Báo cáo hóa học: " Research Article Hölder Regularity of Solutions to Second-Order Elliptic Equations in Nonsmooth Domains Sungwon Cho and Mikhail Safonov" pot

Hindawi Publishing Corporation Boundary Value Problems Volume 2007, Article ID 57928, 24 pages doi:10.1155/2007/57928 Research Article Hölder Regularity of Solutions to Second-Order Elliptic Equations in Nonsmooth Domains Sungwon Cho and Mikhail Safonov Received 16 March 2006; Revised 25 April 2006; Accepted 28 May 2006 Recommended by Ugo Pietro Gianazza We establish the global H ¨ older estimates for solutions to second-order elliptic equations, which vanish on the boundar y, while the right-hand side is allowed to be unbounded. For nondivergence elliptic equations in domains satisfying an exterior cone condition, similar results were obtained by J. H. Michael, who in turn relied on the barrier techniques due to K. Miller. Our approach is based on special growth lemmas, and it works for both divergence and nondivergence, elliptic and parabolic e quations, in domains satisfying a general “exterior measure” condition. Copyright © 2007 S. Cho and M. Safonov. This is an open access article distributed un- der the Creative Commons Attribution License, which permits unrestricted use, distri- bution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In the theory of partial differential equations, it is important to have estimates of solu- tions, which do not depend on the smoothness of the given data. Such kind of estimates include different versions of the maximum principle, which are crucial for investigation of boundary value problems for second-order elliptic and parabolic equations. More del- icate properties of solutions, such as H ¨ older estimates and Harnack inequalities, are very essential for the building of general theory of nonlinear equations (see [1–6]). In this paper, we establish the global H ¨ older regularity of solutions to the Dirichlet problem,orthefirst boundar y value problem, for second-order elliptic equations. We deal with the Dirichlet problem Lu = f in Ω, u = 0on∂Ω. (DP) Here Ω is a bounded open set in R n ,n ≥ 1, satisfying the following “exterior measure” condition (A). This condition appeared in the books [4, 5]. 2 Boundary Value Problems Definit ion 1.1. An open set Ω ⊂ R n satisfies the condition (A) if there exists a constant θ 0 > 0, such that for each y ∈ ∂Ω and r>0, the Lebesgue measure   B r (y) \ Ω   ≥ θ 0   B r   ,(A) where B r (y)istheballofradiusr>0, centered at y. We deal simultaneously with the cases w hen the elliptic operator L in (DP) is either in the divergence form: Lu : =−(D,aDu) =−  i, j D i  a ij D j u  ,(D) or in the nondivergence form: Lu : =−(aD,Du) =−  i, j a ij D ij u, (ND) where D j u := ∂u/∂x j , D ij u := D i D j u,anda = [a ij ] = [a ij (x)] is a matrix function with real entries, which satisfies the uniform ellipticity condition (aξ,ξ) ≥ ν|ξ| 2 ∀ξ ∈ R n , a := max |ξ|≤1 |aξ|≤ν −1 ,(U) with a constant ν ∈ (0,1]. In (D), (ND), and throughout this paper, D is a symbolic col- umn vector with components D i := ∂/∂x i , which helps to write explicit expressions for Lu in a shorter form. Note that the conditions (U) are invariant with respect to rotations in R n ,andν = 1, if and only if −L = Δ :=  i D ii —the Laplace operator. Indeed, from (U) with ν = 1, it follows |ξ| 2 ≤ (aξ,ξ) ≤|aξ|·|ξ|≤|ξ| 2 ∀ξ ∈ R n ; (1.1) hence |aξ|≡|ξ|,(aξ,ξ) ≡|ξ| 2 , which is possible if and only if a = I = the identity matrix, so that L =−Δ. The notations (·,·)and|·|are explained at the end of this section. For operators L in the divergence form (D), it has been proved by Littman et al. [7]that the boundary points of Ω are regular if and only if they are regular for L =−Δ.Inparticu- lar, isolated points cannot be regular in the divergence case (D). On the other hand, from the results by Gilbarg and Serrin in [8, Section 7], it follows that the functions u(x): =|x| γ and γ = const ∈ (0,1) satisfy the equation Lu = 0inΩ :={x ∈ R n :0< |x| < 1}, n ≥ 2, with some operators L in the nondivergence form (ND). For such operators, the bound- ary regularity of solutions to problem (DP) is usually investigated by the standard method of barrier functions. However, this method requires certain smoothness of the boundary ∂Ω.FordomainsΩ satisfying an exterior cone condition, such barrier functions were constructed by M iller [9], and his construction was then widely used by many authors. In par t icular, Michael [10, 11] used Miller’s technique in his general Schauder-type exis- tence theory, which is based on the interior estimates only. One of the key elements in his theory is the following estimate for solutions to problem (DP): sup Ω d −γ |u|≤NF,whereF := sup Ω d 2−γ | f |,(M) S. Cho and M. Safonov 3 d = d(x):= dist(x, ∂Ω), and the constants γ ∈ (0,1) and N>0dependonlyonn, ν,and the characteristics of exterior cones. Note that the function f is allowed to be unbounded near ∂Ω. At about the same time, Gilbarg and H ¨ ormander [12] also used Miller’s barriers in their theory of intermediate Schauder estimates. Once again, Schauder estimates in Lipschitz domains are treated there on the grounds of estimates similar to (M) (see [12, Lemma 7.1 ]). All these results deal with operators L in the nondivergence form (ND). Our method is applied to general domains satisfying the “exterior measure” condition (A), and it works for both divergence and nondivergence equations. However, the natural functional spaces for solutions in these two cases are different. We use the same notation W(Ω) for classes of solutions, which are different in the case (D)or(ND), in order to treat these cases simultaneously. The classes W(Ω) are introduced in Definition 2.1 at the beginning of Section 2. In the rest of Section 2, we discuss the three basic facts: (i) maximum principle (Lemma 2.2), (ii) pointwise estimate (Lemma 2.4), and (iii) growth lemma (Lemma 2.5). Growth lemmas originate from methods of Landis [13]. They were essentially used in the proof of the interior Harnack inequality for solutions to elliptic and parabolic equations in the non-divergence form (ND) (see [3, 14, 15]). One can also use growth lemmas for an alternative proof of Moser’s Harnack inequality in the divergence case (D); see [16, 17]. In Section 3,weproveestimate(M)with0<γ<γ 1 ≤ 1, where γ 1 depends only on the dimension n, the ellipticity constant ν in (U), and the constant θ 0 > 0 in the condition (A). This estimate, together with the interior H ¨ older regularity of solutions implies the g lobal estimates for solutions to problem (DP)intheH ¨ older space C 0,γ (Ω), with an appropriate γ>0. Remark 1.2. Estimate (M) means that from f = O(d γ−2 ), 0 <γ<γ 1 ≤ 1, it follows u = O(d γ ) and in particular, u → 0asd = d(x) → 0 + .Theassumption0<γ<1 is essential even in the one-dimensional case: −u  = f := d γ−2 =  1 −|x|  γ−2 in Ω = (−1,1), u(±1) = 0. (1.2) Indeed, if γ ≤ 0, then any solution to the equation −u  = f blows to +∞ near ∂Ω = { 1,−1}.Ifγ>1, then this problem has a unique solution u,butestimate(M) cannot hold, because it implies the equalities u  (±1) = 0, conflicting the properties u(±1) = 0andu  < 0in( −1,1). Finally, in the case γ = 1, from (M)andu(±1) = 0itfollows|u  (±1)|≤NF, while −u  = d −1 implies that u  (±1) are unbounded. Therefore, the restrictions 0 <γ<1 are necessary for validity of estimate (M). They are also sufficient for operators L in the form (ND) and the boundary ∂Ω of class C 2 (see [10]). In Theorem 3.9, we extend this result to domains Ω satisfying an exterior sphere condition. The proof of this theorem uses elementary comparison arguments only. Basic notations. R n is the n-dimensional Euclidean space, n≥1, with points x=(x 1 , ,x n ) t , where x i are real numbers. Here the symbol t stands for the transposition of vectors, which indicates that vectors in R n are treated as column vectors. For x = (x 1 , ,x n ) t and y = (y 1 , , y n ) t in R n ,thescalar product (x, y):=  x i y i ,thelength of x is |x| := (x, x) 1/2 . For y ∈ R n , r>0, the ball B r (y):={x ∈ R n : |x − y| <r}. Du := (D 1 u, ,D n u) t ∈ R n , where D i := ∂/∂x i . 4 Boundary Value Problems Let Ω be an open set in R n .For1≤ p ≤∞ and k = 0,1, ,W k,p (Ω) denotes the Sobolev space of functions, which belongs to the Lebesgue space L p (Ω) together with all its derivatives of order ≤ k. The norm of functions u ∈ W k,p (Ω)isdefinedasu W k,p (Ω) :=  |l|≤k D l u p,Ω , where summation is taken over all multi-indices (vectors with nonneg- ative integer components) l = (l 1 , ,l n )oforder|l| := l 1 + ···+ l n . In this expression, D l u := D l 1 1 ···D l n n u,and f  p,Ω is the norm of f in L p (Ω), that is,  f  p p,Ω :=  Ω | f | p dx for 1 ≤ p<∞;  f  ∞,Ω := esssup Ω | f |. (1.3) Furthermore, W k,p loc (Ω) denotes the class of functions which belong to W k,p (Ω  )forarbi- trary open subsets Ω  ⊂ Ω  ⊂ Ω. ∂Γ is the boundary of a set Γ in R n , Γ := Γ ∪ ∂Γ is the closure of Γ,anddiamΓ := sup{|x − y| : x, y ∈ Γ}—the diameter of Γ.Moreover,|Γ| :=|Γ| n is the n-dimensional Lebesgue measure of a measurable set Γ in R n . c + := max(c,0), c − := max(−c,0), where c is a real number. “A : = B”or“B =: A” is the definition of A by means of the expression B. N = N(···) denotes a constant depending only on the prescribed quantities, such as n, ν, and so forth, which are specified in the parentheses. Constants N in different expressions may be different. For convenience of cross-references, we assign indices to some of them. 2. Auxiliary statements Let Ω be a bounded open set in R n ,andletL be an elliptic operator in the form (D)or (ND)withcoefficients a ij = a ij (x) satisfying the uniform ellipticity condition (U)with aconstantν ∈ (0,1]. Using the notation for Sobolev spaces W k,p (Ω), we introduce the class of functions W(Ω), which depends on the case ( D)or(ND). Definit ion 2.1. (i) In the divergence case (D), W(Ω): = W 1,2 loc (Ω) ∩ C(Ω). Functions u ∈ W(Ω)and f ∈ L 2 loc (Ω) satisfy Lu :=−( D,aDu) ≤ (≥,=) f in Ω (in a weak sense) if  Ω (Dφ,aDu)dx ≤ (≥,=)  Ω φf dx for any function φ ∈ C ∞ 0 (Ω), φ ≥ 0. (2.1) If Lu = f ,then(2.1) holds for all functions φ ∈ C ∞ 0 (Ω)(φ can change sig n). (ii) In the non-divergence case (ND), W(Ω): = W 2,n loc (Ω) ∩ C(Ω). For u ∈ W(Ω)and measurable functions f on Ω,therelationsLu : =−(aD,Du) ≤ (≥,=) f in Ω (in a s trong sense) are understood almost everywhere (a.e.) in Ω. By approximation, the property (2.1) is easily extended to nonnegative functions φ ∈ W 1,2 (Ω) with compact support in Ω.Ifu ∈ W 1,2 (Ω) ∩ C(Ω), then (2.1)holdstruefor φ ∈ W 1,2 0 (Ω)—the closure of C ∞ 0 (Ω)inW 1,2 (Ω). Lemma 2.2 (maximum principle). Let u be a function in W(Ω) satisfying Lu ≤ 0 in Ω. Then sup Ω u = sup ∂Ω u. (2.2) S. Cho and M. Safonov 5 This is a well-known classical result. It is contained, for example, in [2,Theorem8.1 (case (D)) and Theorem 9.1 (case (ND))]. Since our assumptions in the case (D)are slightly different from those in [2], we give a sketch of the proof. Proof(inthecase(D)). Suppose the equality (2.2) fails, that is, the left-hand side in (2.2) is strictly larger than the right-hand side. Replacing u by u − const, we can assume that the set Ω  := Ω ∩{u>0} is not empty, and u<0on∂Ω. Then automatically u = 0on ∂Ω  . Approximating u + := max (u,0) in W 1,2 (Ω) by functions φ ∈ C ∞ 0 (Ω), one can see that the inequality (2.1) holds with φ = u + and f = 0. This yields ν  Ω  |Du| 2 dx ≤  Ω  (Du,aDu)dx ≤ 0. (2.3) Hence Du = 0andu = const on each open connected component of Ω  .Sinceu = 0 on ∂Ω  ,wemusthaveu ≡ 0inΩ  , in contradiction to our assumption Ω  := Ω ∩{u> 0 } =∅.  Applying this lemma to the function u − v, we immediately get the following. Corollary 2.3 (comparison principle). If u,v ∈ W(Ω) satisfy Lu ≤ Lv in Ω,andu ≤ v on ∂Ω,thenu ≤ v in Ω. Lemma 2.4 (pointwise estimate). (i) For an arbitrary elliptic ope rator L (in the form (D) or (ND)) with coefficients a ij which are defined on a ball B R := B R (x 0 ) ⊂ R n and satisfy (U) w ith a constant ν ∈ (0,1], there exists a function w ∈ W(B R ) such that 0 ≤ w ≤ N 0 R 2 , Lw ≥ 1 in B R ; w = 0 on ∂B R , (2.4) where the constant N 0 = N 0 (n,ν). (ii) Moreover, for an arbitrary open s et Ω ⊆ B R and an arbitrary function u ∈ W(Ω), sup Ω u ≤ sup ∂Ω u + N 0 R 2 · sup Ω (Lu) + . (2.5) Proof. (i) By rescaling x → R −1 x, we reduce the proof to the case R = 1. In the divergence case (D), consider the Dirichlet problem Lw : =−(D,aDw) = 1inB 1 ; w = 0on∂B 1 . (2.6) It is known (see [2, Theorems 8.3 and 8.16]) that there exists a unique solution w to this problem, which belongs to W 1,2 (B 1 ) ∩ C(B 1 ) ⊂ W(B 1 ) and satisfies 0 ≤ w ≤ N 0 = N 0 (n,ν)onB 1 . This function w satisfies all the properties (2.4)(withR = 1). In the nondivergence case (ND), we take w(x): = (2nν) −1 · (1−|x−x 0 | 2 ). Since tra :=  i a ii ≥ nν,wehave Lw : =−(aD,Dw) = (nν) −1 · tra ≥ 1inB 1 , w = 0on∂B 1 , (2.7) so that (2.4) holds with N 0 := supw = (2nν) −1 . 6 Boundary Value Problems (ii) We will compare u = u(x) with the function v = v(x):= sup ∂Ω u +sup Ω (Lu) + · w(x). (2.8) We have Lu ≤ sup Ω (Lu) + ≤ Lv in Ω, u ≤ sup ∂Ω u ≤ v on ∂Ω. (2.9) By the comparison principle, u ≤ v in Ω.Sincew ≤ N 0 R 2 , the inequality (2.5)follows.  Lemma 2.5 (growth lemma). Let x 0 ∈ R n and let r>0 be such that the Lebesgue measure   B r \ Ω   ≥ θ   B r   , θ>0, (2.10) where B r := B r (x 0 ).Thenforanyfunctionu ∈ W(Ω), sat isfying u>0, Lu ≤ 0 in Ω,and u = 0 on (∂Ω) ∩ B 4r , sup B r u ≤ β · sup B 4r u = β · sup ∂B 4r u, (2.11) where β = β(n,ν,θ) ∈ (0,1). Assume that u is extended as u ≡ 0 on B 4r \ Ω, s o that both sides of (2.11) are always well defined. The last equality in (2.11) is a consequence of the maximum principle. In the divergence case (D), Lemma 2.5 (in equivalent formulations) is contained in [13, Chapter 2, Lemma 3.5], or in [17, formula (39)]. In the n ondivergence case (ND), this follows from [15, Corollary 2.1]. In dealing with these references, or more generally, with different versions of growth lemmas, one can always impose the additional simpli- fying assumptions. Assumptions 2.6. (i) The function u is defined on the whole ball B 4r insuchawaythat u ∈ W  B 4r  , Lu ≤ 0inB 4r , (2.12) and Ω : = B 4r ∩{u>0} satisfies   B r \ Ω   =   B r ∩{u ≤ 0}   >θ   B r   , θ>0. (2.13) (ii) All the functions a ij and u belong to C ∞ (B 4r ). Here we show that if the previous lemma is true under these additional assumptions, then it holds true in its original form. We proceed in two steps accordingly to parts (i), extension of u from Ω ∩ B 4r to B 4r , and (ii), approximation of a ij and u by smooth func- tions. (i) Extension to B 4r . Fix ε>0 and choose a function G ∈ C ∞ (R 1 )(dependingonε)such that G,G  ,G  ≥ 0onR 1 , G ≡ 0on(−∞,ε], G  ≡ 1on[2ε,∞). (2.14) S. Cho and M. Safonov 7 Further , define u ε := G(u)inΩ ∩ B 4r , u ε ≡ 0onB 4r \ Ω. (2.15) From the above properties of the function G it follows (u − 2ε) + ≤ u ε ≤ (u − ε) + in Ω. (2.16) Since u = 0 on the set (∂Ω) ∩ B 4r , the function u ε vanishes near this set. Hence in both cases (D)and(ND), we have u ε ∈ W(B 4r )andu ε ≥ 0inB 4r .Moreover,weclaimthat Lu ε ≤ 0inB 4r . In the non-divergence case (ND), this follows immediately from Lu ε ≡ 0 on B 4r \ Ω and Lu ε = LG(u) = G  (u) · Lu − G  (u) · (Du,aDu) ≤ 0inΩ. (2.17) In the divergence case (D), the inequality Lu ε ≤ 0 is understood in a weak sense (2.1). Let φ be an arbitrary nonnegative function in C ∞ 0 (B 4r ). Then the function φ 0 := φ · G  (u) is also non-negative, belongs to W 1 2 (Ω), and has compact support in Ω ∩ B 4r . By approx- imation, we can put φ 0 in place of φ in the inequality (2.1) corresponding to Lu ≤ 0inΩ, that is,   Dφ 0 ,aDu  dx ≤ 0. (2.18) Having in mind that Du ε = DG(u) = G  (u)Du (see [2, Section 7.4]), and Dφ 0 = D  φG  (u)  = G  (u)Dφ + φG  (u)Du, (2.19) we obtain   Dφ,aDu ε  dx =  G  (u) · (Dφ,aDu)dx =   Dφ 0 ,aDu  dx −  φG  (u) · (Du,aDu)dx ≤ 0. (2.20) Since this is true for any φ ∈ C ∞ 0 (B 4r ), φ ≥ 0, it follows Lu ε :=−(D,aDu ε ) ≤ 0inB 4r (in a weak sense). Now suppose that Lemma 2.5 is true under additional Assumptions 2.6(i). For any small ε>0, we can apply this weaker formulation to the function u ε := G(u)inΩ ε := { u ε > 0}∩B 4r .Weknowthatu ε ∈ W(B 4r )andLu ε ≤ 0inB 4r .Moreover,estimate(2.10) for Ω implies a bit stronger estimate (2.13)forΩ ε ⊂ Ω. In addition, obviously u ε = 0on (∂Ω ε ) ∩ B 4r . Hence the functions u ε satisfy estimate (2.11) with the same β = β(n,ν,θ) ∈ (0,1). By virtue of (2.16), u ε → u as ε → 0 + ,uniformlyonΩ, and we get estimate (2.11) under the original assumptions in Lemma 2.5. (ii) Approximation by smooth functions. The additional Assumptions 2.6(i) help in ap- proximation of a ij and u by smooth functions. Note that since both sides of (2.13)are continuous with respect to r,wealsohave   { u ≤ 0}∩B ρ   >θ   B ρ   (2.21) 8 Boundary Value Problems for all ρ<rwhich are close enough to r.Fixsuchρ<rand approximate a ij by convolu- tions a (ε) ij ,0<ε<ε 0 := r − ρ>0, which are defined in a standard way: f (ε) (x):=  η ε ∗ f  (x):=  η ε (x − y) f (y)dy =  f (x − y)η ε (y)dy. (2.22) Here η ε are fixed functions satisfying the properties η ε ∈ C ∞ (R n ), η ε ≥ 0inR n , η ε (x) ≡ 0for|x|≥ε,  η ε dx = 1. (2.23) Then a (ε) ij ∈ C ∞ (B 4ρ ) and the matrices a (ε) := [a (ε) ij ] satisfy the uniform ellipticity condi- tion (U) with the same constant ν. Further, we consider the cases (D)and(ND) separately. Divergence case (D). Denote r 0 := 4ρ + ε 0 < 4r.Fromu ∈ W(B 4r ):= W 1,2 loc (B 4r ) ∩ C(B 4r ) it follows u ∈ W 1,2 (B r 0 ) ∩ C(B r 0 )andaDu ∈ L 2 (B r 0 ). Hence the functions f ε :=−  D,(aDu) (ε)  ∈ C ∞  B 4ρ  ,0<ε<ε 0 . (2.24) Without loss of generality, assume x 0 = 0. Then for fixed x ∈ B 4ρ = B 4ρ (0) and 0 <ε<ε 0 , the function φ(y): = η ε (x − y) is non-negative, belongs to C ∞ , and has compact support in B r 0 .SinceLu :=−(D,aDu) ≤ 0inB r 0 ,andDφ(y) =−Dη ε (x − y), we have f ε (x):=−  D,   η ε (x − y), aDu(y)  dy  =   Dφ(y),aDu(y)  dy ≤ 0 (2.25) for x ∈ B 4ρ and 0 <ε<ε 0 . In terms of Schwartz distributions, this property simply means that Lu ≤ 0 implies f ε = (Lu) (ε) ≤ 0. Next, consider the Dirichlet problem L ε u ε :=−  D, a (ε) Du ε  = f ε in B 4ρ , u ε = u (ε) on ∂B 4ρ , (2.26) where 0 <ε<ε 0 .Herea (ε) , f ε ,andu (ε) belong to C ∞ (B 4ρ ), so that this problem has a unique classical solution u ε , which b elongs to C ∞ (B 4ρ ) (see, e.g., [2, Theorem 6.19]). Then the functions v ε := u ε − u (ε) , g ε := (aDu) (ε) − a (ε) Du (ε) ∈ C ∞  B 4ρ  , (2.27) and v ε = 0on∂B 4ρ . Integrating by par ts over the ball B 4ρ , and then applying the Cauchy- Schwartz inequality, we derive   Dv ε ,a (ε) Du ε  dx =  v ε L ε u ε dx =  v ε f ε dx =   Dv ε ,(aDu) (ε)  dx, ν    Dv ε   2 dx ≤   Dv ε ,a (ε) Dv ε  dx =   Dv ε ,a (ε) Du ε − a (ε) Du (ε)  dx =   Dv ε ,g ε  dx ≤ ν 2    Dv ε   2 dx + 1 2ν  g 2 ε dx. (2.28) S. Cho and M. Safonov 9 It follows  | Dv ε | 2 dx ≤ ν −2  | g ε | 2 dx, and then by the Poincar ´ e inequality,    v ε   2 dx ≤ N(n,ν,ρ) ·  g 2 ε dx,0<ε<ε 0 . (2.29) Further, we will use the property of convolution: for any open set Ω ⊂ R n ,andany bounded open subset Ω  ⊂ Ω  ⊂ Ω,wehave h (ε) −→ h in L p (Ω  )asε −→ 0 + ,ifh ∈ L p (Ω), 1 ≤ p<∞; h (ε) −→ h a.e. in Ω  as ε −→ 0 + ,ifh ∈ L p (Ω), 1 ≤ p ≤∞; h (ε) −→ h in L ∞ (Ω  )asε −→ 0 + ,ifh ∈ C(Ω). (2.30) In our case Ω  := B 4ρ ⊂ Ω := B r 0 .Wewriteg ε = g 1,ε + g 2,ε + g 3,ε ,where g 1,ε := (aDu) (ε) − aDu, g 2,ε := aDu − a (ε) Du, g 3,ε := a (ε) Du − a (ε) Du (ε) . (2.31) From a ∈ L ∞ (B r 0 )andDu,aDu ∈ L 2 (B r 0 ), it follows g 1,ε → 0inL 2 (B 4ρ ). We also have a (ε) → a a.e. in B 4ρ , and by the dominated convergence theorem, g 2,ε → 0inL 2 (B 4ρ ). Finally, since all the matrices a (ε) satisfy (U) with same constant ν,thenormofg 3,ε in L 2 (B 4ρ ),   g 3,ε   2 ≤   a (ε)   ·   Du − Du (ε)   2 ≤ ν −1 ·   Du − (Du) (ε)   2 −→ 0; (2.32) therefore,   g ε   2 ≤   g 1,ε   2 +   g 2,ε   2 +   g 3,ε   2 −→ 0asε −→ 0 + . (2.33) By virtue of (2.29), v ε  2 → 0asε → 0 + . Furthermore, since u ∈ C(B r 0 ), the convolutions u (ε) → u uniformly on B 4ρ , which implies convergence in L 2 (B 4ρ ). Summarizing the above arguments, we obtain u ε = v ε + u (ε) −→ u in L 2  B 4ρ  as ε −→ 0 + . (2.34) Fix a small constant h>0, and note that u ε − u>h on S ε,h,ρ :={u ε >h, u ≤ 0}∩B ρ . (2.35) By virtue of (2.34), the measure   S ε,h,ρ   ≤ h −2  S ε,h,ρ  u ε − u  2 dx ≤ h −2  B ρ  u ε − u  2 dx −→ 0asε −→ 0 + . (2.36) Now from {u ε ≤ h}∩B ρ ⊇ ({u ≤ 0}∩B ρ ) \ S ε,h,ρ and (2.21), it follows    u ε ≤ h  ∩ B ρ   ≥   { u ≤ 0}∩B ρ   −   S ε,h,ρ   >θ   B ρ   , (2.37) provided ε>0 is small enough. 10 Boundary Value Problems Now suppose that Lemma 2.5 is true for smooth a ij and u. We can apply it to the function u ε − h which satisfies L ε (u ε − h) = f ε ≤ 0inB 4ρ .ByLemma 2.2,themaximum of u ε on B 4ρ is attained on the boundary ∂B 4ρ ,sothatforsmallε>0, sup B ρ  u ε − h  ≤ β · sup B 4ρ  u ε − h  <β· sup B 4ρ u ε = β · sup ∂B 4ρ u (ε) ≤ β · sup B 4ρ u (ε) . (2.38) Further, since u is continuous on B ρ ,wehave sup B ρ u<u+ h on an open nonempty set O ⊆ B ρ . (2.39) From the convergence u ε → u in L 2 (B 4ρ ), it follows the convergence in L 1 (O). Using also (2.38) and the uniform convergence u (ε) → u on B 4ρ ,weobtain sup B ρ u< 1 |O|  O (u + h)dx = lim ε→0 + 1 |O|  O  u ε + h  dx = 2h +lim ε→0 + 1 |O|  O  u ε − h  dx ≤ 2h + β · sup B 4ρ u. (2.40) Letting h → 0 + and then ρ → r − , we ar rive at the estimate sup B r u ≤ β · sup B 4r u (2.41) which is equivalent to (2.11) (under additional Assumptions 2.6(i)). Thus we have re- duced Lemma 2.5 for divergence operators (D) to the smooth case. Nondivergence case (ND). We will partially follow the previous arguments, with obvi- ous simplification. Now from u ∈ W(B r ):= W 2,n loc (B r ) ∩ C(B r ), it follows u ∈ W 2,n (B r 0 ) ∩ C(B r 0 ), where r 0 := 4ρ + ε 0 <r.Then f := Lu := (aD,Du) ∈ L n (B 4ρ ), and from f ≤ 0in B r 0 ,itfollows f (ε) ≤ 0inB 4ρ ,for0<ε≤ ε 0 .Forsuchε,theDirichletproblem L ε u ε :=−  a (ε) D, Du ε  = f (ε) in B 4ρ , u ε = u (ε) on ∂B 4ρ , (2.42) has a unique classical solution u ε which belongs to C ∞ (B 4ρ ). Then u ε − u ∈ W 2,n (B 4ρ ) ∩ C(B 4ρ ), and g ε := L ε  u ε − u  −→ 0inL n  B 4ρ  as ε −→ 0 + , (2.43) because g ε = g 1,ε + g 2,ε ,where g 1,ε = L ε u ε − Lu = f (ε) − f −→ 0, g 2,ε = Lu − L ε u =  (a (ε) − a)D,Du  −→ 0. (2.44) In addition, u ε − u = u (ε) − u → 0uniformlyon∂B 4ρ . By the Aleksandrov-type estimate (see, [2, Theorem 9.1]), sup B 4ρ   u ε − u   ≤ sup ∂B 4ρ   u ε − u   + N(n,ν,ρ) ·   g ε   n,B 4ρ −→ 0asε −→ 0 + . (2.45) [...]... Ω0 = Ω , and from Lu ≤ 0 in Ω and u = 0 on ∂Ω , it follows u ≤ 0 in Ω , in contradiction to our assumption u > 0 in Ω Since u is continuous on Ω1 , we can choose a point x0 ∈ Ω1 at which u x0 = M := sup u > 0 Ω1 (3.11) 14 Boundary Value Problems Then u ≤ M on the set Ω ∩ {d = r1 } ⊆ (∂Ω0 ) ∩ (∂Ω1 ) On the remaining part of ∂Ω0 , which is contained in ∂Ω , we have u = 0 < M By the maximum principle,... theorem for elliptic differential equations, ” Communications on Pure and Applied Mathematics, vol 14, pp 577–591, 1961 [20] J Serrin, “Local behavior of solutions of quasi-linear equations, ” Acta Mathematica, vol 111, no 1, pp 247–302, 1964 Sungwon Cho: Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA Email address: cho@ math.msu.edu Mikhail Safonov: School of Mathematics,... previous argument to −u and −λ in place of u and λ correspondingly Since ω(ρ) and ω(4ρ) remain the same after such a substitution, estimate (3.69) holds true in any case Finally, we set γ = γ(n,ν,θ0 ) := min{γ1 , − log4 β } ∈ (0,1], and let 0 < γ < γ, or equivalently, 0 < γ < γ1 and 4γ β < 1 The restriction 0 < γ < γ1 was needed in the case (i), which was based on Corollary 3.6 The inequality τ := 4γ... attained for operators L in the non-divergence form (ND), and domains Ω satisfying the exterior sphere condition which is specified in Definition 3.8 below The argument after Corollary 3.6 shows that under these assumptions, estimate (3.16) in Theorem 3.5 should be true for any constant γ ∈ (0,1) We give a direct proof of this fact in Theorem 3.9 below For domains of class C 2 , one can prove it using... in both divergence and non-divergence forms For this purpose, one can use the “parabolic” versions of growth lemmas in [3, 13, 14], or [16] References [1] A Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, New Jersey, 1964 [2] D Gilbarg and N S Trudinger, Elliptic Partial Differential Equations of Second Order, vol 224 of Fundamental Principles of Mathematical Sciences, Springer,... parabolic equations, ” in Harmonic Analysis and Boundary Value Problems (Fayetteville, AR, 2000), vol 277 of Contemp Math., pp 87–112, American Mathematical Society, Rhode Island, 2001 [17] V A Kondrat’ev and E M Landis, “Qualitative theory of second-order linear partial differential equations, ” in Partial Differential Equations, 3, Itogi Nauki i Tekhniki, pp 99–215, Akad Nauk SSSR Vsesoyuz Inst Nauchn... d(x) ≤ x − z0 − 1 in Ω, r = d x0 = x0 − y0 = x0 − z0 − 1 (3.43) Therefore, d1 (x) := h d(x) ≤ d0 (x) := h x − z0 − 1 = 1 − x − z0 −m in Ω, (3.44) S Cho and M Safonov 19 and d1 (x0 ) = d0 (x0 ) By definition of M1 in (3.41) and the choice of x0 , we get γ γ u ≤ M1 d1 ≤ M1 d0 in Ω, γ γ u x0 = M1 d1 x0 = M1 d0 x0 (3.45) In Remark 2.8, we pointed out that the choice m := nν−2 guarantees the inequality L(|x|−m... − γ)−1 ], and estimate (3.16) follows This approach uses some technical assumptions, such as the existence of solutions ur ∈ W(Ω ) to problems (3.15) with f = fr , and also the possibility of interchanging L with integration with respect to r in (3.20), which implies Lu = c1 ∞ 0 r γ−3 Lur dr = c1 ∞ 0 r γ−3 fr dr = f (3.25) The validation of these assumptions requires some standard work Instead, we... condition in Definition 3.8; in this case this estimate (M) holds true with γ0 = 1 Finally, this estimate together with Lemmas 2.4 and 2.5 imply the global H¨ lder regularity of o solutions to problem (DP), which is contained in Theorem 3.10 Lemma 3.1 Let ω(ρ) be a nonnegative, nondecreasing function on an interval (0,ρ0 ], such that ω(ρ) ≤ q−α ω(qρ) ∀ρ ∈ 0, q−1 ρ0 , (3.1) with some constants q > 1 and α... Springer, Berlin, 2nd edition, 1983 [3] N V Krylov, Second-Order Nonlinear Elliptic and Parabolic Equations, Nauka, Moscow, 1985, English translation: Reidel, Dordrecht, 1987 [4] O A Ladyzhenskaya, V A Solonnikov, and N N Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967, English translation: American Mathematical Society, Rhode Island, 1968 [5] O A Ladyzhenskaya and N N . Hindawi Publishing Corporation Boundary Value Problems Volume 2007, Article ID 57928, 24 pages doi:10.1155/2007/57928 Research Article Hölder Regularity of Solutions to Second-Order Elliptic Equations. for solutions to second-order elliptic equations, which vanish on the boundar y, while the right-hand side is allowed to be unbounded. For nondivergence elliptic equations in domains satisfying. lemmas originate from methods of Landis [13]. They were essentially used in the proof of the interior Harnack inequality for solutions to elliptic and parabolic equations in the non-divergence form