This paper is concerned with a maximum principle for subsolutions, in the class W^{2,p}_loc, of second order linear elliptic equations in nondivergence form in arbitrary open (bounded or not) subsets of R^n, n &gt;=2, when p &gt; n/2

CAVALIERE, Paola

TRANSIRICO, Maria

English

Archivio della Ricerca - Università di Salerno

International Journal of Pure and Applied Mathematics————————————————————————–Volume 57 No. 3 2009, 299-311A STRONG MAXIMUM PRINCIPLE FORLINEAR ELLIPTIC OPERATORSPaola Cavaliere1 §, Maria Transirico21Department of Engineering of Information and Applied MathematicsUniversity of SalernoVia Ponte Don Melillo, Fisciano (Salerno), I-84084, ITALYe-mail: pcavaliere@unisa.it2Department of Mathematics and InformaticsUniversity of SalernoVia Ponte Don Melillo, Fisciano (Salerno), I-84084, ITALYe-mail: mtransirico@unisa.itAbstract: This paper is concerned with a maximum principle for subso-lutions, in the class W 2,ploc, of second order linear elliptic equations in non-divergence form in arbitrary open (bounded or not) subsets of Rn, n ≥ 2, whenp > n2 .The main coefficients are required to be just locally in L∞ ∩ V MO. Auniqueness result for the related homogenous Dirichlet problem is also obtained.AMS Subject Classification: 35J25, 35R05Key Words: elliptic operators, VMO-coefficients1. IntroductionLet Ω be an open bounded set in Rn, n ≥ 2, and consider a uniformly ellipticoperator in non-divergence formL0 =n∑i,j=1aij(x)∂2∂xi∂xj,whose coefficients aij are measurable in Ω. Moreover, let u be a solution of theReceived: October 4, 2006 c© 2009 Academic Publications§Correspondence author300 P. Cavaliere, M. TransiricoDirichlet problemu ∈ W 2,p(Ω) ∩ C0(Ω̄) ,L0u = f ∈ Lp(Ω) ,u|∂Ω = 0 ,(1.1)with p > n2 .If p = n, then the classical Aleksandrov-Bakelman-Pucci estimate assertsthatsupΩ|u| ≤ c ||f ||Lp(Ω) , (1.2)where c ∈ R+ depends only on n, Ω and the ellipticity constant (see, for in-stance, [11], [1], and also [2] for an interesting improving of the above result).However, an example in [10] points out that, when the ellipticity constant issmall enough, the a priori estimate (1.2) fails for Lp-norms, p < n, in place ofthe Ln-norm, and c depending only on n, Ω and the ellipticity constant.On the other side, it is well known that, under additional regularity assump-tions on the coefficients aij, (1.2) still holds for p < n even if with c dependingalso on p and on the regularity of the aij’s. This has been proved in the caseof Hölder continuous coefficients aij(x) in [7], when the aij are continuous in[8] and if the aij belong to W1,n(Ω) in [9] (we refer to [6] for a more recenttreatment of such theory).Therefore, it appears natural to investigate the problem (1.1) when p < nassuming that the coefficients aij belong to a space wider than those alreadyconsidered in the literature. Let us remember that both hypotheses aij uni-formly continuous and aij ∈ W1,n imply aij ∈ V MO (see [5]).Note also that it follows from (1.2) that the problemu ∈ W 2,p(Ω) ∩ C0(Ω̄) ,L0u = 0 ,u|∂Ω = 0 ,(1.3)with p > n2 , has only the zero solution.In this paper we fix an arbitrary open (bounded or not) subset Ω of Rn,n ≥ 2, a real number p > n2 , and we consider the second-order locally uniformlyelliptic differential operatorL =n∑i,j=1aij∂2∂xi∂xj+n∑i=1ai∂∂xi+ a ,where the main coefficients aij are locally bounded and locally V MO, the lowerorders terms ai and a satisfy suitable local summability conditions, and a isA STRONG MAXIMUM PRINCIPLE FOR... 301locally negative. In this situation, we will establish that if u is a solution of theproblemu ∈ W 2,ploc(Ω) , Lu ≥ 0 , (1.4)then u does not have any positive relative maximum in Ω. Then, as an appli-cation of this result, we will show that the problemu ∈ W 2,ploc(Ω) , Lu = 0 ,limx−→x0 u(x) = 0 ∀x0 ∈ ∂Ω ,lim|x|−→+∞ u(x) = 0 if Ω is unbounded,(1.5)admits only the zero solution.2. Some NotationLet Ω be an open subset of Rn. If p > 1 and g ∈ Lp(Ω), a function ωp[g,Ω] :R+ −→ R+ is called a modulus of continuity of g in Lp(Ω) ifsupE∈Σ(Ω)|E|≤t||g||Lp(E) ≤ ωp[g,Ω](t) ∀t ∈ R+ , limt→0+ωp[g,Ω](t) = 0 .If Ω has the property|Ω(x, r)| ≥ Arn ∀x ∈ Ω , ∀r ∈]0, 1] , (2.1)where Ω(x, r) = Ω ∩ B(x, r) and A is some positive constant independent of xand r, one can consider the space BMO(Ω, t), t ∈ R+, consisting of all functionsg in L1loc(Ω̄) such that[g]BMO(Ω,t) = supx∈Ωr∈]0,t]∫—Ω(x,r)|g −∫—Ω(x,r)g| < +∞ .If g ∈ BMO(Ω) = BMO(Ω, tA), withtA = supt∈R+(supx∈Ωr∈]0,t]rn|Ω(x, r)|≤1A),we shall say that g is in V MO(Ω) when [g]BMO(Ω,t) → 0 as t → 0+.Moreover, a function η[g,Ω] : R+ −→ R+ is called a modulus of continuity of gin V MO(Ω) if[g]BMO(Ω,t) ≤ η[g,Ω](t) ∀t ∈ R+ , limt→0+η[g,Ω](t) = 0 .We will also say that g ∈ V MOloc(Ω) if ζg belongs to V MO(Ω) for eachζ ∈ C∞◦ (Ω).302 P. Cavaliere, M. Transirico3. Auxiliary ResultsThroughout this section, let p ∈]n2 ,+∞[and B an arbitrary open ball of Rn,n ≥ 2, of radius d ∈ R+. We consider in B the operatorL =n∑i,j=1aij∂2∂xi∂xj+n∑i=1ai∂∂xi+ a (3.1)making the following assumption on its coefficients:aij = aji ∈ L∞(B) ∩ V MO(B) , i, j = 1, . . . , n ,∃ ν ∈ R+ :n∑i,j=1aij ξi ξj ≥ ν|ξ|2 a.e. in B , ∀ ξ ∈ Rn ,ai ∈ Lr(B), i = 1, . . . , n, r > n if p ≤ n, and r = p if p > n,a ∈ Lp(B), a ≤ 0 a.e. in B .(hB)Under (hB), the operator L from W2,p(B) into Lp(B) is bounded, and thefollowing estimate holds||Lu||Lp(B) ≤ c1||u||W 2,p(B) ∀u ∈ W2,p(B), (3.2)where c1 ∈ R+ depends on n, p, d, ||aij ||L∞(B), ||ai||Lr(B), ||a||Lp(B).We first provide the following “maximum principle”:Lemma 3.1. Assume (hB) and let h ∈ W2,p(B) such that h|∂B ≥ 0. Ifw ∈ W 2,p(B) satisfies{Lw = 0 in B,w|∂B = h|∂B ,(3.3)then w ≥ 0 in B.Proof. If n ≥ 3, the result is already known (see Lemma 3.1 of [3]). In thecase n = 2, we deduce the statement proceeding as in the proof of Lemma 3.1of [3] but applying Theorem 3.5 of [4] whereas in [3] is used Theorem 2.1 of[14].Moreover, we have the following existence and uniqueness result:Lemma 3.2. Assume that (hB) holds. Then the problem{u ∈ W 2,p(B)∩◦W1,p(B) ,Lu = f , f ∈ Lp(B) ,(3.4)is uniquely solvable and the solution u satisfies the a priori bound||u||W 2,p(B) ≤ c||f ||Lp(B) , (3.5)with c ∈ R+ depending on n, p, d, ν, ||aij ||L∞(B), η[p(aij)], ||ai||Lr(B), ||a||Lp(B),A STRONG MAXIMUM PRINCIPLE FOR... 303ωr[ai, B], ωp[a,B], and where p(aij) is an extension of aij to Rn of class L∞(Rn)∩V MO(Rn).Proof. It is enough to observe that an application of Theorem 5.1 of [12]yields that there exist extensions p(aij) of aij to Rn (i, j = 1, ..., n) of classL∞(Rn) ∩ V MO(Rn). Then, if n ≥ 3, the result is a consequence of Theorem2.1 of [14], while it follows from Theorem 3.5 of [4] for n = 2.We are now in position to state the following result.Lemma 3.3. Suppose that condition (hB) is satisfied, and consider prob-lem (3.4). Then, f ≥ 0 implies that u ≤ 0 in B.Proof. If p ≥ n, the statement is a consequence of Theorem III of [11].Thus we confine ourselves to the case n2 < p < n and we proceed in two steps.Step 1. We first assume that f ∈ Ln(B). Denote by ȧ the extension of a toRn with zero values out of B and by (Jk)k∈N a sequence of mollifiers, for anyk we define ak = Jk ∗ ȧ andLk =n∑i,j=1aij∂2∂xi∂xj+n∑i=1ai∂∂xi+ ak .The sequence (ak)k∈N satisfies the following properties:ak −→ a in Lp(B) , (3.6)||ak||Lp(B) ≤ ||a||Lp(B) , ωp[ak, B] ≤ ωp[a,B] , ∀k ∈ N , (3.7)ak ≤ 0 in B , ∀k ∈ N . (3.8)For any k ∈ N, consider now the Dirichlet problem{uk ∈ W2,p(B)∩◦W1,p(B) ,Lkuk = f ∈ Ln(B) .(3.9)By (3.7), (3.8) and Lemma 3.2, problem (3.9) is uniquely solvable and thesolution uk satisfies the a priori bound||uk||W 2,p(B) ≤ c||f ||Lp(B) , (3.10)with c ∈ R+ depending on n, p, d, ν, ||aij ||L∞(B), η[p(aij)], ||ai||Lr(B), ||a||Lp(B),ωr[ai, B], ωp[a,B]. Moreover, from known regularity results (see Lemma 3.3 of[4] if n = 2 and Theorem 2.2 of [13] if n ≥ 3) we deduce that uk ∈ W2,n(B)and then, using again Theorem III of [11], we have thatuk ≤ 0 in B . (3.11)On the other hand, by (3.10) we have that (uk)k∈N is a bounded sequence inW 2,p(B); so there exists a subsequence, that we still denote as (uk)k∈N , such304 P. Cavaliere, M. Transiricothatuk ⇀ u′ in W 2,p(B) ,uk −→ u′ in W 1,q(B) , 1 < q <npn − p ,uk −→ u′ in C(B̄) ,(3.12)with u′ ∈ W 2,p(B) ∩ W 1,q(B). Moreover, from (3.11), one has that u′ ≤ 0 inB.We claim that the sequence (Lkuk)k∈N weakly converges to Lu′ in Lp(B).In fact, for each ϕ ∈ Lp′(B), where 1p+ 1p′= 1, we have∫B|(Lkuk − Lu′)ϕ|dx ≤n∑i,j=1||aij ||L∞(B)∫B|((uk)xixj − u′xixj)ϕ|dx+n∑i=1||ai||Lr(B)||(uk)xi − u′xi||Lq(B)||ϕ||Lp′ (B)+ supB̄|uk − u′| ||a||Lp(B)||ϕ||Lp′ (B) + ||ak − a||Lp(B) supB̄|uk| ||ϕ||Lp′ (B) ,where q = rpr − p . So, coming back to (3.6) and (3.12), the weak convergenceof (Lkuk)k∈N to Lu′ in Lp(B) follows. Therefore Lu′ = f a.e. in B andu′ ∈ W 2,p(B)∩◦W 1,p(B), and hence the uniqueness of the solution of problem(3.4) yields that u = u′, so that u ≤ 0 in B.Step 2. We now consider f ∈ Lp(B). If we define fk = Jk ∗ ḟ , k ∈ N, whereḟ is the extension of f to Rn with zero values out of B, then the sequence(fk)k∈N satisfies the following properties:fk −→ f in Lp(B) , (3.13)||fk||Lp(B) ≤ ||f ||Lp(B) , ∀k ∈ N , (3.14)fk ≥ 0 in B , ∀k ∈ N . (3.15)If, for any k ∈ N, we consider the Dirichlet problem{uk ∈ W2,p(B)∩◦W1,p(B) ,Luk = fk ,(3.16)then Lemma 3.2 assures that (3.16) is uniquely solvable and the solution uksatisfies the a priori bound||uk||W 2,p(B) ≤ c||fk||Lp(B) ≤ c||f ||Lp(B) . (3.17)Moreover, from the previous step, we also have that uk ≤ 0 in B. Then,employing the same arguments of the proof of the previous step, we obtainagain that u ≤ 0 in B.A STRONG MAXIMUM PRINCIPLE FOR... 305Let µ0, µ1, µ2 ∈ R+ such that:n∑i,j=1||aij ||L∞(B) ≤ µ0 , d1−nrn∑i=1||ai||Lr(B) ≤ µ1 , d2−np ||a||Lp(B) ≤ µ2.Using Lemmas 3.1, 3.2 and 3.3, we are able to improve Lemma 3.2 of [3]and also to extend it to the case n = 2.Lemma 3.4. Suppose that condition (hB) holds. If u is a solution of theproblemu ∈ W 2,p(B) ,Lu ≥ f ∈ Lp(B) ,u|∂B ≤ 0 ,(3.18)then there exists C ∈ R+ such thatsupBu ≤ C d2−np ||f−||Lp(B) , (3.19)with C dependent on n, p, ν, µ0, µ1, µ2, [p(aij)]BMO(Rn,·), ωr[d1−nr ai, B],ωp[d2−np a,B], and where p(aij) is an extension of aij to Rn inL∞(Rn) ∩ V MO(Rn).Proof. Let B = B(x0, d), where x0 is the centre of B, and put B∗ =B(x0, 1). Consider the trasformation T : B −→ B∗ defined by the positionT (x) = x0 +x − x0d= z ,and for each function g defined on B, put g∗ = g ◦ T−1.Observe thatL∗u∗ = d2(Lu)∗ ,whereL∗ =n∑i,j=1a∗ij∂2∂zi∂zj+ dn∑i=1a∗i∂∂zi+ d2a∗ .Denote by p(aij) an extension of aij to Rn such thatp(aij) ∈ L∞(Rn) ∩ V MO(Rn),(i, j = 1, . . . , n). Sincep(aij)∗ ∈ L∞(Rn) ∩ V MO(Rn) , p(aij)∗|B∗ = a∗ij, (3.20)we also havea∗ij ∈ L∞(B∗) ∩ V MO(B∗) (3.21)306 P. Cavaliere, M. Transirico(see [12]). Moreover, the hypothesis (hB) yields thata∗ij = a∗ji , i, j = 1, . . . , n ,n∑i,j=1a∗ijξiξj ≥ ν|ξ|2 a.e. in B∗ , ∀ξ ∈ Rn ,a∗i ∈ Lr(B∗), i = 1, . . . , n, a∗ ∈ Lp(B∗), a∗ ≤ 0 a.e. in B∗ .(3.22)Consider now the following problem:{L∗v = h ∈ Lp(B∗) ,v ∈ W 2,p(B∗)∩◦W1,p(B∗) .(3.23)It follows from (3.20), (3.22) and from Lemma 3.2 that there exists a uniquesolution v of (3.23) satisfying the estimate||v||W 2,p(B∗) ≤ K||h||Lp(B∗) , (3.24)where K depends on n, p, ν, ||a∗ij ||L∞(B∗), [p(aij)∗]BMO(Rn,·), ||da∗i ||Lr(B∗),||d2a∗||Lp(B∗), ωr[da∗i , B∗], ωp[d2a∗, B∗]. Hence the constant K depends on n, p,ν, µ0, [p(aij)]BMO(Rn,·), µ1, µ2, ωr[d1−nr ai, B], ωp[d2−np a,B].Thus by (3.24) we obtain that there is K1 ∈ R+, depending on the sameparameters on which K does, such thatmaxB̄∗|v| ≤ K1||h||Lp(B∗) , (3.25)and hence for each z ∈ B∗ there is g(z, ·) ∈ Lp′(B∗)(1p+ 1p′= 1)for whichv(z) = −∫B∗g(z, y)h(y)dy . (3.26)The map g(z, ·) is the Green function for the operator L∗ in B∗. From (3.21),(3.22), (3.26) and from Lemma 3.3 we deduce that∫B∗g(z, y)h̃(y)dy ≥ 0 ∀h̃ ∈ Lp(B∗) , h̃ ≥ 0 ; (3.27)moreover, by (3.25) and (3.26) we have that||g(z, ·)||Lp′ (B∗) ≤ K1 . (3.28)Setting h = L∗u∗ = d2(Lu)∗ in (3.23), we have that v − u∗(∈ W 2,p(B∗)) is asolution of the problem{L∗(v − u∗) = 0 in B∗ ,(v − u∗)|∂B∗ = −u∗|∂B∗ ≥ 0 ,and so it follows from (3.21), (3.22) and from Lemma 3.1 that v − u∗ ≥ 0 inA STRONG MAXIMUM PRINCIPLE FOR... 307B∗. Thus, applying (3.26) with h = L∗u∗, it follows from (3.27) and (3.28) thatu∗(z) ≤ −∫B∗g(z, y)d2(Lu)∗(y)dy (3.29)≤ −d2∫B∗g(z, y)f∗(y)dy ≤ −2d2∫B∗g(z, y)(f∗)−(y)dy≤ 2d2||g(z, ·)||Lp′ (B∗)||(f∗)−||Lp(B∗) ≤ 2d2K1||(f−)∗||Lp(B∗) ∀z ∈ B∗ .It is now easy to deduce the statement from (3.29).4. Main ResultsLet Ω be an open (bounded or not) subset of Rn, n ≥ 2, with the property(2.1), and let p ∈]n2,+∞[. Consider in Ω the operatorL =n∑i,j=1aij∂2∂xi∂xj+n∑i=1ai∂∂xi+ a , (4.1)and putL0 =n∑i,j=1aij∂2∂xi∂xj.On the coefficients of L we make the following assumptions:aij = aji ∈ L∞loc(Ω) ∩ V MOloc(Ω) , i, j = 1, . . . , n ,ai ∈ Lrloc(Ω), i = 1, . . . , n, where r > n if p ≤ n, r = p if p > n,a ∈ Lploc(Ω) ,∃ν ∈ L∞loc(Ω) : ν(x) > 0 a.e. in Ω ,n∑i,j=1aijξiξj ≥ ν(x)|ξ|2 a.e. in Ω , ∀ξ ∈ Rn ,for any open subset E ⊂⊂ Ω , ess infEν > 0 , ess supEa < 0.(h)Our aim is to prove the following result.Theorem 4.1. Assume that (h) holds. If u is a solution of the problemu ∈ W 2,ploc(Ω) , Lu ≥ 0 , (4.2)then u does not have any positive relative maximum in Ω.Proof. By contradiction, suppose that there exists some point of positive308 P. Cavaliere, M. Transiricorelative maximum y in Ω for u. Since u ∈ C0(Ω), there exists d ∈ R+ such that:B(y, d) ⊂⊂ Ω , 0 < u(x) ≤ u(y) ∀x ∈ B(y, d) . (4.3)Puta0 = − ess supB(y,d)a , µ0 = ess supB(y,d)n∑i,j=1aij , ν0 = ess infB(y,d)ν (4.4)and fix α ∈ R+ such thatα ≥(10µ0a0)12. (4.5)For any λ ∈]0, min(1, dα)]we haveB = B(y, αλ) ⊆ B(y, d) . (4.6)Moreover, putϕ(x) = ϕλ(x) = 1 + λ2 −|x − y|2α2, (4.7)x ∈ B(y, αλ) and λ ∈]0, min(1,dα)],and see that1 ≤ ϕ ≤ 1 + λ2 . (4.8)Consider now the map v defined by the positionv(x) = ϕ(x)u(x) − u(y) , x ∈ B(y, αλ) . (4.9)Note thatL0(ϕu) = ϕL0u + uL0ϕ + 2n∑i,j=1aijϕxjuxi ,n∑i=1ai(ϕu)xi = ϕn∑i=1aiuxi + un∑i=1aiϕxi ,soϕLu = ϕL0u + ϕn∑i=1aiuxi + ϕau (4.10)= L0(ϕu) − uL0ϕ − 2n∑i,j=1aijϕxjuxi+n∑i=1ai(ϕu)xi − un∑i=1aiϕxi + aϕu ≥ 0 in B.A STRONG MAXIMUM PRINCIPLE FOR... 309Sinceuxiϕxj =ϕxj !ϕ(ϕu)xi −ϕxiϕxjϕ2(ϕu) ,from (4.10) it follows thatL0(ϕu) +n∑i=1bi(ϕu)xi + cϕu ≥ un∑i=1aiϕxi in B , (4.11)wherebi = ai − 2n∑j=1aijϕxjϕ, i = 1, . . . , n , (4.12)andc = a −n∑i,j=1aijϕxixjϕ+ 2n∑i,j=1aijϕxiϕxjϕ2. (4.13)Coming back to (4.9) and (4.11), we getL0v +n∑i=1bivxi + cv ≥ un∑i=1aiϕxi − cu(y) in B . (4.14)Moreover, as ϕxi = −2(xi−yi)α2, ϕxixj = 0 if i 6= j, ϕxixj = −2α2if i = j, we havec ≤ −a0 + 2∑ni=1 aiiϕ·!1α2+ 8λ2∑ni,j=1 aijϕ2·1α2≤ −a0 + (10n∑i,j=1aij) ·1α2in Band hence it follows from (4.5) that c ≤ 0 in B.Finally, since ϕ|∂B(y,αλ) = 1, v satisfies the following condition:v|∂B = u|∂B − u(y) ≤ 0 . (4.15)Therefore by (4.12)-(4.15) and by Lemma 3.4 we obtain:supΩv ≤ C(αλ)2−np ||(un∑i=1aiϕxi − cu(y))−||Lp(B) , (4.16)where the constant C depends on n, p, ν0, µ0, d1−nr∑ni=1 ||ai||Lr(B(y,d)),d2−np ||a||Lp(B(y,d)), [p(aij)]BMO(Rn,·), ωr[d1−nr ai, B(y, d)], ωp[d2−np a,B(y, d)]. Itfollows from (4.16) that:supΩv ≤ C1λ2−np ||un∑i=1aiϕxi ||Lp(B) , (4.17)with C1 depending on the same parameters as C and on a0. In particular,310 P. Cavaliere, M. Transirico(4.17) yieldsu(y) ≤ C1λ−np u(y)||n∑i=1aiϕxi ||Lp(B) . (4.18)Since |ϕxi | ≤2α· λ, applying the Hölder inequality if p ≤ n, from (4.18) itfollows thatu(y) ≤ C2u(y)λ1−nrn∑i=1||ai||Lr(B) ≤ C2u(y)n∑i=1||ai||Lr(B) , (4.19)with C2 ∈ R+ depending on the same parameters as C1. On the other hand,we can choose λ small enough, such that∑ni=1 ||ai||Lr(B) ≤12C2. Therefore itfollows from (4.19) thatu(y) ≤ 0,that is a contradiction.The following uniqueness result is an obvious consequence of Theorem 4.1.Corollary 4.2. If (h) is satisfied, the problemu ∈ W 2,ploc(Ω) , Lu = 0 ,limx−→x0 u(x) = 0 ∀x0 ∈ ∂Ω ,lim|x|−→+∞ u(x) = 0 if Ω is unbounded,(4.20)has only the zero solution.Proof. If we suppose M = supΩu > 0, it results M = u(y) for some y ∈ Ωand then u has a positive relative maximum in Ω. But this is not possible,by virtue of Theorem 4.1. Hence supΩu ≤ 0. Since −u verifies the sameconditions of u, with the same arguments used in the above case, we have alsothat supΩ(−u) ≤ 0 and then we deduce that u = 0 in Ω.Remark 4.3. 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Vitolo, BMO spaces on domains of Rn,Ricerche Mat., 45 (1996), 355-378.[13] C. Vitanza, W2,p-regularity for a class of elliptic second order equationswith discontinuous coefficients, Matematiche (Catania), 47 (1992), 177-186.[14] C. Vitanza, A new contribution to the W2,p-regularity for a class of el-liptic second order equations with discontinuous coefficients, Matematiche(Catania), 48 (1993), 287-296.312

A strong maximum principle for linear elliptic operators

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A strong maximum principle for linear elliptic operators

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