ON SECOND ORDER ELLIPTIC AND PARABOLIC EQUATIONS OF MIXED TYPE 4 GONGCHENANDMIKHAILSAFONOV 1 0 2 Abstract. It is known that solutions to second order uniformlyelliptic and parabolicequations,eitherindivergenceornondivergence (general)form,are n Hölder continuous and satisfythe interior Harnack inequality. We show that a even intheone-dimensional case(x∈R1),these propertiesarenotpreserved J for equations of mixed divergence-nondivergence structure: for elliptic equa- 2 tions. Di(a1ijDju)+a2ijDiju=0, ] P andparabolicequations A p∂tu=Di(aijDju), wherep=p(t,x)isaboundedstrictlypositivefunction. TheHöldercontinuity . h andHarnackinequalityareknownifpdoesnotdependeitherontoronx. We t essentially use homogenization techniques in our construction. Bibliography: a 23titles. m [ 1 v 1. Introduction 1 5 Leavingasidetheellipticandparabolicequationswith“regular”coefficients,and 3 alsothecasesoflowerdimension,theHo¨lderregularityofsolutionswasfirstproved 0 in 1957by De Giorgi[DG] for uniformly elliptic equations, and soonafterwardsby . 1 Nash [N] for more general uniformly parabolic equations in the divergence form 0 4 Lu:=−∂tu+Di aijDju =0, (D) 1 (cid:0) (cid:1) where ∂ u := ∂u/∂t, D := ∂/∂x for i = 1,2,...,n, and the equation is under- : t i i v stood in the integral sense, i.e. u is a weak solution of (D). Here and throughout i X the paper, we assume the summation convention over repeated indices i,j from 1 to n. The coefficients a =a (t,x) are real Borel measurable functions satisfying r ij ij a the uniform parabolicity condition a ξ ξ ≥ν|ξ|2 for all ξ ∈Rn, and a2 ≤ν−2, (U) ij i j ij Xi,j with ν = const ∈ (0,1]. The elliptic equations D a D u = 0 can be formally i ij j considered as a particular case of (D), in which aij(cid:0)and u d(cid:1)o not depend on t, so that ∂ u=0. t The interior Harnack inequality was first proved in 1961 by Moser [M1] for elliptic equation, and then in 1964 [M2] for more general parabolic equations (D) (see Theorem 1.1 below). 2010 Mathematics Subject Classification. Primary35B27,35B65; Secondary35K10,35M10. Keywordsandphrases. Harnackinequality,qualitativepropertiesofsolutions,equationswith measurablecoefficients,homogenization. 1 2 GONGCHENANDMIKHAILSAFONOV TheHarnackinequalitytogetherwithHo¨lderregularityofsolutionstouniformly parabolic equations in the nondivergence form Lu:=−∂ u+a D u=0 (ND) t ij ij was provedmuch later, at the end of 1970th, by Krylov and Safonov [KS]. See [K, GT,Li, NU] andreferencestherein forfurther history,generalizationsto equations with unbounded lower order terms, and various applications. The method in [KS] is based on some variants of growth lemmas, which were originally introduced by Landis [La]. In its simplest formulation (without involving lower order terms), these results forparaboliccasecanbesummarizedasfollows. Intheellipticcase,onecansimply drop the dependence on t: a =a (x) and u=u(x). ij ij Theorem 1.1 (Harnack inequality). For y ∈ Rn,Y := (s,y) ∈ Rn+1, and r > 0, denote B (y):={x∈Rn : |x−y|1 depends only on n and ν in (U). Theorem 1.2 (Oscillationestimate). Let u be a bounded solution (not necessarily non-negative) of (D) or (ND) in C (Y) for some Y ∈Rn+1 and r >0. Then 2r oscu:=supu−infu≤θ·oscu with a constant θ =θ(n,ν)∈(0,1), (1.3) Cr Cr Cr C2r where C :=C (Y). r r The proof of these theorems, separately in the cases (D) and (ND), is contained in the above mentioned references. Corollary 1.3 (Hölder estimate). Let u be a bounded solution of (D) or (ND) in a cylinder C :=C (Y) for some Y ∈Rn+1 and r >0. Then r r 2ρ α ω :=oscu≤ ω for 0<ρ≤r, (1.4) ρ r Cρ (cid:16) r (cid:17) where α=α(n,ν):=−log θ >0, θ ∈(0,1) is the constant in (1.3). 2 Proof. Using the estimate (1.3) with ρ∈(0,r/2] in place of r, we get ω ≤θω for 0<2ρ≤r. ρ 2ρ For each ρ ∈ (0,r], there is a unique integer k ≥ 0 such that 2−k−1r < ρ ≤ 2−kr. Iterating the previous inequality and using monotonicity of ω , we get ρ ω ≤θω ≤···≤θkω ≤θkω . ρ 2ρ 2kρ r Finally, taking α:=−log θ >0, we obtain 2 2ρ α θk = 2−k α ≤ , r (cid:16) (cid:17) (cid:0) (cid:1) which yields the desired estimate (1.4). (cid:3) ON SECOND ORDER ELLIPTIC AND PARABOLIC EQUATIONS OF MIXED TYPE 3 More recently, Ferretti and Safonov [FS] tried to develop some “unifying” techniques which would equally applicable to equations in both (D) and (ND) forms. Theyfoundoutthethegrowthlemmascanserveasacommongroundfortheproof oftheHarnackinequalityandotherrelatedfacts,thoughthemethodsoftheirproof are completely different in these two cases. A natural question arises, whether or notTheorems1.1and1.2holdtrue,withconstantsindependentonthesmoothness of the coefficients, for solutions of mixed type elliptic equations of “mixed” type D (a1 D u)+a2D u=0, (1.5) i ij j ij ij where matrix functions a1 and a2 satisfy (U). It was shown [FS, Example 1.7] by ij ij direstcalculationthatthis is nottrue eveninthe one-dimensionalcase,when(1.5) is reduced to an ordinary differential equation with highly oscillating coefficients. In Section 2 we discuss this phenomenon from the point of view of homogenization theory. Namely, we use the fact that solutions of differential equations with periodic coefficients can be approximated by solutions of equations with constant coefficients as the period tends to 0. For this purpose, we need to take into con- ∗ sideration periodic solutions to the adjoint equation L v = 0. As a variant of the Fredholm alternative, it is known (see [BJS, Part II, Ch. 3], that in the class of smooth periodic functions, the elliptic equation (possibly of higher order) Lu = f issolvableifandonlyiff isorthogonalinL2 toanynontrivialsolutiontoL∗v =0, ∗ i.e. (f,v)=0,andmoreover,thehomogeneousequationsLu=0andL v =0have same number of linearly independent solutions. For elliptic equations of second order, Lu:=a D u+b D u=0, (1.6) ij ij i i from the strong maximum principle it follows that any periodic solution must be ∗ constant. Moreover, periodic solutions to L v = 0 cannot change sign, because otherwisewecouldchooseapositiveperiodicfunctionf suchthat(f,v)=0. Then we must have a periodic solution to the equation Lu=f >0, which is impossible by the strong maximum principle. InSection3,weextendtheFredholmalternativetosecondorderparabolicequa- tionsintheclassoffunctionswhichareperiodicbothintandx. Finally,inSection 4, we show that even for x∈R1, the solutions of the “mixed” parabolic equations Lu:=−p∂ u+D (a D u)=0 (1.7) t i ij j with p = p(t,x) ∈ [ν,ν−1] do not satisfy the Harnack inequality (1.2) and Ho¨lder estimate (1.4) with constants independent on the smoothness of coefficients. Note that if p = p(t) does not depend on x, then one can simply divide both sides of (1.7) by p and replace a by a /p, so that the equation (1.7) is reduced to the ij ij standard form (D). It is also known from [PE], see also [FS], that the Harnack inequality and other related facts are true if p = p(x) does not depend on t. One of the key observations here is that the function I(t):= p(x)u(t,x)dx Z Rn 4 GONGCHENANDMIKHAILSAFONOV under reasonable assumptions (which allow differentiation and application of the divergence theorem) satisfies ′ I (t)= p∂ udx= D (a D u)dx=0, t i ij j Z Z Rn Rn so that I(t)≡I(0) for t≥0. This argument does not work for p=p(t,x). We essentially use homogenization technique which was introduced by M. I. Freidlin[F]in1964formtheprobabilisticpointofview. Intheanalyticsetting,our methodcanbeconsideredasanapplicationofthemethod of asymptotic expansions (see [BLP, Sec. 1.2] or [JKO, Sec. 1.4], and references therein). More recently, a similartechniquewasusedinamoredifficult(non-periodic)casebyN.Nadirashvili [Nd](seealso[S])forthe proofofnon-uniquenessofweaksolutionstosecondorder elliptic equations a D u=0 with measurable coefficients a . ij ij ij In the parabolic case, our approach to the existence of time-periodic solutions is based on a Fredholm type argument, which was also discussed by G. Lieberman [Li1] for more general equations and domains. For our purposes, it is technically simplerthatanalternativeapproachbasedontheKrein-Rutmantheorem(see,e.g. [H]). We assume that all the functions a ,b ,u,v, etc., are smooth enough, so that ij i all the derivatives in our formulations and proofs are understood in the classical sense. Our goal is to show that for solutions of the equations (1.5) and (1.7), the Harnackinequality,andinfacteventheestimatesforthemodulusofcontinuity,are ingeneralimpossiblewithconstantsindependentonthe smoothnessofcoefficients. BASIC NOTATIONS: x=(x ,x ,...,x ) are vectors or points in Rn, |x| is the length of x∈Rn. 1 2 n The balls B (y) and cylinders C (Y) are defined in (1.1). r r In Sections 2 and 4, S stands for the set of all smooth 1-periodic functions. ThenotationA:=B, orB =:Ameans“A=B by definition”. Throughoutthe paper, N (with indices or without) denotes different constants depending only on the prescribed quantities such that n,ν, etc. This dependence is indicated in the parentheses: N =N(n,ν,...). 2. Elliptic equations of mixed type Inthissection,Sdenotesthesetofallsmooth1-periodicfunctionsonR1. Inthe one-dimensional case, the Fredholm alternative for elliptic operators L in (1.6) is containedinTheorem2.2below. Part(II)ofthistheoremcanalsobeconsideredas a particular case of Theorem 3.1 in the next section, when there is no dependence on t. We still give a direct elementary proof, because some of its details, such as Corollary 2.4, are used below. We precede it with a simple lemma. Lemma 2.1. (I) For arbitrary continuous functions a = a(x) > 0 and b = b(x) on R1, the function x y b(z) u (x):= e−F(y)dy, where F(y):= dz (2.1) 0 Z Z a(z) 0 0 ON SECOND ORDER ELLIPTIC AND PARABOLIC EQUATIONS OF MIXED TYPE 5 ′′ ′ ′ satisfies Lu :=au +bu =0; u (0)=0, u (0)=1. 0 0 0 0 0 (II) In addition tothefunctions a and bin theprevious part (I), let acontinuous function f =f(x) on R1 be given. Then the function x y eF(z)f(z) U(x):= e−F(y) dzdy, (2.2) Z Z a(z) 0 0 ′′ ′ ′ satisfies LU :=aU +bU =f; U(0)=U (0)=0. Note that the functions u and U in this lemma are not 1-periodic in general. 0 Theorem 2.2 (Fredholm alternative). Consider the linear differential equation ′′ ′ Lu:=au +bu =f (2.3) witha,b,f ∈S,where a=a(x)>0,anditsformally adjoint homogeneousequation ∗ ′′ ′ L v =(av) −(bv) =0. (2.4) We have the following properties. (I) The equation (2.4) has a nontrivial solution v ∈ S which is unique up to a multiplicative constant. Moreover, v(x )=0 at some point x if and only if v ≡0. 0 0 The solution is uniquely defined by the normalization condition 1 v(x)dx=1. (2.5) Z 0 (II) The equation (2.3) has a solution in S if and only if 1 f0 :=(f,v):= fvdx=0, (2.6) Z 0 where v ∈Ssatisfies (2.4)–(2.5). The solution is uniqueupto an additive constant. Proof. (I) Obviously, the equation (2.4) is equivalent to ′ (av) +bv =c=const. (2.7) Fora giveninitial value v(0) the unique solutionto this equationonR1 is givenby the expression x a(x)v(x)=e−F(x) a(0)v(0)+c eF(y)dy , (2.8) (cid:20) Z (cid:21) 0 wherethe functionF isdefinedin(2.1). Thereisauniquechoiceoftheconstant c which guarantees the equality v(1)=v(0). In this case, since a,b,f are 1-periodic, bothfunctions v(x) andv(x+1)satisfy (2.7)withthesameinitialconditionatthe point x=0. By uniqueness for the Cauchy (initial value) problem, we have v(x+1)≡v(x) on R1, i.e. v ∈S. (2.9) Moreover, from v(0) = v(1) = 0 it follows c = 0 and v ≡ 0. By periodicity, same is true with any point x ∈ R1 in place of 0: from v(x ) = v(x + 1) = 0 it 0 0 0 followsv ≡0. Inparticular,any non-trivialsolutionv cannotchange sign. Finally, multiplying v by an appropriateconstant, we geta solutionv ∈Ssatisfying (2.5). 6 GONGCHENANDMIKHAILSAFONOV (II) For an arbitrary smooth function u, integrating by parts, or equivalently, using the identity ′ ∗ ′ ′ w =vLu−uL v =vLu, where w :=uav−(av)u+buv, (2.10) we obtain 1 (Lu,v):= vLudx=w(1)−w(0). (2.11) Z 0 Therefore, if u ∈ S satisfies Lu = f, then by periodicity, w(1) = w(0), hence (f,v)=(Lu,v)=0. Now suppose that f ∈ S satisfies (f,v) = 0. We take u := U +λu , where u 0 0 and U are defined in (2.1)–(2.2), and a constant λ is chosen in such a way that u(1)=0. Then Lu=f and u(0)=u(1)=0. ′ From (f,v) = 0 and (2.11) it follows w(0) = w(1), which in turn implies u(0) = ′ u(1). Therefore, the functions u(x) and u(x+1) satisfy the equation Lu = f of ′ secondorderwithsameinitialconditionsfor uandu atthe pointx=0. Similarly to (2.9), u(x+1)≡u(x) on R1, i.e. u∈S. (2.12) The existence of a solution u ∈ S of Lu = f is proved. The uniqueness up to an additive constant follows from the fact that the difference u˜ := u −u of two 1 2 solutions of Lu = f satisfies the homogeneous equation Lu˜ = 0. Since u˜ is also periodic, by the maximum principle we must have u˜ = u −u = const. Theorem 1 2 is proved. (cid:3) Corollary 2.3. For an arbitrary f ∈ S, there exists a unique, up to an additive constant, solution in S of the equation Lu=f −f0, where f0 :=(f,v), (2.13) and v ∈S is defined by (2.4)–(2.5). Corollary 2.4. In a special case f =b, 1 1 b(x) b0 :=(b,v):= bvdx>0 if and only if F(1):= dx>0. (2.14) Z Z a(x) 0 0 Proof. Since a,b,v∈S, from (2.7) it follows b0 :=(b,v)=c. In order to guarantee the equality v(1)=v(0) in (2.8), we must have sign c=signF(1). (cid:3) Theorem 2.5. Let a,b∈S, a=a(x)>0, and 1 b(x) F(1):= dx>0. (2.15) Z a(x) 0 For ε > 0, denote aε(x) := a(ε−1x), bε(x) := b(ε−1x), and let u (x) be a solution ε to the Dirichlet problem Lεu :=aεu′′+ε−1bεu′ =0; u (0)=0, u (1)=1. (2.16) ε ε ε ε ε Then u →1 as ε→0+ uniformly on every interval [δ,1], 0<δ <1. ε ON SECOND ORDER ELLIPTIC AND PARABOLIC EQUATIONS OF MIXED TYPE 7 Proof. Denote a0 := (a,v), b0 := (b,v), where v is defined by (2.4)–(2.5). We know from (2.14) that b0 > 0. By Corollary 2.3, there exist functions A,B ∈ S satisfying LA=a−a0, LB =b−b0. Denote Aε(x):=A(ε−1x), Bε(x):=B(ε−1x). Then LεAε =ε−2(aε−a0), LεBε =ε−2(bε−b0). (2.17) For fixed K =const>1 and small ε>0, consider the functions 1−e−Kx h (x):= , and w :=u −h +g , (2.18) K 1−e−K K,ε ε K K,ε where g :=ε2 Aεh′′ +ε−1Bεh′ . K,ε K K Using (2.16), (2.17), and the element(cid:0)ary equality (cid:1) Lε(f f )=f Lεf +f Lεf +2aεf′f′, 1 2 2 1 1 2 1 2 we can write Lεw =−Lεh +I +I +I , K,ε K 1 2 3 where I := (aε−a0)h′′ +ε−1(bε−b0)h′ =Lεh −a0h′′ −ε−1b h′ , 1 K K K K 0 K I := ε2 AεLεh′′ +ε−1BεLεh′ , 2 K K I := 2ε(cid:0)2aε (Aε)′h′′′ +ε−1(Bε)′(cid:1)h′′ . 3 K K SinceAandB aresm(cid:0)ooth,thederivatives(Aε(cid:1))′ and(Bε)′ areoforderε−1. There- fore, for fixed K > 1, after cancelation of terms ±Lεh in the expression for K Lεw , the remaining terms are bounded by constants independent of ε, except K,ε for a negative term −ε−1b h′ of order ε−1. This term guarantees that for small 0 K ε>0, we have Lεw <0 on [0,1]. K,ε Forarbitraryδ andδ in(0,1),onecanchooseK >1suchthat1≥h ≥1−δ 0 K 0 on [δ,1], and then choose a small ε>0 such that Lεw <0 and |g |≤δ on [0,1]. K,ε K,ε 0 We have w = g ≥ −δ at the ends of the interval [0,1]. By the maximum K,ε K,ε 0 principle, w ≥−δ on [0,1], and K,ε 0 1≥u =h +w −g ≥1−3δ on [δ,1]. ε K K,ε K,ε 0 Since δ >0 can be made arbitrarily small, the theorem is proved. (cid:3) 0 As an easy application of this theorem, consider the following generalization of Example 1.7 in [FS]. Corollary 2.6. Let a and a be even positive functions in S such that 1 2 1 ′ a (x) 1 dx>0. (2.19) Z a (x)+a (x) 1 2 0 For ε>0 and j =1,2, denote aε(x):=a (ε−1x), and let u (x) be solutions to the j j ε Dirichlet problems Lεu :=(aεu′)′+aεu′′ =0 in (−1,1); u (−1)=−1, u (1)=1. (2.20) ε 1 ε 2 ε ε ε 8 GONGCHENANDMIKHAILSAFONOV Then 1 if x>0, u →signx:=0 if x=0, as ε→0+, (2.21) ε  −1 if x<0, uniformly on every set [−1,−δ]∪[δ,1], 0<δ <1. Proof. It is easy to see that the equation Lεu = 0 in (2.20) can be written in ε the form (2.16) with a(x) := a (x)+a (x) and b(x) := a′(x). Since aε are even 1 2 1 j functions, the functions v (x) := −u (−x) also satisfy (2.20). By uniqueness, we ε ε must have v ≡ u , i.e. u (x) are odd functions. In particular, u (0) = 0, so that ε ε ε ε u satisfy (2.16). Now (2.21) follows immediately from the previous theorem. (cid:3) ε Remark 2.7. The property (2.21) implies that for solutions u , ε > 0, to the ε equation (2.20) of mixed type, there are no uniform estimates for the modulus of continuity at the point x = 0. Moreover, the non-negative functions u˜ := 1+u ε ε also satisfy Lεu˜ = 0 in (−1,1), but the Harnack inequality (1.2) fails for u = ε u˜ , Y =Y =0, r=1/2, because it easily brings to a contradiction: ε r 1 1=u˜ (0)≤ sup u˜ ≤N · inf u˜ =N ·u˜ − →0 as ε→0+. ε ε ε ε (−1/2,1/2) (−1/2,1/2) (cid:16) 2(cid:17) Remark 2.8. Thereis alotofflexibility inthechoiceoffunctions a (x) satisfying j (2.19). We can assume that |a −1| ≤ δ with an arbitrary small δ > 0. For j 0 0 example, one can take a :=1+η , a :=1−η +η , where 0≤η ,η ≤δ , 1 1 2 1 2 1 2 0 both functions η and η are even, belong to S, vanish near integers x ∈ Z, not 1 2 ′ identically zero, and η has compact support in the set {x : η (x) > 0}. Then 1 2 ′ η η ≥0, and (2.19) holds true: 1 2 1 1 1 ′ ′ a (x) dη η η 1 dx= 1 = 1 2 dx>0. Z a (x)+a (x) Z 2+η Z (2+η )2 1 2 2 2 0 0 0 Remark 2.9. The assumption (2.19) in Corollary 2.6 is sharp in the following sense: if we replace “>” by “<”, then instead of (2.21), we will have convergence u → 0 uniformly on every interval [−δ,δ], 0 < δ < 1. For the proof of this fact, ε one can note that if in the formulation of Theorem 2.5, the assumption F(1) > 0 is replaced by F(1) < 0, then u → 0 as ε → 0+ uniformly on every interval ε [0,δ], 0< δ < 1. In turn, the proof of this statement only requires substitution of ′′ ′ the function h in (2.18), which satisfies h +Kh =0, by K K K eKx−1 ′′ ′ h (x):= satisfying h −Kh =0. K eK −1 K K 3. Existence of periodic solutions to parabolic equations In this section, we discuss the existence of solutions to second order parabolic equations Lu:=−p∂ u+a D u+b D u=f, (3.1) t ij ij i i all functions p,a ,b ,f, and u are periodic both in t and x, a satisfy the uniform ij j ij parabolicitycondition(U)withaconstantν ∈(0,1],andalsop=p(t,x)∈[ν,ν−1]. Thisrequirescertainconditionsonthesefunctions,whichareindifferentsituations ON SECOND ORDER ELLIPTIC AND PARABOLIC EQUATIONS OF MIXED TYPE 9 areusuallyreferredtoastheFredholmalternative. Forellipticequations,including higher order case, these conditions are presented from the point of view of Fourier analysis in Part 2, Ch. 3 of the book [BJS]. One can adjust the general approach there to our equations (3.1). However, since we need it only in a very particular case, we give a proof for completeness, and also as an application of the Harnack inequality. Formally, by dividing both parts of (3.1) by p, one can reduce this equationtothe casep=1,withν2 inplaceofν. However,thisisnotthe casewith ∗ the adjoint equation L v =0 in (3.2) below. Let l and l be given positive constants. Denote 0 K :=[0,l]n ⊂Rn, K :=[0,l ]×K ⊂Rn+1. 0 0 0 We say that the function u=u(t,x)=u(t,x ,...,x ) is K-periodic if it is defined 1 n on the whole space Rn+1, l -periodic with respect to t, and l-periodic with respect 0 to each of variables x ,...,x . Let S(K) denote the class of all K-periodic func- 1 n tions u(t,x) on Rn+1, which are continuous in Rn+1 together with all derivatives ∂ u,D u,D u. t i ij Roughly speaking, the Fredholm alternative states that the equation (3.1) has a solution in S(K) if and only if f is orthogonal in L2(K) to solutions v of the corresponding homogeneous adjoint equation ∗ L v :=∂ (pv)+D (a v)−D (b v)=0. (3.2) t ij ij i i For simplicity, we assume that all the functions p,a ,b ,u,v belong to S(K), so ij i that we deal with classical solutions of equations (3.1) and (3.2). Theorem 3.1. Let a satisfy the uniform parabolicity condition (U), p, a , and ij ij b belong to S(K), and let v ∈ S(K) be a strictly positive solution of (3.2). Then i for a given function f ∈ S(K), the equation (3.1) has a solution u ∈ S(K) if and only if (f,v):= (fv)(t,x)dxdt =0. (3.3) ZZ K Proof. Let u∈S(K) satisfy the equation (3.1). Consider the function I(t):= (puv)(t,x)dx. (3.4) Z K0 By the smoothness assumptions and equalities (3.1)–(3.2), we obtain ′ I (t)= ∂ (puv)(t,x)dx= vp∂ u+u∂ (pv) dx t t t Z Z (cid:0) (cid:1) K0 K0 = v a D u+b D u−f +u −D (a v)+D (b v) dx. ij ij i i ij ij i i Z h (cid:0) (cid:1) (cid:0) (cid:1)i K0 Since all the functions p,a ,b ,u,v are l-periodic with respect to x ,x , after ap- ij i i j plying Fubini’s theorem and integrating by parts, the terms with a and b cancel ij i and we get ′ I (t)=− (fv)(t,x)dx. (3.5) Z K0 10 GONGCHENANDMIKHAILSAFONOV The function I(t) is l -periodic, because u and v are l -periodic with respect to t. 0 0 This implies for an arbitrary T T+l0 T+l0 ′ (fv)(t,x)dx dt=− I (t)dt=I(T)−I(T +l )=0. (3.6) 0 Z (cid:20)Z (cid:21) Z T K0 T In particular, taking T = 0, we get the desired equality (3.3), and the theorem is proved in one direction. For the proof in the opposite direction, fix a function f ∈S(K) satisfying (3.3), and let u =u (t,x) be a bounded solution to the Cauchy problem 0 0 Lu (t,x)=f(t,x) in Rn+1 :={t>0, x∈Rn}; u (0,x)≡0. 0 + 0 It is well known (see, e.g. [Fr, LSU]) that there exists a classical solution to this problem, which is unique in the class of bounded functions. From the uniqueness and l-periodicity of p,a ,b ,f with respect to x , (k = 1,...,n), it follows that ij i k u (t,x+le ) ≡ u (t,x), i.e. the solution u is also l-periodic with respect to x . 0 k 0 0 k Note that we did not use periodicity with respect to t in the proof of the equality (3.5) for the function I(t) in (3.4). Hence the same equality holds true with u 0 instead of u, namely, ′ I (t):= (pu v)(t,x)dx satisfies I (t)=− (fv)(t,x)dx. 0 Z 0 0 Z K0 K0 In addition, we have I(0)=0 and the equality (3.3). Therefore, l0 l0 ′ I (l )= I (t)dt=− (fv)(t,x)dx dt=− (fv)(t,x)dxdt=0. 0 0 Z 0 Z (cid:20)Z (cid:21) ZZ 0 0 K0 K ′ Moreover,since f and v are l -periodic with respect to t, the function I (t) is also 0 0 l -periodic, and we obtain 0 I (kl ):= (pu v)(kl ,x)dx=0 for k =0,1,2,.... (3.7) 0 0 0 0 Z K0 Further, consider the functions u (t,x):=u (t+kl ,x), w (t,x):=u (t,x)−u (t,x) on Rn+1 (3.8) k 0 0 k k+1 k + for k = 0,1,2,.... Since all the given functions are l -periodic with respect to t, 0 we have Lw =0 in Rn+1 for k =0,1,2,.... k + We want to show that the functions u converge uniformly on Rn+1 to a solution k + u ∈ S(K) of the equation (3.1). For this purpose, we will use the estimate for oscillation similar to (1.3) applied to the functions w . Denote k c := osc w := sup w − inf w for k =0,1,2,.... k k k k Rn+1 Rn+1 Rn+1 + + + By the maximum principle and l-periodicity with respect to x ,...,x , we have 1 n c =oscw (0,x)=oscw (0,x) for k =0,1,2,.... k k k Rn K0

On Second Order Elliptic and Parabolic Equations of Mixed Type

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