ dxdt = Z T Z Ω ( u n − ˜ u n )∆ ϕ dxdt → n → ∞ . Let us define µ n as µ n ( t ) = µ i,n if t ∈ [( i − τ n , iτ n ) . (3.31) 27 roof of Theorem . Let u be the function in Theorem 3.1. To begin with, we prove that u isa weak solution of (P). Since u i,n and V i,n satisfy Z Ω [ V i,n ( ϕ − u i,n ) + ∆ u i,n ∆( ϕ − u i,n )] dx ≥ ϕ ∈ K , we observe that Z T Z Ω [ V n ( w − ˜ u n ) + ∆˜ u n ∆( w − ˜ u n )] dxdt = n X i =1 Z iτ n ( i − τ n Z Ω [ V i,n ( w − u i,n ) + ∆ u i,n ∆( w − u i,n )] dxdt ≥ , i.e., Z T Z Ω [ V n w + ∆˜ u n ∆ w ] dxdt ≥ Z T Z Ω h V n ˜ u n + | ∆˜ u n | i dxdt for all w ∈ K . (3.32)It follows from (3.6) that Z T Z Ω V n w dxdt → Z T Z Ω u t w dxdt as n → + ∞ . (3.33)Moreover Lemma 3.1 gives us that Z T Z Ω ∆˜ u n ∆ w dxdt → Z T Z Ω ∆ u ∆ w dxdt as n → + ∞ , (3.34)and lim inf n → + ∞ Z T Z Ω | ∆˜ u n | dxdt ≥ Z T Z Ω | ∆ u | dxdt. (3.35)Combining (3.2) with (3.30), we have˜ u n → u as n → + ∞ in L (0 , T ; L (Ω)) . (3.36)Then (3.6) and (3.36) imply that Z T Z Ω V n ˜ u n dxdt → Z T Z Ω u t u dxdt as n → + ∞ , (3.37)e.g., see [27], Proposition 23.9. By virtue of (3.32)–(3.35) and (3.37), we assert that Z T Z Ω [ u t ( w − u ) + ∆ u ∆( w − u )] dxdt ≥ w ∈ K , (3.38)i.e., u is a weak solution of (P).For any ϕ ∈ C ∞ c (Ω × (0 , T )) with ϕ ≥
0, we verify that w := u + ϕ ∈ K . Hence it followsfrom (3.38) that Z T Z Ω [ u t ( x, t ) ϕ ( x, t ) + ∆ u ( x, t )∆ ϕ ( x, t )] dxdt ≥ . (3.39) 28ince ϕ is arbitrary, (3.39) implies that u t ( x, t ) + ∆ u ( x, t ) ≥ × (0 , T ) , (3.40)where ∆ u is written in the sense of distribution. Moreover, the regularity of u follows fromTheorems 3.1–3.3.We now prove (1.7). By (3.31) and Theorem 2.3, we observe that k µ n k L ([0 ,T ]; M (Ω)) := Z T (cid:18)Z Ω dµ n (cid:19) dt (3.41) = n X i =1 Z iτ n ( i − τ n (cid:18)Z Ω dµ i,n (cid:19) dt = τ n n X i =1 µ i,n (Ω) < C. This implies that µ n ⇀ µ weakly in L (0 , T ; M (Ω))up to a subsequence. Setting µ := u t + ∆ u, we observe from (3.40) that µ is a measure on Ω × (0 , T ), and there holds µ = µ by uniquenessof the limit. Since µ n converges to µ weakly in L (0 , T ; M (Ω)), it follows from (3.41) that k µ k L (0 ,T ; M (Ω)) ≤ lim inf n →∞ k µ n k L (0 ,T ; M (Ω)) ≤ C. This is equivalent to (1.7), and implies that µ is a positive Radon measure on Ω for a.e. t ∈ (0 , T ).Finally, when N ≤
3, we prove that u satisfies the problem (P) in the sense of distribution.To prove this assertion, it is sufficient to show that, if u > f , then u t + ∆ u = 0 holds. Let usset N := { ( x, t ) ∈ Ω × (0 , T ) : u ( x, t ) > f ( x ) } . Since u is continuous in Ω × (0 , T ) by Theorems 3.2 and 3.3, N is an open set, so that, for any( x , t ) ∈ N , there exist δ > W × ( t , t ) of ( x , t ) such that u ( x, t ) − f ( x ) > δ in W × ( t , t ) . (3.42)Lemma 3.1 implies that there exists a number N > u n ( x, t ) > u ( x, t ) − δ W × ( t , t ) for any n > N. Combining this with (3.42), we have, for any n > N ,˜ u n ( x, t ) > f ( x ) + δ W × ( t , t ) . (3.43)Let ζ ∈ C ∞ ( W × ( t , t )) with 0 ≤ ζ ≤ δ/
2. Then (3.43) asserts that ψ ( x, t ) := ˜ u n ( x, t ) − ζ ( x, t ) ∈ K for each t ∈ [0 , T ] . ψ as ϕ in (2.81) and integrating it with respect to t on (0 , T ), we obtain Z T Z Ω ∆ u i,n ( x ) ζ ( x, t ) dxdt ≤ − Z T Z Ω V i,n ( x ) ζ ( x, t ) dxdt. (3.44)From the definition (3.31), the inequality can be reduced to n X i =1 Z iτ n ( i − τ n Z Ω ζ ( x, t ) dµ n dt ≤ . (3.45)Since µ n ≥
0, we see that the integral in (3.45) must be equal to 0, i.e., µ n ( W × ( t , t )) = 0 . (3.46)It follows from (3.41) that k µ n k M (Ω × (0 ,T )) := Z T Z Ω dµ n dt < C. Thus we deduce that µ n converges to µ t weakly in M (Ω × (0 , T )), i.e., Z T Z Ω ϕ ( x, t ) dµ n dt → Z T Z Ω ϕ ( x, t ) dµdt for any ϕ ∈ C ∞ (Ω × (0 , T )). This fact also yields that k µ k M (Ω × (0 ,T )) ≤ lim inf n → + ∞ k µ n k M (Ω × (0 ,T )) . (3.47)Combining (3.46) with (3.47), we conclude that µ ( W × ( t , t )) = 0 , (3.48)which completes the proof. References [1] R.A. Adams and J.J.F. Fournier,
Sobolev spaces , Pure and Applied Mathematics ,Academic Press, Amsterdam, 2003.[2] S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of ellipticpartial differential equations satisfying general boundary conditions. I , Comm. Pure Appl.Math. (1959) 623–727.[3] L. Ambrosio, Minimizing movements , Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) (1995), 191–246.[4] L. Ambrosio, N. Gigli, and G. Savar`e, Gradient Flow , Birkh¨auser, 2008.[5] G. Barbatis,
Explicit estimates on the fundamental solution of higher-order parabolic equa-tions with measurable coefficients , J. Differential Equations (2001), 442–463.[6] H. Brezis,
Op´erateurs Maximaux Monotones et Semi-groupes de Contractions dans les Es-paces de Hilbert , North-Holland, Amsterdam/London, 1973.307] H. Brezis and G. Stampacchia,
Remarks on some fourth order variational inequalities , Ann.Sc. Norm. Super. Pisa Cl. Sci. (1977), 363–371.[8] L. A. Caffarelli, The obstacle problem revisited , Jour. Fourier Anal. Appl. (1998), no. 4-5,383–402.[9] L. A. Caffarelli and A. Friedman, The obstacle problem for the biharmonic operator , Ann.Sc. Norm. Super. Pisa Cl. Sci. (4) (1979), 151–184.[10] L. A. Caffarelli, A. Friedman and A. Torelli, The two-obstacle problem for the biharmonicoperator , Pacific J. Math. (1982), no. 2, 325–335.[11] G. Caristi and E. Mitidieri,
Existence and nonexistence of global solutions of higher-orderparabolic problems with slow decay initial data , J. Math. Anal. Appl. (2003), 710–722.[12] L. A. Caffarelli, A. Petrosyan and H. Shahgholian,
Regularity of a free boundary in parabolicpotential theory , J. Amer. Math. Soc. (2004), no. 4, 827–869.[13] J. W. Cholewa and A. Rodriguez-Bernal, Linear and semilinear higher order parabolicequations in R N , Nonlinear Anal. (2012), 194–210.[14] C. Elliott and B. Stinner, Modeling and computation of two phase geometric biomembranesusing surface finite elements , J. Comput. Phys. (2010), no. 18, 6585–6612.[15] J. Frehse,
On the regularity of the solution of the biharmonic variational inequality ,Manuscripta Math. (1973), 91–103.[16] F. Gazzola, On the moments of solutions to linear parabolic equations involving the bihar-monic operator , Discrete Contin. Dyn. Syst. (2013), no. 8, 3583–3597.[17] F. Gazzola and H.-C. Grunau, Eventual local positivity for a biharmonic heat equation in R n , Discrete Contin. Dyn. Syst. Ser. S (2008), no. 1, 83–87.[18] F. Gazzola, H.-C. Grunau, G. Sweers, Polyharmonic boundary value problems. Positivitypreserving and nonlinear higher order elliptic equations in bounded domains , Lecture Notesin Mathematics , Springer-Verlag, Berlin, 2010.[19] V. A. Galaktionov and P. J. Harwin,
Non-uniqueness and global similarity solutions for ahigher-order semilinear parabolic equation , Nonlinearity (2005), 717–746.[20] V. A. Galaktionov and S. I. Pohozaev, Existence and blow-up for higher-order semilin-ear parabolic equations: majorizing order-preserving operators , Indiana Univ. Math. J. (2002), no. 6, 1321–1338.[21] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order ,Springer, 1998.[22] N. S. Landkof,
Foundations of Modern Potential Theory , Springer-Verlag, New York, 1972.[23] P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The limitcase. I. , Rev. Mat. Iber. (1985), no. 1, 145–201.[24] B. Schild, On the coincident set in biharmonic variational inequalities with thin obstacle ,Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) (1986), no.4, 559–616.3125] L. Schwartz, Th´eorie des Distributions , Herman, Paris, 1957.[26] C. Baiocchi, F. Gastaldi and F. Tomarelli,