Boundary value problems for parabolic operators in a time-varying domain
aa r X i v : . [ m a t h . A P ] D ec BOUNDARY VALUE PROBLEMS FOR PARABOLICOPERATORS IN A TIME-VARYING DOMAIN
SUNGWON CHO, HONGJIE DONG, AND DOYOON KIM
Abstract.
We prove the existence of unique solutions to the Dirichlet bound-ary value problems for linear second-order uniformly parabolic operators ineither divergence or non-divergence form with boundary blowup low-order co-efficients. The domain is possibly time varying, non-smooth, and satisfies theexterior measure condition. Introduction
In this paper we consider parabolic operators in divergence form Lu = D t ( u ) − D j ( a ij D i u ) + b i D i u − D i ( c i u ) + c u (D)and in non-divergence form Lu = D t u − a ij D ij u + b i D i u + c u (ND)in a time-varying domain Q in R n +1 , n ≥
1, with boundary blowup low-ordercoefficients. Here and in the sequel, D i := ∂∂x i , D ij := D i D j , i, j = 1 , . . . , n, D t := ∂∂t , some derivatives in parentheses in divergence form are understood in the weak sense,and summation over repeated indices is assumed. For convenience of notation, inthe sequel we set c i = 0 , i = 1 , . . . , n, in the non-divergence case.With the operator L in (D) or (ND), we study the following boundary valueproblems of Dirichlet type: (cid:26) Lu = f in Q,u = g on ∂ p Q, (DP)where f = f − D i f i for the divergence case and ∂ p Q is the parabolic boundaryof Q (see Definition 2.1 below). We prove that there exist unique solutions to theDirichlet problems (DP) when the domain satisfies the exterior measure conditionand the boundary data is zero ( g ≡ W , p , p > ( n + 2) /
2. While
Mathematics Subject Classification.
Key words and phrases.
Parabolic Dirichlet boundary value problems, Time-varying domain,Exterior measure condition, Vanishing mean oscillation, Blowup low-order coefficients.S. Cho was supported by Basic Science Research Program through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2012-0003253).H. Dong was partially supported by the NSF under agreement DMS-1056737.D. Kim was supported by Basic Science Research Program through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2011-0013960). in the divergence case, they are understood in the weak sense. In both cases,solutions are continuous up to the boundary. The coefficients which we considerhave two features. First, concerning the leading coefficients, while in the divergencecase we do not impose any regularity assumptions, in the non-divergence case weassume that they have vanishing mean oscillations (VMO) with respect to thespacial variables and merely measurable with respect to the time variable. Second,the lower-order coefficients may blow up near the boundary with a certain optimalgrowth condition.Indeed, there is an extensive literature on the existence of solutions to theboundary value problem (DP) in a straight cylindrical domain with lower-ordercoefficients which are bounded or in certain Sobolev spaces. See, for instance,[28, 31, 22, 1, 32, 27] and the references therein.Regarding non-divergence form parabolic equations in time-varying domains (ormore general degenerate elliptic-parabolic equations) one may find related results inFichera [14, 15], Oleinik [37, 38], Kohn–Nirenberg [21], and Krylov [25], where, un-der certain assumptions, the existence, uniqueness, and regularity of solutions werediscussed. The solutions to parabolic equations in non-divergence form consideredhere are called L p -strong solutions in [8], where the authors treated various types ofsolutions to nonlinear equations. We note that in [8] Crandall, Kocan, and ´Swi‘echconsidered equations in a cylindrical domain satisfying a uniform exterior cone con-dition and the L p -strong solutions are locally in W , p , p > p , where p > ( n + 2) / n + 1. We also mention that in [33] Lieberman treated a similarproblem for a non-divergence elliptic operator in a cylindrical domain with blowuplower-order coefficients in weighed H¨older spaces.As noted above, in the non-divergence case we assume that the leading coeffi-cients are in a class of VMO functions. The study of elliptic and parabolic equationswith VMO coefficients was initiated by Chiarenza, Frasca, and Longo [4], and con-tinued in [5] and [1]. The class of leading coefficients in this paper was introducedby Krylov [26] in the context of parabolic equations in the whole space. See also[19, 20, 9, 10] for further development, the results of which we shall use in the proof.For divergence form equations in time-varying domains, Yong [43] proved theunique existence of weak solutions by using a penalization method when the do-main satisfies the exterior measure condition (Definition 1.2) and its cross sectionat time t is simply connected. He considered equations with a non-zero initial con-dition, and coefficients and the data f , f i are in some suitable Lebesgue spaces,so that the weak solutions are actually H¨older continuous up to the boundary.Later in [2] Brown et al. obtained a similar solvability result in a parabolic Lips-chitz time-varying domain with bounded measurable coefficients and more generalsquare integrable data. One may refer to Lions [35] for existence results in anabstract framework. Recently, Byun and Wang [3] obtained certain L p -estimatesfor equations in time-varying δ -Reifenberg domains. We also refer the reader to[34, 30, 17, 39, 36] and the references therein for other results about boundaryvalue problems in time-varying domains. Of course, the boundary value problemin curvilinear cylinders can be deduced from the estimates in “straight cylinders”by using a change of variables as long as the domains are sufficiently regular.For the Laplace operator, we recall that a necessary and sufficient condition forthe solvability of the corresponding boundary value problem to (DP) is the cele-brated Wiener’s criterion. See, for example, [42, 18, 29]. As to the heat equation, ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 3 an analogous result was established by Evans and Gariepy [12]. We are going touse this result in our proofs below.To formulate our main results, we introduce some notation, function spaces, andassumptions. A typical point in R n +1 is denoted by X = ( x, t ), where x ∈ R n and t ∈ R . The parabolic distance between points X = ( x, t ) and Y = ( y, s ) in R n +1 is | X − Y | := max {| x − y | , | t − s | } . For any Y = ( y, s ) ∈ R n +1 and r >
0, we set B r ( y ) := { x ∈ R n : | x − y | < r } and C r ( Y ) := B r ( y ) × ( s − r , s ) = { X = ( x, t ) ∈ R n +1 : | X − Y | < r, t < s } to be a ball in R n and a standard parabolic cylinder in R n +1 , respectively. We alsoset ˆ C r ( Y ) = B r ( y ) × ( s − r , s + r ). Let | Γ | := | Γ | n +1 be the n + 1-dimensionalLebesgue measure of a set Γ in R n +1 . For any real number c , denote c + := max ( c, c − := max ( − c, a ij are defined on R n +1 and satisfy the followinguniform ellipticity condition: there exists a constant ν ∈ (0 ,
1] such that ν | ξ | ≤ n X i,j =1 a ij ( X ) ξ i ξ j , sup i,j | a ij ( X ) | ≤ ν − (UE)for all X ∈ R n +1 and ξ = ( ξ , ...ξ n ) ∈ R n . In the non-divergence case, we imposethe following vanishing mean oscillation (VMO) condition on a ij with respect to x . We denote ω a ( R ) := sup r ∈ (0 ,R ] sup ( x ,t ) ∈ R d +1 sup i,j – Z C r ( t ,x ) | a ij ( x, t ) − – Z B r ( x ) a ij ( y, t ) dy | dX, where – Z C f ( Y ) dY is the average of f over C . Assumption . We have ω a ( R ) → R → + . Note that, under this assumption, no regularity is reguired for a ij as functions of t . For instance, ω a ( R ) = 0 if a ij = a ij ( t ). In the non-divergence case, without lossof generality, we may assume a ij = a ji . However, we do not impose such conditionin the divergence case.For lower-order coefficients, we assume the following: Z Q ( c ( X ) φ ( X ) + c i ( X ) D i φ ( X )) dX ≥ , ∀ φ ≥ , φ ∈ C ∞ ( Q ) (1)in the divergence case, and c ( X ) ≥ Q in the non-divergence case. Note that the above two conditions can be collectivelyreferred to the following unified condition: L ≥ , (2)which implies the maximum principle for L . The lower-order coefficients b i and c i are allowed to blow up near the boundary under a certain growth condition, stated S. CHO, H.DONG, AND D. KIM in the theorem below. In light of the example before the proof of Theorem 1.3, thisgrowth condition is optimal.We impose the following exterior measure condition (or condition (A)) on thedomain.
Definition 1.2.
An open set Q ⊂ R n +1 satisfies the condition (A) if there existsa constant θ ∈ (0 ,
1) such that for any X = ( x, t ) ∈ ∂ p Q and r >
0, we have | C r ( X ) \ Q | > θ | C r | .We deal simultaneously with both cases of divergence (D) and non-divergence(ND) form. In fact, the first author and Safonov took a unified approach andobtained global a priori H¨older estimates in [7, Corollary 3.6, Theorem 3.10] forelliptic equations without lower-order terms and with locally bounded right-handside, and in [6, Theorem 3.4, Theorem 4.2] for the parabolic case. For this approach,it is convenient to introduce the solution space W ( Q ), which varies according to(D) and (ND). Let p ∈ ( n +22 , ∞ ) be a fixed constant. We use the notation: W ( Q ) := W ND ( Q ) := W , p , loc ( Q ) ∩ C ( ¯ Q ) in the case (ND), W ( Q ) := W D ( Q ) := H ( Q ) ∩ C ( ¯ Q ) in the case (D),where ¯ Q := Q ∪ ∂Q and H ( Q ) = { u ∈ L ( Q ) | D i u ∈ L , loc ( Q ) ,D t ( u ) = g + D i g i for some g ∈ L , loc ( Q ) , g i ∈ L , loc ( Q ) } . Here f ∈ L p, loc ( Q ), p > f ∈ L p ( Q ′ ) for any open set Q ′ ⋐ p Q .Throughout the paper we use Q ′ ⋐ p Q to indicate that Q ′ ⊂ Q and dist( Q ′ , ∂ p Q ) >
0. In both cases, the functions u ∈ W are continuous on ¯ Q . In addition, in thenon-divergence case (ND), the functions u have strong derivatives D i u , D ij u , D t u in the Lebesgue space L p , loc ( Q ). In this case, the relations Lu = f or Lu ≤ f areunderstood in the almost everywhere sense in Q . In the divergence case (D), thefunctions u ∈ W have weak (generalized) derivatives D i u and D t u , and Lu = ( ≤ , ≥ ) f − D i f i for f , f i ∈ L , loc ( Q ) is understood in the following weak sense: Z Q ( − uD t φ + a ij D i uD j φ + b i D i uφ + c i uD i φ + c uφ − f φ − f i D i φ ) dX = ( ≤ , ≥ )0for any nonnegative function φ ∈ C ∞ ( Q ).Regarding the data, we consider more general function spaces than those in [7]and [6]. We first define, for β ∈ (0 , p > k f k F β,p ( Q ) := sup Y ∈ ∂ p Q,r> – Z ˆ C r ( Y ) ∩ Q | d − β ( X ) f ( X ) | p dX ! /p , ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 5 where d ( X ) := dist( X, ∂ p Q ), and we say f ∈ F β,p ( Q ) when k f k F β,p ( Q ) < ∞ . Notethat if f ∈ F β,p ( Q ), then f ∈ L p, loc ( Q ). For some β ∈ (0 , F ( Q ) = F β ( Q ):= { f = ( f , f , . . . , f n ) : f ∈ F β,p ( Q ) , f i ∈ F β, p ( Q ) , i = 1 , . . . , n } , k f k F ( Q ) := k f k F β,p ( Q ) + n X i =1 k f i k F β, p ( Q ) for the divergence case, and F ( Q ) = F β ( Q ) := F β,p ( Q ) for the non-divergencecase.Now we are ready to state the main results of this paper: Theorem 1.3.
Let p ∈ ( n +22 , ∞ ) , L be a uniformly parabolic operator in eitherdivergence (D) or non-divergence form (ND) satisfying (2) , and Q be a boundeddomain in R n +1 satisfying the measure condition (A) with a constant θ ∈ (0 , .Suppose that c ∈ L ∞ , loc ( Q ) and | b i | , | c i | = o ( d − ) , d = d ( X ) , i.e., there exists a nondecreasing function γ on ¯ R + such that γ ( d ) → as d → and | b i | , | c i | ≤ d − γ ( d ) . For the non-divergence case, we further assume that the coefficients a ij satisfyAssumption 1.1. Let β := β ( n, ν, θ ) be the constant from Proposition 4.1 below.Then, for any f ∈ F β ( Q ) , β ∈ (0 , β ) , there exists a unique solution u ∈ W ( Q ) to the Dirichlet problem (DP) when g ≡ . It is worth noting that from the proofs below the solution u is globally H¨oldercontinuous in ¯ Q and, in the divergence case, D t ( u ) = g + D i g i for some g , g i ∈ L , loc ( Q ). The corresponding results for elliptic operators are also obtained byfollowing the proofs in this paper.Here we illustrate the idea in the proof of Theorem 1.3. Our proof relies onthe growth lemma, from which we deduce an a priori uniform boundary estimatein Proposition 4.1. To prove the existence result, in the non-divergence case, firstwe approximate the operator L by a sequence of operators L k which become theheat operator near the boundary and coincide with the original operator L in theinterior of the domain. We then find a sequence of solutions u k correspondingto the operators L k by Perron’s method, which requires barrier functions and thesolvability in cylindrical domains. We construct certain barrier functions by usingthe result of Evans and Gariepy [12] mentioned above and an idea in Krylov [24].Under Assumption 1.1, the W , p solvability of non-divergence form parabolic equa-tions in cylindrical domains is also available in the literature. By using the a prioriboundary estimate and the interior W , p estimate, we are able to show that alonga subsequence u k converge locally uniformly to a solution u ∈ W ( Q ) of the originalequation. The divergence case is a bit more involved. We additionally take mollifi-cations of the coefficients and data, and rewrite the approximating equations intonon-divergence form equations, for which the solvability has already been proved.We then show the convergence of a subsequence of u k to a solution u ∈ W ( Q ) ofthe original equation by again using the a priori boundary estimate and the interiorDe Giorgi–Nash–Moser estimate. S. CHO, H.DONG, AND D. KIM
The remaining part of the paper is organized as follows. We present severalauxiliary results in the next section including a version of the maximum principlefor solutions in W ( Q ). Section 3 is devoted to a growth lemma (Lemma 3.1) anda pointwise estimate (Lemma 3.3). In Section 4, we obtain an a priori boundaryestimate which is crucial in our argument, and in Section 5 we complete the proof ofthe main results. In the Appendix, we show that any domains satisfying the exteriormeasure condition also satisfy Wiener’s criterion, which is used in the constructionof the barrier functions. 2. Auxiliary results
This section is devoted to some auxiliary results. First we recall the followingstandard definition.
Definition 2.1.
Let Q be an open set in R n +1 . The parabolic boundary ∂ p Q of Q is the set of all points X = ( x , t ) ∈ ∂Q such that there exists a continuousfunction x = x ( t ) on an interval [ t , t + δ ) with values in R n satisfying x ( t ) = x and ( x ( t ) , t ) ∈ Q for all t ∈ ( t , t + δ ). Here x = x ( t ) and δ > X .We denote ¯ ∂ p Q to be the closure of ∂ p Q in ∂Q . By the continuity, it is easilyseen that the condition (A) is satisfied for any X ∈ ¯ ∂ p Q . The next lemma followsfrom Lemma 2.3 of [6], which reads that ∂Q \ ¯ ∂ p Q is locally flat. Lemma 2.2.
Let Q be an open set in R n +1 and X = ( x , t ) ∈ ∂Q \ ¯ ∂ p Q . Thenthere exists r > such that B r ( x ) × { t = t } ⊂ ∂Q \ ¯ ∂ p Q , and B r ( x ) × ( t , t + r ) ⊂ R n +1 \ ¯ Q, B r ( x ) × ( t − r , t ) ⊂ Q. For any interior point X ∈ Q , we have the following measure condition forsufficiently large r . Lemma 2.3.
Assume that Q satisfies the measure condition (A) with a constant θ ∈ (0 , . Let X = ( x, t ) ∈ Q and denote ρ = d ( X ) . Then for any r ≥ ρ/θ , wehave | C r ( X ) \ Q | > θ − n − | C r | . (3) Proof.
By a scaling argument, without loss of generality we may assume that ρ = 1.Let Y = ( y, s ) be a point on ¯ ∂ p Q such that d ( X, Y ) = 1. In the case when s ≤ t ,we have C r/ ( Y ) ⊂ C r ( X ). By using the condition (A) at Y , | C r ( X ) \ Q | ≥ | C r/ ( Y ) \ Q | > θ | C r/ | = θ − n − | C r | , (4)which gives (3).Next we consider the case when s > t . We claim that C r/ ( X ) \ Q is not empty.Otherwise, we would have C r/ ( X ) ⊂ Q . Since ρ = 1 and r ≥ /θ , | C r/ ( Y ) \ Q | ≤ | C r/ ( Y ) \ C r/ ( X ) | ≤ θ | C r/ | , which contradicts with the condition (A) at Y . Now we fix a point Y ∈ C r/ ( X ) \ Q .Denote { Y τ | Y τ = (1 − τ ) X + τ Y , τ ∈ [0 , } be the line segment connecting X and Y . Let τ ∗ be the smallest number in (0 ,
1) such that Y τ ∗ ∈ R d +1 \ Q . Clearly, wehave Y τ ∗ ∈ ∂ p Q and C r/ ( Y τ ∗ ) ⊂ C r ( X ). By using the condition (A) at Y τ ∗ , weobtain (4) with Y τ ∗ in place of Y . The lemma is proved. (cid:3) In the remaining part of this section, we do not impose the condition (A) on Q .The following lemma is useful in approximating u + by smooth functions. ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 7
Lemma 2.4.
Let G ∈ C ∞ ( R ) and u ∈ W ( Q ) . Then v := G ( u ) ∈ W ( Q ) . Inaddition, assume G ′ , G ′′ ≥ on R and G (0) = 0 . Then, for a function f definedin Q such that f ∈ L p , loc ( Q ) in the case (ND) , or f = f − D i f i , f ∈ L , loc ( Q ) , f i ∈ L , loc ( Q ) in the case (D) , satisfying Lu ≤ f , we have Lv ≤ F in Q where F := F − D i F i ,F = G ′ ( u ) f − G ′′ ( u ) a ij D i uD j u + G ′′ ( u ) f i D i u ∈ L , loc ( Q ) ,F i = G ′ ( u ) f i ∈ L , loc ( Q ) , for (D) and F := G ′ ( u ) f − G ′′ ( u ) a ij D i uD j u for (ND) . In particular, we have Lv ≤ in Q provided that Lu ≤ in Q .Proof. Clearly, in both cases v ∈ C ( ¯ Q ). First we consider the non-divergence case.We have D i v = G ′ ( u ) D i u, v t = G ′ ( u ) u t , D ij v = G ′ ( u ) D ij u + G ′′ ( u ) D i uD j u. Since G ′ ( u ) , G ′′ ( u ) ∈ C ( ¯ Q ), we get D i v, v t ∈ L p , loc ( Q ). By the parabolic Sobolevembedding theorem (see, for instance, [28, Chap. II]), D i u ∈ L p, loc ( Q ) with p = p ( n + 2) / ( n + 2 − p ) > p , which implies that D ij v ∈ L p , loc ( Q ). Therefore, v ∈ W ( Q ). Since G ′ ≥ c ≥
0, we get Lv = LG ( u ) = G ′ ( u ) Lu − G ′′ ( u ) a ij D i uD j u + c ( G ( u ) − G ′ ( u ) u ) ≤ F in Q, where we have used the simple inequality G ( u ) ≤ G ′ ( u ) u (5)because G (0) = 0 and G is convex.In the divergence case, by the definition of the space H ( Q ), we have D i v = G ′ ( u ) D i u ∈ L , loc ( Q ) and D t ( v ) = G ′ ( u ) u t = G ′ ( u )( g + D i g i )provided that D t ( u ) = g + D i g i for some g ∈ L , loc ( Q ) and g i ∈ L , loc ( Q ). Then D t ( v ) = ˜ g + D i ˜ g i , where ˜ g = G ′ ( u ) g − G ′′ ( u ) g i D i u, ˜ g i = G ′ ( u ) g i . It is easily seen that ˜ g ∈ L , loc ( Q ) and ˜ g i ∈ L , loc ( Q ). Therefore, v ∈ W ( Q ). Toshow the desired inequality, it suffices to prove that for any positive φ ∈ C ∞ ( Q ), Z Q − G ( u ) φ t + G ′ ( u ) (cid:0) a ij D i uD j φ + b i D i uφ (cid:1) + c i G ( u ) D i φ + c G ( u ) φ − F φ − F i D i φ dX ≤ . S. CHO, H.DONG, AND D. KIM
From (1) and (5), we have Z Q c i G ( u ) D i φ + c G ( u ) φ dX = Z Q c i D i ( G ( u ) φ ) − c i G ′ ( u ) φD i u + c G ( u ) φ dX ≤ Z Q c i D i ( G ′ ( u ) uφ ) − c i G ′ ( u ) φD i u + c G ′ ( u ) uφ dX = Z Q c i uD i ( G ′ ( u ) φ ) + c G ′ ( u ) uφ dX. Therefore, we only need to show that Z Q − G ( u ) φ t + a ij D i uD j ψ + b i ψD i u + c i uD i ψ + c uψ − f ψ − f i D i ψ dX ≤ , (6)where ψ := G ′ ( u ) φ . We use a standard mollification argument. Define ψ ε = G ′ ( u ( ε ) ) φ , where u ( ε ) is the standard mollification of u . By the definition of a weaksolution, Z Q − uD t ψ ε + a ij D i uD j ψ ε + b i ψ ε D i u + c i uD i ψ ε + c uψ ε − f ψ ε − f i D i ψ ε dX ≤ . (7)Let Q ′ = supp ψ ⋐ p Q . Since u ( ε ) → u in C ( ¯ Q ′ ) and Du ( ε ) → Du in L ( Q ′ ), wehave ψ ε → ψ in C ( ¯ Q ′ ) , Dψ ε → Dψ in L ( Q ′ ) . From this together with u ∈ W ( Q ), we see that the left-hand side of (7) convergesto that of (6) as ε →
0. In particular, − Z Q uD t ψ ε dX = Z Q g ψ ε dX − Z Q g i D i ψ ε dX = Z Q (cid:16) G ′ ( u ( ε ) ) g − G ′′ ( u ( ε ) ) g i D i u ( ε ) (cid:17) φ dX − Z Q G ′ ( u ( ε ) ) g i D i φ dX → Z Q (˜ g φ − ˜ g i D i φ ) dX = − Z Q G ( u ) φ t dX. This completes the proof of (6). The second assertion follows from the first oneby taking f = 0 and using G ′′ ≥ (cid:3) The following lemma allows us to reduce our consideration to functions definedon a standard cylinder C r ( X ) rather than on a general open set Q ⊂ R n +1 . Foroperators without lower-order terms, a similar result is claimed in Theorem 2.6 of[6], the proof of which, however, contains a flaw. Lemma 2.5.
Let Q be an open set in R n +1 and u ∈ W ( Q ) satisfy Lu ≤ in Q with an operator L in the form (ND) or (D) . Suppose u ≤ on ¯ C r ∩ ¯ ∂ p Q , where C r := C r ( X ) , X ∈ Q . Then for any ε > , there exists a function u ε ∈ W ( C r ) which vanishes in a neighborhood of ¯ C r ∩ ∂Q and satisfies u ε ≥ , Lu ε ≤ in C r , u ε ≡ in C r \ Q, and ( u − ε ) + ( X ) ≤ u ε ( X ) , u ε ≤ u + in Q. ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 9
Proof.
The idea of the proof is to modify u so that it vanishes near ¯ C r ∩ ∂Q . For ε >
0, we choose a convex non-decreasing nonnegative function G ε ∈ C ∞ ( R ) suchthat ( s − ε ) + ≤ G ε ( s ) ≤ ( s − ε/ + on R . We first modify u near ¯ C r ∩ ¯ ∂ p Q . ByLemma 2.4, we have v ε := G ε ( u ) ∈ W ( Q ) and it satisfies v ε ≥ , Lv ε ≤ , ( u − ε ) + ≤ v ε ≤ ( u − ε/ + in Q. Since u ≤ C r ∩ ¯ ∂ p Q , v ε vanishes in a neighborhood of ¯ C r ∩ ¯ ∂ p Q in ¯ Q . Now wemodify u near ¯ C r ∩ (cid:0) ∂Q \ ¯ ∂ p Q (cid:1) . Thanks to Lemma 2.2, ∂Q \ ¯ ∂ p Q = [ α ∈A S α × { t = t α } , where A is an index set. Here, for each α ∈ A , S α is an open set in R n , t α ∈ R ,and ∂S α × { t = t α } ⊂ ¯ ∂ p Q. In fact, A is at most countable (see Remark 2.6). However, we will not use this inthe proof below. Since u is uniformly continuous in ¯ Q and u ≤ C r ∩ ¯ ∂ p Q , wecan find δ > u ≤ ε/ { X ∈ ¯ C r ∩ ¯ Q | dist( X, ¯ C r ∩ ¯ ∂ p Q ) < δ } . Clearly,the set { X ∈ ¯ C r ∩ ∂Q | dist( X, ¯ C r ∩ ¯ ∂ p Q ) ≥ δ/ } ⊂ ∂Q \ ¯ ∂ p Q is compact, which has a finite covering by S α k × { t = t α k } , where α , . . . , α M ∈ A and M ∈ N . Denote S δα = (cid:8) X ∈ S α × { t = t α } ∩ ¯ C r | dist( X, ¯ C r ∩ ¯ ∂ p Q ) ≥ δ/ (cid:9) . By Lemma 2.2, there is a small constant δ ∈ (0 , δ/
2) such that S δα k × [ t α k − δ , t α k ) ⊂ ¯ C r ∩ Q, k = 1 , . . . , M, and these sets do not intersect each other. We choose a smooth function η = η ( x, t )in ¯ C r ∩ Q satisfying the following three properties:(i) 0 ≤ η ≤ C r ∩ Q ;(ii) For each k = 1 , . . . , M , in S δα k × ( t α k − δ , t α k ) the function η is independent of x , non-increasing in t , and η = 0 in S δα k × ( t α k − δ / , t α k );(iii) η = 1 in ¯ C r ∩ Q \ S Mk =1 (cid:0) S α k × ( t α k − δ , t α k ) (cid:1) , which contains X .Observe that D t η ≤ Dη = 0 in { X ∈ ¯ C r ∩ Q | dist( X, ¯ C r ∩ ¯ ∂ p Q ) ≥ δ } ⊃ { X ∈ ¯ C r ∩ Q | v ε ( X ) > } . Consequently, u ε := v ε η satisfies Lu ε = ηLv ε + v ε D t η ≤ C r ∩ Q . Noting that u ε vanishes in a neighborhood of ¯ C r ∩ ∂Q in ¯ Q , we can extend u ε to be zero in C r \ Q . It is now straightforward to check that u ε satisfies all the properties in thelemma. (cid:3) Remark . One example of space-time domains with infinitely many flat portionsof the non-parabolic boundary can be obtained by connecting a sequence of shrink-ing cubes by triangular prisms as in Figure 1 infinitely many times. Note that ∂Q \ ¯ ∂ p Q is a countable union of the top surfaces of these cubes and the domainsatisfies the exterior measure condition.Finally, we prove a version of the maximum principle for solutions in W ( Q ). Figure 1.
Lemma 2.7 (Maximum principle) . Let u ∈ W ( Q ) and Lu ≤ in Q , where L is a uniformly parabolic operator in either divergence (D) or non-divergence form (ND) satisfying (2) with locally bounded lower order coefficients (i.e., bounded onsets Q ′ ⋐ p Q ). For the non-divergence case, we further assume that the coefficients a ij satisfy Assumption 1.1. Then sup Q u ≤ sup ∂ p Q u ∨ . (8) Similarly, if Lu ≥ , then inf Q u ≥ inf ∂ p Q u ∧ . Proof.
In the divergence case, this is classical. See, for instance, [32, Sec. 6.7].Next we treat the non-divergence case. It suffices to prove (8). Due to (2), wemay assume sup ∂ p Q u ≤
0. We first consider the special case when Q = Ω × (0 , T )is a cylindrical domain, Ω is a bounded C , domain in R n , and the coefficients areall bounded. In this case, similar estimates can be found in [27, Theorem 11.8.1]for elliptic operators with continuous coefficients. The proof there is based on the W p estimate for elliptic operators. Since the W , p estimate for parabolic operatorsin cylindrical domains with VMO coefficients is available (cf. [20] and [10]), bythe same argument one can derive the maximum principle for L . It is standard toextend to the general case by a contradiction argument. Suppose that sup Q u > Q is bounded, we may assume that Q ⊂ { t > t } for some t ∈ R . Thenfor δ > v := u − δ ( t − t ) attains its maximum M > X ∈ ¯ Q \ ¯ ∂ p Q and satisfies Lv ≤ − δ in Q . Take a small r > C r ( X ) ⋐ p Q and a smooth function η such that η ( X ) = 1 and η = 0 on ∂ p C r ( X ).It is easily seen that for sufficiently small ε >
0, we have L ( v + εη ) ≤ C r ( X ) , v + εη ≤ M on ∂ p C r ( X ) . By the maximum principle in cylindrical domains proved above, we have M + ε = v ( X ) + εη ( X ) ≤ M ;a contradiction. The lemma is proved. (cid:3) Corollary 2.8 (Comparison principle) . Let u, v ∈ W ( Q ) and, under the sameassumptions on L as in Lemma 2.7, Lu ≥ Lv in Q , u ≥ v on ∂ p Q . Then u ≥ v in Q .Proof. We apply Lemma 2.7 to u − v . (cid:3) ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 11
By the comparison principle, it is immediate to see that a solution to (DP) isunique if it exists.3.
Growth lemma and pointwise estimates
The following growth lemma is in the spirit of the book by Landis [31, Section1.4]. In one form or another, the growth lemma was used in the proofs of Harnackinequalities for solutions to elliptic and parabolic equations. See, for instance, [40]and the references therein.
Lemma 3.1 (Growth Lemma) . Let r ∈ (0 , ∞ ) be a constant and Q ⊂ R n +1 be abounded domain. Suppose X = ( x , t ) ∈ ¯ Q \ ¯ ∂ p Q and | C r \ Q | > θ | C r | , < θ < , C r := C r ( X ) for some r ∈ (0 , r ] . Also, let L be the uniformly parabolic operator in (D) or (ND) satisfying (2) with | b i | , | c i | ≤ K .Then for any function u ∈ W ( Q r ) , Q r := C r ∩ Q , satisfying Lu ≤ in Q r and u ≤ on C r ∩ ∂ p Q, we have u ( X ) ≤ β sup C r u + (9) with constant β = β ( n, ν, θ, K , r ) ∈ (0 , . Assume that u is extended as u ≡ on C r \ Q , so that the right-hand side of (9) isalways well defined.Proof. First we make some reductions. By considering slightly smaller cylinders C r − ε ( x , t − ε ) and letting ε →
0, without loss of generality we may assume that u ≤ C r ∩ ¯ ∂ p Q and X ∈ Q . In particular, if X ∈ ∂Q \ ¯ ∂ p Q , then by Lemma 2.2( x , t − ε ) ∈ Q with a sufficiently small ε >
0. Next by a scaling and a translation,we further assume X = (0 ,
0) and r = 1. Note that the dependence of β on r comes from the scaling argument. Thanks to Lemma 2.5 with Q = Q , there exists u ε such that ( u − ε ) + ( X ) ≤ u ε ( X ) , u ε ≤ u + in Q u ε ∈ W ( C ) , u ε ≥ , Lu ε ≤ C , u ε ≡ C \ Q. We claim that u ε ( X ) ≤ β sup C ( u ε ) + . Once this is proved, using u ε ≤ u + in Q and u ε ≡ C \ Q , we obtain sup C ( u ε ) + ≤ sup C u + . Taking ε → + , we willeventually get (9). Now u ε will be denoted by u for the rest of proof. Moreover,since u ≥ C and L ≥
0, by considering L − c + D i c i = D t − D j ( a ij D i ) + ( b i − c i ) D i instead of L , we only need to treat the case c = c i = 0.We first treat the case when u and the coefficients are smooth. In this case,for operators without the lower-order terms, the lemma was proved in [13] under aslightly weaker condition. The general case follows by using an idea in Remark 6.2there. Indeed, we introduce a new variable x ∈ R and define v ( x , x, t ) = e x + At (cid:0) sup C u − u ( x, t ) (cid:1) , ˜ C r = ( − r, r ) × C r , ˜ Q = R × Q. Set a i = 0 , a i = − b i , i = 1 , . . . , n, a = A in the divergence case, and a i = a i = − b i / , i = 1 , . . . , n, a = A in the non-divergence case. Here we choose A = A ( n, ν, K ) > n +1) × ( n +1) matrix ( a ij ) ni,j =0 satisfies (UE) with a possibly smallerconstant ν . Then it is easy to check that v satisfies L v = v t − n X i,j =0 D j ( a ij D i v ) ≥ L v = v t − n X i,j =0 a ij D ij v ≥ v . Indeed, wecan find a small constant δ ∈ (0 , /
4) depending only on n and θ such that2 δ + (1 − δ ) ≤ , | ˜ C − δ (0 , − s ) \ ˜ Q | > θ | ˜ C | / , where s = 1 − (1 − δ ) . Moreover, since u = 0 in C \ Q , we have v ≥ e − − A sup C u in ˜ C − δ (0 , − s ) \ ˜ Q . Thus we get v ≥ βe − − A sup C u in ˜ C δ for some β = β ( n, ν, θ, K ) ∈ (0 , β = 1 − βe − − A . Wenote that the proof above only requires Lu ≤ C , u ≤ C \ Q. (10)Next we remove the smoothness assumption on u and the coefficients by usingan approximation argument. We take a small δ depending on n and θ such that | C − δ \ Q | ≥ θ | C − δ | /
2. For ε ∈ (0 , δ ), let u ( ε ) , a ( ε ) ij , and b ( ε ) i be the standardmollification of u , a ij , and b i .By the Lebesgue lemma,( a ( ε ) ij , b ( ε ) i ) → ( a ij , b i ) a.e. in C − δ . Non-divergence case:
Clearly, Lu ∈ L p , loc ( C ). Since a ( ε ) ij , b ( ε ) i , and u ( ε ) aresmooth, the Dirichlet problem L ε v ε := v εt − a ( ε ) ij D ij v ε + b ( ε ) i D i v ε = ( Lu ) ( ε ) in C − δ with the boundary condition v ε = u ( ε ) on ∂ p C − δ has a unique smooth solution v ε in C − δ . Moreover, L ε ( v ε − u ( ε ) ) = ( Lu ) ( ε ) − Lu + Lu − L ε u ( ε ) → L p ( C − δ ) . By Lemma 3.3 below and the uniform continuity of u , for any h > | v ε − u | ≤ h in C − δ provided that ε is sufficiently small. Because L ε ( v ε − h ) = ( Lu ) ( ε ) ≤ C − δ and v ε − h ≤ u = 0 in C − δ \ Q, ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 13 (10) is satisfied with v ε − h in place of u and C − δ in place of C . Since v ε − h , a ( ε ) ij , and b ( ε ) i are smooth, by the proof above, we get u − h ≤ v ε − h ≤ β sup C − δ ( v ε − h ) ≤ β sup C u + in C δ (1 − δ ) . Letting h → δ → Divergence case:
Let D t ( u ) − D j ( a ij D i u ) + b i D i u =: f − D i f i . By a property of convolution, we have D t u ( ε ) − D j (cid:16) ( a ij D i u ) ( ε ) (cid:17) + ( b i D i u ) ( ε ) = f ( ε )0 − D i f ( ε ) i . Similar to the non-divergence case, let v ε be the weak solution of L ε v ε := D t v ε − D j (cid:16) a ( ε ) ij D i v ε (cid:17) + b ( ε ) i D i v ε = f ( ε )0 − D i f ( ε ) i in C − δ with the boundary condition v ε = u ( ε ) on ∂ p C − δ . Note that L ε u ( ε ) = D j (cid:16) ( a ij D i u ) ( ε ) (cid:17) − ( b i D i u ) ( ε ) + f ( ε )0 − D i f ( ε ) i − D j (cid:16) a ( ε ) ij D i u ( ε ) (cid:17) + b ( ε ) i D i u ( ε ) . Thus, L ε ( v ε − u ( ε ) ) = D j (cid:16) a ( ε ) ij D i u ( ε ) − ( a ij D i u ) ( ε ) (cid:17) + ( b i D i u ) ( ε ) − b ( ε ) i D i u ( ε ) . It is easy to see that a ( ε ) ij D i u ( ε ) − ( a ij D i u ) ( ε ) → , ( b i D i u ) ( ε ) − b ( ε ) i D i u ( ε ) → L ( C − δ ). Due to the energy inequality, we have v ε → u in L ( C − δ ), and, afterextracting a subsequence, v ε k → u (a.e.) in C − δ for a decreasing sequence ε k ց h >
0, denote S ε,h, − δ := { v ε > h, u ≤ } ∩ C − δ . By Chebyshev’s inequality, as ε → | S ε,h, − δ | ≤ h Z C − δ ( v ε − u ) dX → . Thus, for ε sufficiently small, we have |{ v ε ≤ h } ∩ C − δ | ≥ |{ u ≤ } ∩ C − δ | − | S ε,h, − δ | > θ | C − δ | . Since v ε , a ( ε ) ij , and b ( ε ) i are smooth, and ( Lu ) ( ε ) ≤ C − δ , from the proof aboveand the maximum principle, we reachsup C δ − δ ) ( v ε − h ) ≤ β sup C − δ ( v ε − h ) ≤ β sup ∂ p C − δ ( v ε − h ) = β sup ∂ p C − δ u ( ε ) − h ≤ β sup C u. Bearing in mind that v ε k → u (a.e.) in C − δ as k → ∞ and u is continuous, we getsup C δ − δ ) ( u − h ) ≤ β sup C u. Since h is an arbitrary positive constant, we obtain the desired estimate. (cid:3) Remark . If we additionally assume u = 0 on ∂ p Q , then under the conditions ofLemma 3.1, from u ( X ) ≤ β sup C r u + we can derivesup Q ∩ ˆ C δ r ( X ) u ≤ β sup Q ∩ ˆ C r ( X ) u + with constant δ = δ ( n, θ ) ∈ (0 , / , β = β ( n, ν, θ/ , K , r ) ∈ (0 , . Moreover, in this case X can be any point on ¯ Q . For this, note that the measurecondition | C r ( X ) \ Q | > θ | C r | holds for any X ∈ ˆ C δ r ( X ) with small δ depending on n and θ .In the sequel, N = N ( · · · ) denotes a constant depending only on the prescribedquantities, such as n , ν , etc., which are specified in the parentheses, and the valueof N may change from line to line. Lemma 3.3 (Pointwise estimate) . Let p ∈ ( n +22 , ∞ ) , R > , and L be auniformly parabolic operator (in the form (D) or (ND) ) defined on a cylinder C R := C R ( X ) ⊂ R n +1 , R ≤ R , which satisfies (2) and | b i | , | c i | , | c | ≤ K . Inthe non-divergence case we further assume that a ij satisfy Assumption 1.1. Thenfor any f = f − D i f i , where f ∈ L p ( C R ) and f i ∈ L p ( C R ) for the divergencecase, and f ∈ L p ( C R ) for the non-divergence case, there exists a unique solution w ∈ W ( C R ) to the equation Lw = f in C R , w = 0 on ∂ p C R . Moreover, w is H¨older continuous in C R and | w | ≤ N (cid:16) k f k L p ( C R ) + k f i k L p ( C R ) (cid:17) for the divergence case , (11) where N = N ( n, ν, K , R , p ) , and | w | ≤ N k f k L p ( C R ) for the non-divergence case , (12) where N = N ( n, ν, K , R , p , ω a ) .Proof. In the divergence case, the unique existence of w in H ( C R ) is classical bynoting that f i ∈ L ( C R ) and f ∈ L n +2) / ( n +4) ( C R ). The H¨older continuity and(11) are due to the parabolic De Giorgi–Nash–Moser estimate. See, for instance,[32, Chap. 6]. In the non-divergence case, for operators without the lower-orderterms, the unique solvability was first established in [1] under the assumption thatthe leading coefficients are VMO with respect to both x and t . In the general case,the unique solvability in W , p ( C R ) can be found in [10]; see Theorem 6 and Remark1 there. Recall that p > ( n + 2) /
2. The H¨older continuity and (12) then followfrom the parabolic Sobolev embedding theorem. The lemma is proved. (cid:3)
We remark that if p ≥ n + 1, by the parabolic Alexandrov–Bakelman–Pucciestimate (cf. [23, 41]), the constant N can be taken to be independent of theregularity of a ij . ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 15 Weighted uniform estimate
The a priori estimate in Proposition 4.1 below is a key ingredient in the proof ofthe existence of solutions. The idea of the proof is based on [7, Theorem 3.5]. Ourcase is much more involved because the coefficients of the lower-order terms mayblow up near the boundary. We use a re-scaling argument, for which we introducesome additional notation. For any ρ >
0, we denote u ρ ( X ) = u ( ρx, ρ t ) in ρ − Q := { ( ρ − x, ρ − t ) | ( x, t ) ∈ Q } . For the divergence case (D), by a simple scaling, L ρ u ρ := D t ( u ρ ) − D j ( a ρij D i u ρ ) + ρb ρi D i u ρ − ρD i ( c ρi u ρ ) + ρ c ρ u ρ = ρ f ρ − ρD i f ρi , where a ρij ( X ) = a ij ( ρx, ρ t ), etc. For the non-divergence case (ND), we have L ρ u ρ := u ρt − a ρij D ij u ρ + ρb ρi D i u ρ + ρ c ρ u ρ = ρ f ρ . Denote d ρ ( X ) = dist( X, ∂ p ( ρ − Q )) = ρ − d ( ρx, ρ t ) . Next we modify the operators near the parabolic boundary. For any ρ > η ρ such that η ρ = 0 in { d < ρ/ } , η ρ = 1 in { d > ρ } , and its modulus of continuity ω η ( r ) ≤ N r/ρ . For ε ∈ (0 , a ρ,εij = η ρε ( ρx, ρ t ) a ρij + (cid:0) − η ρε ( ρx, ρ t ) (cid:1) δ ij , which satisfy (UE), and the modulus of continuity ω a ρ,ε has an upper bound ω a ρ,ε ( r ) ≤ N ( ω a ( ρr ) + r/ε ) . Note that this is independent of ρ for ρ ∈ (0 , b ρ,εi = η ρε ( ρx, ρ t ) b ρi , c ρ,εi = η ρε ( ρx, ρ t ) c ρi ,c ρ,ε = η ρε ( ρx, ρ t ) c ρ + D i η ρε ( ρx, ρ t ) c ρi , f ρ,ε = η ρε ( ρx, ρ t ) f ρ . Then b ρ,εi , c ρ,εi , c ρ,ε ∈ L ∞ ( R n +1 ). We define an operator L ρ,ε on R n +1 : L ρ,ε w := D t ( w ) − D j ( a ρ,εij D i w ) + ρb ρ,εi D i w − ρD i ( c ρ,εi w ) + ρ c ρ,ε w in the divergence case, and L ρ,ε w := w t − a ρ,εij D ij w + ρb ρ,εi D i w + ρ c ρ,ε w in the non-divergence case. It is easily seen that L ρ,ε ≥ Proposition 4.1.
Let p ∈ ( n +22 , ∞ ) be a constant, Q be a bounded domain in R n +1 satisfying the measure condition (A) with a constant θ > , and L be a uniformlyparabolic operator in the form of (D) or (ND) satisfying the same assumptions forthe coefficients a ij , b i , c i , and c as in Theorem 1.3. Then there exists a constant β = β ( n, ν, θ ) ∈ (0 , such that the following hold true:(i) For β ∈ (0 , β ) and an open subset Q ′ ⋐ p Q , if u ∈ W ( Q ′ ) , f ∈ F ( Q ) , and Lu = f in Q ′ , u = 0 on ∂ p Q ′ , then sup Q ′ d − β u ≤ N k f k F ( Q ) , where d = d ( X ) , N = N ( n, ν, θ , γ, diam( Q ) , β, p ) , which also depends on ω ∗ a inthe non-divergence case. Here ω ∗ a = max (cid:8) sup ρ ∈ (0 ,ρ ) ω a ρ,ε , ω a ,ρ / (cid:9) , where ε ∈ (0 , depending only on n , ν , θ , and β , and ρ ∈ (0 , depending onlyon the same parameters and γ .(ii) The same estimate holds true for Q ′ = Q if in addition we assume that f vanishes near ∂ p Q , and b i and c i are bounded (for instance, by ) near ∂ p Q .Proof. We first prove the assertion (i). We assume that the set Q ′ ∩ { u > } isnonempty; otherwise there is nothing to prove. The assumption dist ( Q ′ , ∂ p Q ) > d − β u ∈ C ( ¯ Q ′ ), and hence there is a point X ∈ ¯ Q ′ at which d − β ( X ) u ( X ) = M := sup Q ′ d − β u > . (13)Set ρ := d ( X ) := dist ( X , ∂ p Q ) >
0, and choose a point Y ∈ ¯ ∂ p Q for which d ( X ) = | X − Y | . Without loss of generality, we assume X = (0 ,
0) and Y =( y , s ).Now we set R := 4 /θ and β := − log R β >
0, where β = β ( n, ν, θ / n +2 , , /θ )is the constant from Lemma 3.1. Since 0 < β < β and R >
1, we have1 − ( R + 1) β β > − ( R + 1) β − β > . Now we take ε ∈ (0 , n , ν , θ , and β , such that ε β + ( R + 1) β β < . (14)Denote ρ − Q ′ := { ( ρ − x, ρ − t ) | ( x, t ) ∈ Q ′ } and Q r = C r ∩ ( ρ − Q ). Observethat | ρb ρi ( X ) | , | ρc ρi ( X ) | ≤ ρ o (cid:0) d − ( ρx, ρ t ) (cid:1) = ρ o ( ρ − d − ρ ( x, t )) ≤ ε − γ (( R + 1) ρ )in { X ∈ Q R | d ρ ( X ) ≥ ε } , where we used Q R ⊂ { X ∈ ρ − Q | d ρ ( X ) < R + 1 } .We choose ρ ∈ (0 ,
1) depending only on n , ν , θ , β , and γ such that ε − γ (( R + 1) ρ ) ≤ . Next, we consider two cases: (i) ρ < ρ and (ii) ρ ≥ ρ . Case 1: ρ < ρ . We note that f ρ,ε ∈ L p ( C R ) in the non-divergence case, and f ρ,ε ∈ L p ( C R ), f ρ,εi ∈ L p ( C R ) in the divergence case. By Lemma 3.3, thereexists a unique solution w ∈ W ( C R ) to L ρ,ε w = ρ f ρ,ε in the non-divergence case(or L ρ,ε w = ρ f ρ,ε − ρD i ( f ρ,εi ) in the divergence case) with the zero boundarycondition w = 0 on ∂ p C R . Moreover, w satisfies the property (11) (or (12)).Consider the function v ρ ( X ) := u ρ ( X ) − ( ερ ) β M − w ( X ) − sup C R | w | on ρ − Q ′ ∩ C R , (15)which may be extended continuously outside ρ − Q ′ ∩ C R , and define V := ρ − Q ′ ∩ { d ρ > ε } ∩ { v ρ > } . We see that M = sup X ∈ Q ′ d − β ( X ) u ( X ) = ρ − β sup X ∈ ρ − Q ′ d − βρ ( X ) u ρ ( X ) , which implies that u ρ = d βρ d − βρ u ρ ≤ d βρ ρ β M ≤ ( ερ ) β M, when d ρ ≤ ε. ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 17
We first assume v ρ (0) >
0. By Lemma 2.3 applied to ρ − Q , we have | C R \ V | ≥ | C R \ ρ − Q | ≥ − n − θ | C R | . From u ρ = 0 on ∂ p ( ρ − Q ′ ) and u ρ ≤ ( ερ ) β M on { d ρ ≤ ε } , it follows v ρ = 0 on C R ∩ ∂ p V . Moreover, 0 ∈ ¯ V \ ¯ ∂ p V because 0 ∈ ρ − Q ′ ∩ { d ρ > ε } ∩ { v ρ > } and0 / ∈ ¯ ∂ p (cid:0) ρ − Q ′ (cid:1) . We also observe that L ρ,ε v ρ ≤ V , where we use the condition(2), the definition of w , and fact that L ρ,ε coincides with L ρ in { d ρ ≥ ε } ∩ ρ − Q .Then by Lemma 3.1 with Q = V , we obtain v ρ (0) ≤ β sup V ∩ C R v ρ ≤ β sup V ∩ C R u ρ , where β = β ( n, ν, θ / n +2 , , R ) ∈ (0 , d ρ ( X ) ≤ R + 1 on C R , we seethat u ρ ≤ (( R + 1) ρ ) β M on V ∩ C R , which implies v ρ (0) ≤ β (( R + 1) ρ ) β M . Ofcourse, the last estimate also holds in the case v ρ (0) ≤
0. From this estimate,together with (13) and (15), it follows that ρ β M = u (0) = u ρ (0) ≤ ρ β (cid:2) ε β + ( R + 1) β β (cid:3) M + 2 sup C R | w | . By the property (11) or (12) of the function w on C R we have | w | ≤ N k ρ ˜ f ρ k L p ( C R ) + N n X i =1 k ρ ˜ f ρi k L p ( C R ) in the divergence case, where N = N ( n, ν, θ , p ) >
0, and | w | ≤ N k ρ ˜ f ρ k L p ( C R ) in the non-divergence case, where N = N ( n, ν, θ , p , ω a ρ,ε ) > k ρ ˜ f ρ k L p ( C R ) = ρ Z Q R ∩{ d ρ ≥ ε } | f ρ ( Y ) | p dY ! /p = ρ ρ − n − Z Q Rρ ∩{ d ≥ ερ } | f ( X ) | p dX ! /p ≤ ρ − ( n +2) /p + β − ε β − Z Q Rρ ∩{ d ≥ ερ } | d − β ( X ) f ( X ) | p dX ! /p ≤ ρ β − ( n +2) /p ε β − k f k F β,p ( Q ) (2 | C ( R +1) ρ | ) /p . Similarly, in the divergence case, k ρ f ρ k L p ( Q ) ≤ ρ β − ( n +2) /p ε β − k f k F β,p ( Q ) (2 | C ( R +1) ρ | ) /p , k ρf ρi k L p ( Q ) ≤ ρ β − ( n +2) / (2 p ) ε β − k f i k F β +1 , p ( Q ) (2 | C ( R +1) ρ | ) / (2 p ) . Therefore, M ≤ (cid:2) ε β + ( R + 1) β β (cid:3) M + N ε β − k f k F ( Q ) . From this inequality and (14), it follows that M ≤ N k f k F β,q ( Q ) , where N = N ( n, ν, θ , β, p ), which also depends on ω a ρ,ε in the non-divergence case. Case 2: ρ ≥ ρ . In this case, it suffices to bound | u (0) | in terms of f . Let R = diam( Q ) and Q ρ = Q ′ ∩ { X ∈ Q : dist( X, ∂ p Q ) > ρ/ } . In the non-divergence case, let w be the unique solution in W ( C R ) of L ,ρ / w = f ,ρ / in C R , w = 0 on ∂ p C R . By Lemma 3.3, we have | w | ≤ N k f k L p ( Q ρ ) , (16)where N = N ( n, ν, θ , γ, β, p , diam( Q ) , ω a ,ρ / ).Because L ,ρ / coincides with L in Q ρ , L (cid:16) sup ∂ p Q ρ u + + w + N k f k L p ( Q ρ ) (cid:17) ≥ f in Q ρ . Moreover, sup ∂ p Q ρ u + + w + N k f k L p ( Q ρ ) ≥ u on ∂ p Q ρ . By the comparison principle Corollary 2.8 and (16), we have u (0) ≤ sup ∂ p Q ρ u + + N k f k L p ( Q ρ ) . Since u = 0 on ∂ p Q ′ , we observe thatsup ∂ p Q ρ u + ≤ ( ρ/ β M. Thus, M = ρ − β u (0) ≤ − β M + N ρ − β k f k L p ( Q ρ ) ≤ − β M + N ρ − β (cid:16) ρ (cid:17) β − k f k F ( Q ) | Q | /p ≤ − β M + N ρ − k f k F ( Q ) , which gives the desired estimate.The divergence case is similar. This finishes the proof of the first assertion.For the assertion (ii), we find r ∈ (0 ,
1) such that f = 0 and | b i | , | c i | ≤ { X ∈ Q : d ( X ) ≤ r } . Then we take β := log δ β , where β = β ( n, ν, θ / , , δ = δ ( n, θ ) are the constants from Remark 3.2. We observe that by Remark3.2 and an iteration argument, we have | u ( X ) | ≤ N ( d ( X )) β ∀ X ∈ ¯ Q. Thus, for any β ∈ (0 , β ) and X ∈ ∂ p Q ,lim Q ∋ X → X d − β u ( X ) = 0 , and d − β u ∈ C ( ¯ Q ). Then we argue as in the proof of the first assertion with Q inplace of Q ′ with the smaller β between the above β and the one from the proofof assertion (i). The lemma is proved. (cid:3) The following 1-D example suggests that the growth condition on b i and c i nearthe boundary in the proposition above is in some sense optimal even in the ellipticcase. Example . We take a nonnegative smooth function η on R such that η ( x ) = 1for x ≤ / η = 0 for x ≥ /
2. Consider u ( x ) := − η ( x )(ln x ) − . Clearly, u ∈ C ([0 , / ∩ C ∞ ((0 , / u (0) = u (1 /
2) = 0. By a direct computation, itis easily seen that u satisfies u ′′ + bu ′ = f in (0 , / , ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 19 where b ( x ) := x − (cid:0) x ) − (cid:1) for x ∈ (0 , /
3) and f ∈ L ∞ ([0 , | b | ∼ x − near 0 and lim x ց d − β u = ∞ for any β > Proofs of the main theorems
Now we are ready to prove the main results of the paper, Theorem 1.3.
Proof of Theorem 1.3.
We first prove the non-divergence case. Recall the notationintroduced at the beginning of the previous section. For k = 1 , , . . . , denote Q k = { X ∈ Q | dist( X, ∂ p Q ) > /k } ,L k = L , /k , and f k = f , /k . We shall find a unique solution u k to the equation L k u k = f k in Q, u k | ∂ p Q = 0 (17)using Perron’s method. Since the W , p -solvability in cylindrical domains for para-bolic operators with VMO coefficients is available (cf. [10]) and f k are bounded in L p ( Q ), in order to find a unique solution u k ∈ W ( Q ) to the equation (17) usingPerron’s method, it suffices to construct a barrier function ψ k ∈ W ( Q ) satisfying L k ψ k ≥ Q, ψ k = 0 on ∂ p Q. (18)For this purpose, we use an idea in [24]. Let w ∈ W ( Q ) be the solution to the heatequation w t − ∆ w = 1 in Q, w = 0 on ∂ p Q. (19)The existence of such w is due to [12] and the fact that any domain satisfyingthe exterior measure condition also satisfies Wiener’s criterion. See Appendix A.By the maximum principle, w is strictly positive in Q . Denote R = diam( Q ) andwithout loss of generality, we assume Q ⊂ C R . Let v = cosh( µR ) − cosh( µ | x | ).Then v ≥ Q , and a straightforward calculation shows that L k v ≥ Q (20)for µ sufficiently large. Denote F ( x, y ) := min { x, y } and let F ( ε ) be the standardmollification of F . Clearly, for any ε > F ( ε ) is a smooth function and D x F ( ε ) ≥ , D y F ( ε ) ≥ , D x F ( ε ) + D y F ( ε ) = 1 , D F ( ε ) ≤ . (21)Now we choose λ sufficiently large such that λw ≥ v in Q k , and let ψ k = F ( ε ) ((1 + λ ) w, v ) . Then using the definition of L k , (19), (20), and (21), it is easily seen that ψ k satisfies(18) if ε is sufficiently small.By Proposition 4.1 (ii), there exists N = N ( n, ν, θ , γ, diam( Q ) , β, p , ω ∗ a k ) suchthat, for any β ∈ (0 , β ), sup Q d − β u k ≤ N k f k F ( Q ) , where β = β ( n, ν, θ ) ∈ (0 ,
1] is the constant from Proposition 4.1. Note that bythe definition of L k the choice of β can be made independent of k ∈ N . By usingthe simple equality χ (cid:0) χ A + (1 − χ ) B (cid:1) + (1 − χ ) B = χ χ A + (1 − χ χ ) B and the definition of ω ∗ a k , we see that ω ∗ a k has an upper bound ω ∗ a k ( r ) ≤ N ( ω a ( r ) + r/ε + r/ρ ) , which is independent of k . Here ε and ρ are constants from Proposition 4.1. With − u k in place of u k , we obtaininf Q d − β u k ≥ − N k f k F ( Q ) . Therefore, for any X ∈ Q , we have | u k | ≤ N d β k f k F ( Q ) , (22)where N is independent of k .For an arbitrary parabolic cylinder C ⋐ p Q , thanks to the local W , p estimateand the Sobolev embedding theorem, each u k is H¨older continuous in C with a uni-form H¨older norm. Then by applying the Arzela–Ascoli theorem to the sequence { u k } on each Q m , m ∈ N , and using Cantor’s diagonal argument, we find a sub-sequence, still denoted by { u k } , converging locally uniformly to a function u in Q .Moreover, by the inequality (22), u ∈ C ( ¯ Q ) with u = 0 on ∂ p Q . Now we show that Lu = f in C . One can find another parabolic cylinder C ′ such that C ⋐ p C ′ ⋐ p Q .Then for sufficiently large k , we have L ( u k +1 − u k ) = 0in C ′ . By the interior W , p estimate, we have k u k +1 − u k k W , p ( C ) ≤ N k u k +1 − u k k L p ( C ′ ) . From this inequality and the fact that u k converges uniformly to u in C ′ , it followsthat u ∈ W , p ( C ) and Lu = f in C . Since C ⋐ p Q is arbitrary, we have provedthat u ∈ W ( Q ) and Lu = f in Q . The uniqueness assertion follows immediatelyfrom the comparison principle, Corollary 2.8.Now we deal with the divergence case. Take the standard convolutions of coeffi-cients and data functions f with a non-negative mollifier, so that the mollifications a ( k ) ij , b ( k ) i , c ( k ) i , c ( k )0 , f ( k )0 , and f ( k ) i are in C ∞ ( ¯ Q k ). In particular, the mollifiedcoefficients converge to their original ones locally in L ( Q ). Define a kij := (cid:0) a ( k ) ij (cid:1) , /k , b ki := (cid:0) b ( k ) i (cid:1) , /k ,c ki := (cid:0) c ( k ) i (cid:1) , /k , c k := (cid:0) c ( k )0 (cid:1) , /k ,f k := (cid:0) f ( k )0 (cid:1) , /k , f ki := (cid:0) f ( k ) i (cid:1) , /k , where each term is in C ∞ ( ¯ Q ). Let L k be the divergence type operator with thecoefficients a kij , b ki , c ki , and c k . Then we turn L k into a non-divergence type operator˜ L k by defining ˜ L k = ∂∂t − ˜ a kij D ij + ˜ b ki D i + ˜ c k , where ˜ a kij = a kij , ˜ b ki = − D j a kij + b ki − c ki , ˜ c k = c k − D i c ki . Note that, due to the condition (1), we have ˜ c k ≥
0. Indeed, for any nonnegativefunction φ ∈ C ∞ ( Q ), Z Q ˜ c k φ dX = Z Q (cid:0) c k φ + c ki D i φ (cid:1) dX = Z Q (cid:16) c ( k )0 η /k φ + c ( k ) i D i (cid:16) η /k φ (cid:17)(cid:17) dX ≥ , ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 21 which implies ˜ c k ≥ Q . Then using the argument described above for thenon-divergence case, we find a unique u k satisfying ˜ L k u k = f k − D i f ki in Q and u k | ∂ p Q = 0. Since the coefficients and f k − D i f ki are infinitely differentiable, u k isinfinitely differentiable in Q and continuous on ¯ Q . Furthermore, u k also satisfiesthe divergence type equation L k u k = f k − D i f ki in Q . That is, Z Q (cid:0) − u k D t φ + a kij D i u k D j φ + b ki D i u k φ + c ki u k D i φ + c k u k φ (cid:1) dX = Z Q (cid:0) f k φ + f ki D i φ (cid:1) dX (23)for any φ ∈ C ∞ ( Q ). Similarly as in the non-divergence case above, with the DeGiorgi–Nash–Moser estimate we find u ∈ C ( ¯ Q ) with u = 0 on ∂ p Q , which is thelocal uniform limit of { u k } . Let us now prove that u belongs to H ( Q ) and satisfiesthe desired equation. For cylinders C ⋐ p C ′ ⋐ p Q , by the standard local L -estimate, k Du k k L ( C ) ≤ N k u k k L ( C ′ ) + N k f k k L n +2) / ( n +4) ( C ′ ) + N k f ki k L ( C ′ ) , where N is independent of k ∈ N . Moreover, for sufficiently large k , the right-handside is dominated by a quantity independent of k , so after taking a subsequence, Du k converges to Du weakly. Then Z Q a kij D i u k D j φ dX → Z Q a ij D i uD j φ dX, where φ ∈ C ∞ ( C ). More precisely, (cid:12)(cid:12)(cid:12)(cid:12)Z Q a kij D i u k D j φ dX − Z Q a ij D i u k D j φ dX (cid:12)(cid:12)(cid:12)(cid:12) ≤ k a kij − a ij k L ( C ) k Du k D j φ k L ( C ) → a kij converges to a ij locally in L and the L ( C )-norm of Du k D j φ is uni-formly bounded, and Z Q a ij D i u k D j φ dX → Z Q a ij D i uD j φ dX because Du k converges to Du weakly and a ij D j φ ∈ L ( C ). The same reasoningis applied to the other terms in (23). Therefore, by letting k → ∞ in (23), wesee that u satisfies the divergence type equation Lu = f − D i f i . The fact that u ∈ H ( Q ) follows from Du ∈ L , loc ( Q ) and the equation. The uniqueness followsfrom Corollary 2.8 as in the non-divergence case. (cid:3) Remark . For the non-divergence case, Assumption 1.1 is needed for the exis-tence of the solution u k during the proof. If one can generalize the solvability insmooth domains with more general coefficients (see, for instance, [19, 20, 9]), ourmain results can also be improved. Remark . Following the proof of Theorem 1.3, one can obtain a correspondingresult for elliptic equations in a domain Ω in R n satisfying the exterior measurecondition. The solution space is W (Ω) := W p (Ω) ∩ C ( ¯Ω) in the non-divergencecase for p ∈ ( n/ , ∞ ), and W (Ω) := W (Ω) ∩ C ( ¯Ω) in the divergence case. Weleave the details to the interested reader. Appendix A. Regularity of (A) domains
In this appendix, we will show that any (A) domain is regular, i.e., it satisfiesWiener’s criterion. We recall the definition of (A) domains:
Definition A.1.
An open set Q ⊂ R n +1 satisfies the condition (A) if there existsa constant θ ∈ (0 ,
1) such that for any Y ∈ ∂ p Q and r >
0, we have | C r ( Y ) \ Q | >θ | C r | .Now we introduce some standard notation and definitions. One may consult, forinstance, [11].Let F be the fundamental solution of the heat equation, namely, F ( x, t ) = (4 πt ) − n/ exp (cid:18) − | x | t (cid:19) , t > , , t ≤ , and E ( x, t, r ) be a heat ball of level set r centered at ( x, t ), E ( x, t, r ) := { ( y, s ) ∈ R n +1 | F ( x − y, t − s ) ≥ r } . Now we list some useful properties of heat balls: E ( r ) := E (0 , , r ) = { ( − x, − t ) ∈ R n +1 | F ( x, t ) ≥ r } ,F ( r − /n x, r − /n t ) = rF ( x, t ) , | E ( r ) | = r − − /n | E (1) | < ∞ , ( y, s ) ∈ E (1) ⇔ ( x, t ) ∈ E ( r ) , ( x, t ) = ( r − /n y, r − /n s ) , (24) E (1) = n ( x, t ) ∈ R n +1 | t < , ( − πt ) − n/ e | x | t ≥ o . Definition A.2.
An open set Q ⊂ R n +1 satisfies the condition (B) if there existsa constant θ ∈ (0 ,
1) such that, for any Y ∈ ∂ p Q and λ >
1, there exists k ∈ N satisfying |{ ( λ k ) k +1 ≥ F ( Y − Y ) ≥ ( λ k ) k } \ Q | > θ |{ ( λ k ) k +1 ≥ F ( Y − Y ) ≥ ( λ k ) k }| for all k ∈ N .We will show in Theorem A.5 that the condition (A) implies the condition (B).For each − π < t <
0, ( t, x ) ∈ E (1) if and only if | x | ≤ nt ln( − πt ) . This implies that in a neighborhood of the origin, the boundary of E (1) can berepresented as t = Φ( x ), where Φ is a C function and satisfiesΦ(0) = | D Φ(0) | = | D Φ(0) | = 0 . (25) Lemma A.3.
For any θ > , there exists θ = θ ( n, θ ) ∈ (0 , such that, for any r > , | C θr − /n \ E ( r ) | ≤ θ | C θr − /n | . (26) Proof.
Due to the scaling property of x → rx , t → r x of the standard cylinderand the heat ball, it suffices to prove (26) when r = 1. In this case, (26) followsimmediately from (25). (cid:3) ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 23
Lemma A.4.
Let λ > and θ ∈ (0 , be fixed numbers. Then one can choose alarge k = k ( n, λ, θ, θ ) ∈ N such that, for any k ∈ N , (cid:12)(cid:12) E (cid:0) ( λ k ) k +1 (cid:1)(cid:12)(cid:12) ≤ θ | C θλ − k k/n | . Proof.
Set r := λ k . Then (cid:12)(cid:12) E (cid:0) ( λ k ) k +1 (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) E (cid:0) r k +1 (cid:1)(cid:12)(cid:12) = r ( k +1)( − − /n ) | E (1) | and | C θλ − k k/n | = | C θr − k/n | = (cid:16) θr − k/n (cid:17) n +2 | C | . Thus, (cid:12)(cid:12) E (cid:0) r k +1 (cid:1)(cid:12)(cid:12) | C θr − k/n | = r − − /n | E (1) | θ n +2 | C | = 1( λ k ) /n | E (1) | θ n +2 | C | , which can be made small if λ > k is sufficiently large. The lemma isproved. (cid:3) Theorem A.5 (Condition (A) implies Condition (B)) . Let Q be a (A)-domainin R n +1 . For any Y ∈ ∂ p Q , and λ > , there exist k = k ( n, λ, θ ) ∈ N and θ = θ ( n, θ ) ∈ (0 , such that (cid:12)(cid:12)(cid:8) ( λ k ) k +1 ≥ F ( Y − Y ) ≥ ( λ k ) k (cid:9) \ Q (cid:12)(cid:12) ≥ θ (cid:12)(cid:12)(cid:8) ( λ k ) k +1 ≥ F ( Y − Y ) ≥ ( λ k ) k (cid:9)(cid:12)(cid:12) for all k ∈ N .Proof. Without loss of generality, we assume that Y = (0 , ∈ ∂ p Q . For a given λ >
1, take k from Lemma A.4, where θ = θ ( n, θ ) is a number from Lemma A.3.Set r = λ k . Then (cid:8) ( λ k ) k +1 > F ( − Y ) ≥ ( λ k ) k (cid:9) = (cid:8) r k +1 > F ( − Y ) ≥ r k (cid:9) = E ( r k ) \ E ( r k +1 ) . Thus (cid:8) ( λ k ) k +1 > F ( − Y ) ≥ ( λ k ) k (cid:9) \ Q = (cid:2) E ( r k ) \ Q (cid:3) \ (cid:2) E ( r k +1 ) \ Q (cid:3) and (cid:12)(cid:12)(cid:8) ( λ k ) k +1 ≥ F ( − Y ) ≥ ( λ k ) k (cid:9) \ Q (cid:12)(cid:12) ≥ (cid:12)(cid:12) E ( r k ) \ Q (cid:12)(cid:12) − (cid:12)(cid:12) E ( r k +1 ) \ Q (cid:12)(cid:12) ≥ (cid:12)(cid:12) E ( r k ) \ Q (cid:12)(cid:12) − (cid:12)(cid:12) E ( r k +1 ) (cid:12)(cid:12) ≥ (cid:12)(cid:12) E ( r k ) \ Q (cid:12)(cid:12) − θ | C θr − k/n | , (27)where the last inequality is due to Lemma A.4.On the other hand, | C θr − k/n \ Q | = (cid:12)(cid:12)(cid:0) C θr − k/n ∩ E ( r k ) (cid:1) \ Q (cid:12)(cid:12) + (cid:12)(cid:12)(cid:0) C θr − k/n \ E ( r k ) (cid:1) \ Q (cid:12)(cid:12) ≤ (cid:12)(cid:12) E ( r k ) \ Q (cid:12)(cid:12) + (cid:12)(cid:12) C θr − k/n \ E ( r k ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) E ( r k ) \ Q (cid:12)(cid:12) + θ | C θr − k/n | , where the last inequality is due to Lemma A.3. Along with (27) and Definition 1.2, (cid:12)(cid:12)(cid:8) ( λ k ) k +1 ≥ F ( − Y ) ≥ ( λ k ) k (cid:9) \ Q (cid:12)(cid:12) ≥ | C θr − k/n \ Q | − θ | C θr − k/n |≥ θ | C θr − k/n | = θ | C θr − k/n || E ( r k ) | (cid:12)(cid:12) E ( r k ) (cid:12)(cid:12) = θ θ n +2 | C (1) || E (1) | (cid:12)(cid:12) E ( r k ) (cid:12)(cid:12) ≥ θ (cid:12)(cid:12)(cid:8) ( λ k ) k +1 ≥ F ( Y − Y ) ≥ ( λ k ) k (cid:9)(cid:12)(cid:12) , where θ = θ θ n +2 | C (1) || E (1) | . That is, θ depends only on n and θ . (cid:3) We denote V ( R n +1 ) = { u | ∇ u ∈ L ( R n +1 ) , k u ( · , t ) k L ( R n ) ∈ L ∞ ( R ) } . For a compact set K in R n +1 , recall the thermal capacity:cap( K ) = sup { µ ( R n +1 ) | µ ∈ M ( K ) , F ∗ µ ≤ } , where M ( K ) is the set of all nonnegative Radon measure supported in K , and theparabolic capacity:Γ( K ) = inf (cid:26) sup t Z R n u ( x, t ) dx + Z R Z R n |∇ u | dX (cid:27) , where the function u is taken over all functions in V ( R n +1 ) with compact supportsuch that K ⊂ int { X : u ( X ) ≥ } .The following result can be found in [16]. Lemma A.6.
For any compact set K in R n +1 ,cap ( K ) ≥
12 Γ( K ) . As a consequence, we have
Lemma A.7.
For any set K in R n +1 , we have, for some N > ,cap ( K ) ≥ N | K | nn +2 . Proof.
From Lemma A.6, cap( K ) ≥
12 Γ( K ) . By the parabolic type Gagliardo–Nirenberg–Sobolev inequality (see, for instance,[32, Theorem IV.6.9]), for any u ∈ V ( R n +1 ) with compact support such that K ⊂ int { X : u ( X ) ≥ } ,sup t Z R n u ( x, t ) dx + Z R Z R n |∇ u | dX ≥ N (cid:18)Z | u | n +2) n dX (cid:19) nn +2 ≥ N | K | nn +2 . By the definition of Γ( K ), the lemma follows. (cid:3) Finally, we show that the condition (B) implies Wiener’s criterion.
Theorem A.8.
Let Q satisfy the condition (B). Then any point on ∂ p Q is regular.Namely, X ∈ ∂ p Q satisfies the following Wiener’s criterion from [12, Theorem 1] :for any λ > , ∞ X k =1 λ k cap ( Q c ∩ { λ k +1 ≥ F ( X − X ) ≥ λ k } ) = ∞ . ARABOLIC OPERATORS IN A TIME VARYING DOMAIN 25
Proof.
We take the constant k from Theorem A.5. It then follows from TheoremA.5, Lemma A.7, and (24) that ∞ X k =1 λ k cap( Q c ∩ { λ k +1 ≥ F ( X − X ) ≥ λ k } ) ≥ λ k ∞ X k =1 λ k k cap( Q c ∩ { λ k ( k +1) ≥ F ( X − X ) ≥ λ k k } ) ≥ Nλ k ∞ X k =1 λ k k | Q c ∩ { λ k ( k +1) ≥ F ( X − X ) ≥ λ k k }| nn +2 ≥ Nλ k ∞ X k =1 λ k k θ nn +2 |{ λ k ( k +1) ≥ F ( X − X ) ≥ λ k k }| nn +2 = Nλ k ∞ X k =1 θ nn +2 |{ λ k ≥ F ( X − X ) ≥ }| nn +2 = ∞ . The theorem is proved. (cid:3)
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E-mail address : [email protected] (H. Dong) Division of Applied Mathematics, Brown University, 182 George Street,Providence, RI 02912, USA
E-mail address : Hongjie [email protected] (D. Kim)
Department of Applied Mathematics, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do 446-701, Republic of Korea
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