Boundary value problems for elliptic operators with singular drift terms. Josef Kirsch

Transcription

1 This thesis has been submitted in fulfilment of the requirements for a postgraduate degree e.g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use: This work is protected by copyright and other intellectual property rights, which are retained by the thesis author, unless otherwise stated. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given.

2 Boundary value problems for elliptic operators with singular drift terms Josef Kirsch Doctor of Philosophy University of Edinburgh 0

3 Declaration I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. Josef Kirsch)

4 Abstract Let be a Lipschitz domain in R n, n 3, and L = diva B be a second order elliptic operator in divergence form with real coefficients such that A is a bounded elliptic matrix and the vector field B L loc ) is divergence free and satisfies the growth condition distx, ) BX) ε for ε small in a neighbourhood of. For these elliptic operators we will study on the basis of the theory for elliptic operators without drift terms the Dirichlet problem for boundary data in L p ), < p <, and the regularity problem for boundary data in W,p ) and HS. The main result of this thesis is that the solvability of the regularity problem for boundary data in HS implies the solvability of the adjoint Dirichlet problem for boundary data in L p ) and the solvability of the regularity problem with boundary data in W,p ) for some < p <. In [KP93] C.E. Kenig and J. Pipher have proven for elliptic operators without drift terms that the solvability of the regularity problem with boundary data in W,p ) implies the solvability with boundary data in HS. Thus the result of C.E. Kenig and J. Pipher and our main result complement a result in [DKP0], where it was shown for elliptic operators without drift terms that the Dirichlet problem with boundary data in BMO is solvable if and only if it is solvable for boundary data in L p ) for some < p <. In order to prove the main result we will prove for the elliptic operators L the existence of a Green s function, the doubling property of the elliptic measure and a comparison principle for weak solutions, which are well known results for elliptic operators without drift terms. Moreover, the solvability of the continuous Dirichlet problem will be established for elliptic operators L = diva + B) + C + D with B, C, D L loc ) such that in a small neighbourhood of we have that distx, ) BX) + CX) + DX) ) ε for ε small and that the vector field B satisfies B φ C φ for all φ W, 0 of that neighbourhood. 3

5 Contents Abstract 3 Introduction 5 Elliptic Operators in Divergence Form 9. The spaces W k,p and Lipschitz Domains The Continuous Dirichlet Problem Behaviour of weak solutions at the Boundary An Approximation Argument Green s Function 9 3. The Existence of a Green s Function A Representation of the Green s operator based on the Green s Function The Elliptic Measure Definition of the Elliptic Measure The Doubling Property and a Comparison Theorem for L O The Dirichlet Problem for boundary data in L p ) 4 5. Definition of the D) p condition A p weights and the reverse Hölder class B p Some Basics of the classes A p and B p A L log L characterization of A Consequences of the D) p condition The Regularity Problem for boundary data in W,p ) and HS The Hardy Sobolev space HS The Regularity Problem for boundary data in HS Consequences of the R) HS -condition R) HS implies D ) p for some < p < A new proof for: R) p implies R) HS R) HS implies R) p for some < p < The R) C q condition for q < A uniform bound for u) r in HS A note on the Neumann Problem Open Problems 90 A Appendix 9 4

6 Chapter Introduction In this thesis, we will study boundary value problems for elliptic operators L with real coefficients in divergence form with singular drift terms and the corresponding real-valued scalar solutions on a bounded Lipschitz domain R n, n 3. For elliptic operators L 0 of the form L 0 = diva, where the matrix A = a ij X)) has real, bounded measurable coefficients such that there exists λ > 0 with λ ξ ij a ijx)ξ i ξ j for all ξ R n and almost every X, the Lax-Milgram Theorem implies that for every f W, ) there exists a unique weak solution u W, ) with boundary data f, i.e. A u ϕ = 0 for all ϕ C0 ) and Tru) f on, where Tr is the trace operator. This means that the Dirichlet problem L 0 u = 0 in u f on is solvable for boundary data in W, ), where the last equality is to be understood in the trace sense. The question if solvability still holds for other classes of boundary values was extensively studied. In [LSW63], it was shown that the continuous Dirichlet problem is solvable for elliptic operators L 0, i.e. for every f C 0 ) there exists a unique u W, loc ) C0 ) such that L 0 u = 0 in and u f on. What about boundary data in L p )? Historically, the study of the Dirichlet problem with boundary data in L p ) for elliptic operators of L 0 -type was initiated by B.E.J. Dahlberg in [Dah77], where the Laplacian on Lipschitz domains was considered the pullback of the Laplacian on a Lipschitz domain leads to an elliptic operator of the form L 0 ). As one can see for example in [Dah79], the study of Dirichlet boundary data in L p ) is related to the study of the non-tangential maximal function. Due to the fact that the continuous Dirichlet problem is solvable for L 0 -type elliptic operators, one defines that the Dirichlet problem with boundary data in L p ) is solvable for L 0 abbreviated D) p ), if for every f C 0 ) the weak solution u to the problem L 0 u = 0 in and u f on satisfies u L p ) C f L p ), where ) denotes the non-tangential maximal function see Definitions 5.. and 5..). This D) p condition allows one to conclude that for every f L p ) there exists a unique u W, loc ) such that L 0u = 0 and u converges non-tangentially almost everywhere to f on. Thus the question of interest is for which classes of elliptic operators L 0 the D) p -condition holds. Apart from the Dirichlet boundary value problem with data in L p of great interests are also other boundary value problems in particular the L p Neumann problem and Dirichlet regularity problem or just regularity problem) where the data are in W,p ) = {f L p ); T f L p )}, 5

7 where the unit vectors tangential to the boundary of at Q are T i Q), or sometimes T Q) to denote the family of these, and T f p ) p = i fq) T i Q) p dσq)) p. The most classical method for solving these types of boundary value problems at least for symmetric operators with coefficients of sufficient smoothness) is the method of layer potentials [FJR78] for the Laplacian in R n and [MT99], [MT0], [MT00] for variable coefficients operators. What has been observed are intriguing relationships between various boundary value problems. Of particular note is the duality between the L p Dirichlet boundary value problem and W,p regularity problem p denotes the conjugate exponent of p in the whole thesis, i.e. p + p = ). It turns out that the L p Dirichlet boundary value problem is solvable if and only if the W,p regularity problem is solvable for the same operator assuming symmetry and sufficient smoothness of the coefficients). If one does not assume any restrictions on the smoothness of the coefficients nor the symmetry, one has to follow a different path bypassing the shortfalls of the layer potential methods. This path uses some new methods see for example [KKPT00], Theorem.3), certain fundamental properties of weak solutions of elliptic partial differential equations e.g. the maximum principle, the Harnack inequality) and very sophisticated way of integration by parts. For example in [KKPT00], where the study of non-symmetric divergence form operators was initiated, it is shown in two dimensions that the Dirichlet problem for boundary data in L p for some possibly large) < p < is solvable if the matrix A is independent in one of the variables. The results in the literature on the Dirichlet problem for boundary data in L p ) can be categorized into three different types see Chapter 5 for examples and the related papers): the solvability of the Dirichlet problem for boundary data in L p ) for a certain class of operators is proven directly. perturbation results: under the assumption of the D) p condition for one specific elliptic operator, the solvability of the Dirichlet problem with boundary data in L q ) is proven for a class of elliptic operators which are perturbations of that specific elliptic operator. The index q might be equal to p or larger. consequences of the D) p condition are proven, e.g. an interpolation and extrapolation property, i.e. that D) p implies D) q for q p ε, ) and some ε > 0, which means that solvability is an open property with respect to the index p on, ). In contrast to the Dirichlet problem the regularity problem imposes some regularity on the boundary data, e.g. f W,p ) or f HS ). The study of the regularity problem for elliptic operators L 0 = diva with A as above and symmetric was started by C.E. Kenig and J. Pipher in [KP93]. They say that the regularity problem for boundary data in W,p ) is solvable abbreviated by R) p ) if for every f W,p ) C 0 ) the weak solution u to L 0 u = 0 in and u f on satisfies N u) Lp ) + u Lp ) C f W,p ), where N ) is a variant of the non-tangential maximal function. As for the Dirichlet problem, the R) p condition allows to conclude that for every f W,p ) there exists a unique u W, loc ) such that L 0u = 0, u converges non-tangentially to f almost everywhere and N u) Lp ) + u Lp ) C f W,p a ). The results on the regularity problem can be categorized into the same three categories as for the Dirichlet problem. In this thesis we are mostly dealing with two things: first, we will extend several well known results for elliptic operators without drift terms or with drift terms in L to singular drift terms and, second, we will study consequences of the R) HS condition, which we will define in Definition 6... In [GT0] it is shown that the continuous Dirichlet problem for elliptic operators of the form Lu = diva u + Bu) + C u + Du is solvable for bounded B, C and D. In [HL0] S. Hofmann and J.L. Lewis consider singular drift terms and it is shown there that the continuous Dirichlet problem is solvable for elliptic operators of the above form with B, D 0 and the vector field C satisfies the growth condition distx, ) CX) C for a constant C > 0 and distx, ) CX) dx is a Carleson measure. By applying a scaling argument see the proof of Theorem..) to the results 6

8 in [GT0] and an integral version of Hardy s inequality see Lemma..) we are able to prove the solvability of the continuous Dirichlet problem for a similar, but different class of coefficients than in [HL0]. We show that the continuous Dirichlet problem is solvable for singular drift terms of the form B, C, D L loc ) such that in a neighbourhood of one has distx, ) BX) + CX) + DX) ) ε for ε small and B φ C φ for all φ W, 0 of that neighbourhood. The family of these elliptic operators is denoted by O. Thus compared to [HL0] we need the smallness of ε, but not the Carleson measure condition. For elliptic operators without drift terms and no assumption on the smoothness of the matrix A properties like the existence of a Green s function, see [GW8], and the doubling property of the elliptic measure, see [CFMS8], are well known and these properties are essential in the study of the theory of boundary values in L p for elliptic operators without drift terms and no assumption on the smoothness of the matrix A. We will adapt the proofs for these well known properties to the subset O 0 O, where L O is in O 0 if C, D 0 and B is divergence free. The major difficulty, why we have to restrict ourselves to this subset, is to prove the existence of a Green s function for L O and the adjoint L of L. The assumption that C 0 D and B is divergence free will make this possible. The main motivation for this thesis is the paper [DKP0] by M. Dindoš, C.E. Kenig and J. Pipher. They define the D) BMO condition for elliptic operators without drift terms), which is the endpoint at for the D) p condition, and they prove that the D) p condition is an open condition with respect to the index p on, ], where D) is to be understood as D) BMO. Precisely, they show that D) BMO holds if and only if the corresponding elliptic measure is in A and therefore if and only if D) p holds for some < p <. Motivated by the duality of BMO and H, we define for L O 0 the solvability of the regularity problem for boundary data in HS ), abbreviated by R) HS, which can be seen as the natural extension of the definition of the R) p condition for < p < given in [KP93]. With Theorem 0. in [BB0] and the methods in [KP93] we will show that under the assumption of the R) HS condition for every f HS exists a unique weak solution u W, loc ) with boundary data f. Moreover, in [KP93] it is shown that R) p for < p < implies R) HS this result is contained in the proof of Theorem 5. of [KP93]) and R) p+ε for some ε > 0 for symmetric elliptic operators without drift terms. By the characterisation of HS in terms of a maximal function in [BD09] and the methods used in [KP93] and [She07] we will be able to proof the extrapolation property of the R) p condition at the endpoint R) HS. Namely, we show that R) HS implies R) p for some < p <. Thus the result of C.E. Kenig and J. Pipher and our extrapolation result, which is the main result of this thesis, complement the result in [DKP0]. The second motivation for investigating the R) HS condition is the above mentioned duality of the Dirichlet problem with boundary data in L p ) and the regularity problem with boundary data in W,p ). Since we do not assume symmetry of the matrix A, the question precisely is, if the D ) p condition, which is the D) p condition for the adjoint elliptic operator, is equivalent to the R) p condition the direction that R) p implies D ) p is proven in [KP93]). The fact that most of the proven results for the Dirichlet problem have been proven for the regularity problem as well compare for example [Dah79] and [KP93] or [KKPT00] and [KR09]) emphasizes the interest on that open problem. With the aid of Z. Shen s main result in [She07], we are able to simplify the requirements for a possible proof of that duality see Corollary 6.3.3). Outline In Chapter, we will follow the ideas and proofs in [GT0], Chapter 8, and will combine them with a variant of Hardy s inequality to show that the continuous Dirichlet problem on a Lipschitz domain for elliptic operators L O is solvable. Further, we will follow [CFMS8] and [HL0] to extend some results regarding the behaviour of weak solutions for L O at the boundary. In the last section of Chapter we will prove an approximation argument for elliptic operators in O under the additional assumption that constant functions are weak solutions. This approximation argument originates in [KP93] for L 0 = diva and A symmetric and is extended to possibly non-symmetric A in [KKPT00]. In Chapter 3, we will prove the existence of a Green s function for elliptic operators in O 0 by 7

9 adjusting the corresponding proof in [GW8], which deals with operators without drift terms. In Chapter 4, we will introduce the harmonic measure and, as in [CFMS8], we will prove the doubling property for elliptic measures and a comparison Theorem for weak solutions corresponding to elliptic operators in O 0, which are well-known results for elliptic operators without drift terms. In Chapter 5, we will look at the Dirichlet problem for boundary data in L p ) and we will introduce the A p -weights of Muckenhoupt. We will give a detailed proof based on Young s inequality for Orlicz spaces of a L log L characterization for A the endpoint of Gehring s Lemma). In Chapter 6, we will introduce the regularity problem on Lipschitz domains for boundary data in the Hardy Sobolev space HS. We will show that, under the assumption that R) HS holds, for every f HS exists a unique weak solution u such that u converges non-tangentially to f almost everywhere. Further, we will show that R) HS implies D ) p and R) p for some < p <. In addition, we will introduce the R) C q condition for q < and look at the extrapolation property of the Neumann problem on the Hardy space H the extrapolation property for N) p with < p < is proven in [KP93]). In the last chapter, Chapter 7, we summarize some open problems, which appear in this thesis and make suggestions for some further investigations. Acknowledgements First and foremost, I wish to thank my supervisor Martin Dindoš for his encouragement and for many helpful discussions and suggestions. Thank you for investing so much time and energy into my supervision and for allowing me all the freedom to work in my own way! Secondly, I am grateful to Jill Pipher for her s about the endpoint of Gehring s Lemma. Finally, I would like to thank the University of Edinburgh and the Maxwell Institute for their financial support. 8

10 Chapter Elliptic Operators in Divergence Form In this chapter we consider elliptic operators in divergence form with singular drift terms on a bounded Lipschitz domain R n, n 3, i.e. operators of the form LuX) = divax) ux) + BX)uX)) + CX) ux) + DX)uX).) where A is a real bounded elliptic matrix and real B, C, D L loc ) such that in a neighbourhood of one has distx, ) BX) + CX) + DX) ) ε for ε small and B φ C φ for all φ W, 0 in that neighbourhood see the beginning of section. for the precise definition). A function u W,, loc ) the function space Wloc ) and the domain are defined in section.) is called a weak solution subsolution, supersolution) for L if Lu = 0 0, 0) in the weak sense, i.e. Lu, v) = A u v + u B v C u v Duv = 0 0, 0).) for all v C0) with v 0. Using a scaling argument we see that a weak solution u for L O can locally on a ball centred at X with radius α = distx, )/ be seen like a weak solution u α for L α on a ball of radius one which is approximately one away from the boundary, where L α has bounded coefficients B α, C α, D α on that ball see Theorem.. for the precise argument). This means that we can use the local results from [GT0], Chapter 8, where operators as in O but with bounded B, C, D are considered. One can see section. as a generalization of Chapter 8 in [GT0], especially since we will adapt the proofs from [GT0] with the aid of Lemma.. to show that the continuous Dirichlet problem is solvable for our type of elliptic operators. For elliptic operators with bounded drift terms it is shown in [GT0] and for example in [Ken94] where the proof is given for operators of the form diva, but can easily be extended to operators of the form.) with bounded drift terms) that weak solutions in the interior of are Hölder continuous, satisfy the Harnack principle and the Cacciopoli inequality. Moreover, it is shown that if Dv B v) 0 or Dv + C v) 0 for all non-negative v C 0), the maximum principle holds, and therefore that under the additional assumption that satisfies an exterior cone condition at every Q, one can find a unique u W, loc ) C0 ) for every g C 0 ) with Lu = 0 in u g on. This means that the continuous Dirichlet problem is solvable. In [HL0] S. Hofmann and J.L. Lewis consider operators of the form Lu = diva u + C u for A as above and the vector field C satisfies distx, ) CX) c for some c > 0 and 9

11 distx, ) CX) dx is a Carleson measure. Among other things they show that the continuous Dirichlet problem is solvable for these operators see [KP0] as well). Thus if one compares the operators in [HL0] and our operators in O one sees that we require the smallness of ε, but we do not impose the Carleson measure condition. The main result of this chapter is that the continuous Dirichlet problem is solvable for operators in O. We will start this chapter by defining the spaces W k,p ) and the domain. Then we show the solvability of the continuous Dirichlet problem for elliptic operators in O. Furthermore, we will give detailed proof for some results about the behaviour of weak solutions at the boundary, which are well known for operators without drift terms. At the end of this chapter we prove an approximation argument for elliptic operators in a subclass of O. This approximation argument originates in [KP93], section 7, for elliptic operators of the form L 0 = diva and A symmetric and is extended to non-symmetric A in [KKPT00], page 57. The proof given in [KKPT00] relies on the coercivity of L 0. We will bypass the lack of coercivity by the usage of the Green s operator.. The spaces W k,p and Lipschitz Domains In this section, we summarize well-known facts about the function spaces W k,p, Lipschitz domains and the trace operator. Let R n be bounded and open. The dimension n will be larger than or equal to 3 in the whole thesis, if it is not stated otherwise. For u L loc ), we say that v is the αth -weak derivative of u with α = α,, α n ) a multi-index) if vϕ = ) α uϕ α) for all ϕ C α 0 ) where α = n j= α j. We denote the weak derivative v of u by D α u. Therefore D α u is defined almost everywhere. A function is called weakly differentiable of order k if all α th -weak derivatives exist for all α k. An integration by parts argument shows that, if u C k ), the weak derivative of u coincides with the derivative of u for all α with α k. The subspace of L loc ) of all weakly differentiable functions of order k for k an integer is denoted by W k ) and we define the space W k,p ), p <, by W k,p ) = {u W k ) : D α u L p ), α k} and a norm on W k,p ) by u W k,p ) = α k D α u L p ). The completion of C 0 ) in W k,p ) is denoted by W k,p 0 ). We will use the following notation throughout the thesis: Q, P denote points on and X, Y in. For Z R n and R > 0, the open ball centred at Z with radius R is denoted by B R Z) and rb) denotes the radius of the ball B. In addition R Q) = B R Q), T R Q) = B R Q), δx) = distx, ), ) β = {X : δx) < β}, β = \ ) β. For f L E) with E a measurable set with positive Lebesgue measure, i.e. E > 0, we write fe) = E f and f E = ffl E f = E E f. Definition.. C k,α -Domain; [GT0], page 94). We call a bounded domain R n a C k,α - domain, if at each point Q there exists a ball B rq) Q) and a one-to-one mapping ψ Q 0

12 onto the unit ball B 0) R n such that: ψ Q B rq) Q) ) = B 0) R n + ψ Q B rq) Q) ) = B 0) R n + ψ Q C k,α B rq) Q)), ψ Q Ck,α B 0)), where R n + = {x, t) : x R n, t > 0}. Lemma... Let be a C 0, -domain. Then there exists a finite sequence {P j } j and a constant R 0 > 0 such that for every Q there exists a P j with T R0 Q) T rpj)p j ), where rp j ) is given by Definition... Hence there exists a finite sequence {Q k } k such that ) R0/ k T R 0 Q k ). Proof. Since is compact we can find a finite sequence {P j } such that j P j R n + B 0)) =. ψ All ψ Pj, ψ P j are Lipschitz continuous and therefore, we can find a uniform constant M such that X Y ψ P M j X) ψ P j Y ) M X Y for all X, Y B 0). This means that the image of a ball B r Z) B 0) under any ψ P j a ball with radius at least M r. Thus for Z Rn + B 0), the set ψ ball B ψ M P j Z)). Hence, for every Q there exists j such that B M which implies that T Q) T rp M j)ψ P j 0)). The choice R 0 = M contains P j B Z)) contains the Q) ψ P j B 0)), finishes the first part of the proof. We have ) R0/ Q B R 0 Q). A compactness argument justifies the existence of a finite sequence {Q k } k such that ) R0/ k T R 0 Q k ), which completes the proof. Definition... A C 0, -domain is called a Lipschitz domain. For a Lipschitz domain, we see that is locally the region above a Lipschitz graph ϕ and so for Q = x, ϕx )) we define A R Q) = x, ϕx ) + R) and for X we define X such that A R X) = X for an appropriate R. Thus A R Q) and X are well defined in each B R0 Q k ), where R 0 and Q k are as in Lemma... This means that A R Q) and X depend on k, but we will omit the index k to maintain an easy readable notation. The next lemma follows from the Hardy inequality, which originates in [Har0], 4). An integral version of Hardy s inequality says that if f is a non-negative integrable function then 0 t t 0 fs) ds)p dt p p )p 0 ft) p dt for p >. Lemma... Let be a Lipschitz domain and B be a non-negative measurable function in with BX) ε δx) for some ε > 0 in ) β and BX) C in β. Then for ϕ W,s 0 ) non-negative, v W,s ), < s <, and any R < min{β, R 0 }, Q 0 we have T R Q Bϕ)s 0) C s ε s T R Q ϕ s 0) v Bϕφ ε φ v s s + C s,ε ε s ) β ϕ s φ s + ϕ s φ s ) + C s,ε β ϕφ s where φ is any non-negative, smooth function in R n and ε > 0. Proof. The first inequality follows from the integral version of the Hardy inequality and the second from the first and an application of Young s inequality. In order to introduce the trace operator, let us observe that.3)

13 Theorem..3 [GT0], Theorem 7.5). Let be a C k, -domain. Then C ) is dense in W k,p ), p <. For φ C ) the map Tr : C ) C 0 ) defined by Trφ)Q) = φq) for Q is well defined and is called the trace operator. From the results in [Ada75], section VII, the trace operator is bounded from W k,p ) to W k p,p ) for a C k, -domain for the definition of fractional order Sobolev spaces see [Ada75]). We know from Theorem..3 that C ) is dense in W k,p ) and so one can extend the trace operator to a bounded linear operator, which we call Tr as well, on W k,p ) with Tr : W,p ) W p,p ) Tru) W p,p ) C u W,p ) for < p < and a Lipschitz domain. Roughly speaking, this means that by going to the boundary, one looses,p p-th derivatives. By Theorem 7.55 in [Ada75] W0 ), which was defined as the completion of C0 ) under the norm of W k,p ), coincides with {φ W,p ) : Trφ) = 0}. In section., we will need the following results about traces. Lemma..4. Let be a Lipschitz domain, u W, 0 ) and v W, ), then uv W, 0 ). Proof. There exist u k C0 ) and v k C ) with u k u, v k v in W, ) and u k W, 0 ) C u W, 0 ). Since u kv k uv u k v k v + v u u k we get u k v k uv in W, ). Additionally, u k v k C0 ) and so uv W, 0 ). Lemma..5. Let be a Lipschitz domain and u W,p ) be non-negative almost everywhere. Assume that u 0 on understood in the trace sense) then u W,p 0 ). Proof. By the technique of mollifiers, we can choose non-negative φ k C ) with φ k u in W,p ). The definition of the trace operator implies T rφ k ) 0 and the boundedness of the trace operator gives T ru φ k ) 0 as k. Hence Tru) 0 and therefore W p ) T ru) = 0. Lemma..6. Let u W,p ) L ) and M = sup u, where the sup is understood in the trace sense, then max{u, M} M W,p 0 ). Proof. By Lemma..5 we have M <. Moreover, max{u, M} M 0 in. The definition of M implies max{u, M} M 0 on and, by Lemma..5, max{u, M} M W,p 0 ).. The Continuous Dirichlet Problem In this section, we will use Lemma.. to extend the results in [GT0], Section 8, from bounded to singular drift terms. We deal with elliptic operators in divergence form L as in.) on a bounded Lipschitz domain R n see Definition..), n 3, with real, measurable coefficients that satisfy : there exists λ > 0 such that AX)ξ, ξ λ ξ for all ξ R n and almost every X. there exists β > 0, ε > 0 and M > 0, with ε small such that δx) BX) + CX) + DX) ) ε for almost all X ) β A L ), B L β ), C L β ), D L β ) M the vector field B satisfies ) β B ϕ M ϕ L ) β ) for all ϕ C 0 ) β ) In order not to confuse readers, which are familiar with the notation used in the literature, we keep the notation which is used in the literature, although this leads to a clash in the notation. For example D can denote a function or the derivative. It will be always clear from the context or will be explained, which meaning is to be considered.

14 positive constants are supersolutions, i.e. Dv B v) 0 for all non-negative v C0) this implies by the result in [GT0] that a local maximum principle holds) The family of elliptic operators that satisfy the above criteria is denoted by O = Oλ, β, ε, M). A function u W, loc ) is called a weak solution subsolution, supersolution) for L O if Lu = 0 0, 0) in the weak sense, i.e. Lu, v) = A u v + u B v C u v Duv = 0 0, 0) for all v C 0) with v 0. The subfamily of operators in O with C 0 D and the vector field B being divergence free in the sense of distributions is denoted by O 0. Thus an operator L O 0 is of the form Lu = diva u + Bu) = diva u B u. The symbol is used as an abbreviation for the following: We will write f g for two functions defined on a given set E, in words that f is comparable with g, if there exists a constant C which depends on λ, β, ε, M, and n such that C f g Cf. We use C ζ or Cζ) if we would like to emphasize that the constant C depends on the parameter ζ. Let us make a few comments on the restrictions imposed on the operators in O: For the proofs in this section it will be essential that for L O the corresponding bilinear form L : W, ) W, 0 ) R is bounded. This is the main reason for the restrictions imposed on the drift terms. We will give brief examples to show that the growth condition of C and D cannot be relaxed to allow a growth like δx) for some ε > 0. To illustrate the +ε ideas, let us assume that we are on 0, ) R and that we restrict the view to the region close to 0. We will write 0 to mean the integral over the interval 0, c 0) for some c 0 > 0 small. If we assume that Cx) = x and we choose fx) = x and gx) = x ε where f symbolises +ε the function in W, ) and g the function in W, 0 )) then 0 Cx)f x)gx) = 0 x =. Similar thoughts work for D if we take f instead. For the term involving B it is different, since the derivative is applied to the function from W, 0 ). Thus if we assume that Bx) = x, f and gx) = x, then 0 Bx)fx)g x) = 0 x =. This example shows that in order for L : W, ) W, 0 ) R to be bounded we need an additional assumption on B. For φ C ) and ψ C0 ) we have B φ ψ = B φψ) φ B ψ. The second term is bounded by C φ W, ) ψ W, 0 ). Therefore for L to be a bounded bilinear functional on W, ) W, 0 ) the vector field B has to satisfy ), β B ϕ M ϕ W, 0 ) β ) for all ϕ W0 ) β ) and some M > 0. Obviously, bounded vector fields B and vector fields B with bounded divergence use the Poincare inequality) satisfy this assumption with a possibly different M). Thus our assumption on B is weaker than B being bounded or having bounded divergence. Let us give an example of an unbounded vector field B with unbounded divergence that satisfies the assumption as well. For this let = [0, ] [0, ] R be the unit square and define Bx, y) = x α, x ) for some 0 < α <. Then for any φ C0 ) we have B φ = xα x φx, y). Thus B φ φ for all φ W, 0 ), but B and the divergence of B are unbounded. If we choose α = 0, then we see that B grows like δx) for x 0 and B has zero divergence. The goal of this section is to prove that the continuous Dirichlet problem for elliptic operators in O is solvable for ε small enough. If one looks at the proofs in [GT0] needed to prove the solvability of the continuous Dirichlet problem for operators with bounded drift terms, one realizes that they can be used in combination with Lemma.. almost equally well for our class of elliptic operators. We start with an interior result. From the results for bounded coefficients in [GT0] and for example [Ken94] we deduce the following: Theorem... Let u W, loc ) be a non-negative weak solution for L O in the Lipschitz domain. Then u is Hölder continuous and it satisfies the Harnack principle and the Cacciopoli inequality in the interior of. Proof. The proof follows from a scaling argument. For X in let α = δx) and B α be the ball 3

15 with radius α and centre X. We write f α X) = fαx). Then for φ W, 0 ) and X = αz we get 0 = A u φ + u B φ C u φ Duφ = α A α u α φ α α + u α B α φ α C α u α φ α D α u α φ α α, where α = {Z R n : αz }. Hence u α is a solution to L α u = diva α u + αb α u) + αc α u + α D α u on α. The ball Bα is transformed by the change of variables X = αz to B, a ball with radius and distance to the boundary comparable to one. Since for example α C α Z) = α CαZ) for all Z B, the coefficients A α,, D α are bounded on B. The results for bounded coefficients are applicable. Theorem.. still holds if B, C, D satisfy δx) DX) C and δx) BX) + CX) ) C for some possibly large) C > 0. But we will see in the following proofs that the methods we use rely heavily on the smallness of ε and the boundedness of the bilinear functional L for L O to prove that the continuous Dirichlet problem for L O is solvable. Following Lemma 3.38 in [HL0], we get: Theorem.. Maximum Principle). Let u, v W, loc ) be two weak solutions for L O with lim sup X Q u v)x) 0 in the Lipschitz domain, then u v in. Proof. Since the coefficients of L are locally in L, we get u, v C 0 ). Using a compactness argument, we get the existence of a δ = δε) > 0 for every ε > 0 such that u v ε in ) δ. Assume that there exists X such that u v)x) > ε. Then δx) > δ and by the local maximum principle we get sup δ )u v) > ε, which is a contradiction. Lemma..3. Let be a Lipschitz domain and L O. Then L, ) is a bounded bilinear functional on W, ) W, 0 ), i.e. for any ϕ W, ) we have that Lϕ which we will sometimes denote as F ϕ ) is in W, 0 )) with Lϕ W, 0 )) C ϕ W, ). Proof. It is enough to show that L, ) is a bounded bilinear functional on W, ) W, 0 ). Let ψ be smooth with ψ in ) β/4, ψ 0 in β and ψ C β. Then Lϕ, v) = = A ϕ v + B ϕv) B + C)ϕ v Dϕv A ϕ v + B ϕvψ) + B ϕv ψ)) B + C) ϕ v Dϕv. Except for the term involving A we split the integral into = ) β + β. On β, we use the L bounds of the coefficients. On ) β, we use the assumption on B, the Cauchy Schwarz inequality, and Lemma.. to get B vϕψ) C ϕvψ) L ) β ) C ϕ W, ) v W, 0 ), ) β B + C) ϕ v C ϕ W, ) v W, 0 ). Similar thoughts work for the other terms. Thus Lϕ, v) C ϕ W, ) v W, Lϕ W, 0 )) C ϕ W, ). 0 ) and so Lemma..4. Let be a Lipschitz domain. For L O with ε sufficiently small, there exists σ = σo) > 0 such that the corresponding bilinear form L satisfies Lu, u) λ u σ u, where λ is the ellipticity constant of the matrix A. 4

16 Proof. Using Lemma.., we see that Lu, u) λ u u B C u Du λ u λ u C λ B C 0 ) u C λ B C u β β D u D u ) β β 9λ u C λ ε u 5M u, 0 i.e. for ε small enough, we get Lu, u) λ u 5M u. The proofs of the following two theorems use the ideas of the proof of Theorem 8.3 in [GT0]. Theorem..5. Let be a Lipschitz domain and L O. Then for F W, 0 )), there exists a unique w W, 0 ) such that Lw = F. Additionally, w W, 0 ) C F W, 0 )). If F = F ϕ then w W, 0 ) C ϕ W, ). The bounded linear operator that maps W, F w W, 0 ) is called Green s operator and will be denoted by G. 0 )) Proof. Lemma..4 implies that there exists σ = σo) such that the bilinear form L σ corresponding to L σ w = Lw σw is bounded and coercive. Define the embedding I : W, 0 ) W, 0 )) by Iwv) = wv. Then I is a compact operator. The equation Lw = F for F W, 0 )) and w W, 0 ) is equivalent to L σ w + σiw = F. By the Lax-Milgram Theorem the operator L σ is invertible with bounded inverse Applying L σ L σ : W, 0 )) W, 0 ). to both sides from the left, we get the equivalent formulation Since I is compact and L σ is compact. w + σl σ Iw = L σ F..4) is continuous, the operator σl σ I) : W, 0 ) W, 0 ) σ I)w = 0 is equivalent to Lw = 0. Thus Moreover, the equation w σl the maximum principle, Theorem.., implies that only the trivial solution satisfies w σl σ I)w = 0. The Fredholm alternative, see Theorem A.0.4 in the Appendix, implies that there exists a unique function w W, 0 ) such that.4) holds and the operator id σl σ I)) has a bounded inverse. Thus w = id σl σ I)) L σ F. with w W, 0 ) C F W, 0 )). If F = F ϕ we get by Lemma..3 that w W, 0 ) C ϕ W, ) and so the proof is complete. Theorem..6. Let be a Lipschitz domain and L O. Then for every ϕ W, ), there exists a unique u W, ) such that Lu = 0 and u ϕ on. Furthermore, u W, ) C ϕ W, ). Proof. Let w W, 0 ) be the solution to Lw = Lϕ. The existence of w is guaranteed by Lemma..3 and Theorem..5. Define u = ϕ w. So Lu = 0 in and u ϕ on. Lemma..3 implies u W, ) C ϕ W, ). The maximum principle gives uniqueness. An important step for the proof of the solvability of the continuous Dirichlet problem for elliptic operators with bounded drift terms in [GT0] is Theorem 8.5 in [GT0]. We will combine the proof of Theorem 8.5 in [GT0] with Lemma.. to get the result which is 5

17 needed to conclude the solvability of the continuous Dirichlet problem for L O. Moreover we include a new result, which follows from the same methods, for indices smaller one, which will be used later to deal with R) q for q <. Theorem..7. Let be a Lipschitz domain, L O, Q, 0 < R < min{β/4, R 0 } and u W, ). If u is a subsolution in, then sup u + M C p ) p u + M p B R Q) for any p >, where M = sup BR Q) u + and { u + M = sup{u M, M} M x x / If u is a supersolution, which is non-negative in B 4R Q), then ) p u m p B R Q) C p inf B R Q) u m for any p < n n where m = inf B 4R Q) u and { u inf{ux), m} mx) = m x x / If u is a weak solution which is non-negative on B 4R Q), then for all q > 0. sup u + M C q ) q u + M q B R Q) Proof. We will prove the subsolution and supersolution results first and, at the end, will mention the changes for the result about weak solutions. By a scaling argument, we can assume that R =. Let ψ C0B 4 Q)) and ū = u + M, if u is a subsolution, and ū = u m, if u is a supersolution. For β R\{0}, we define { ψ ū β M β ) if β > 0 v = ψ ū β m β ) if β < 0. In the whole proof, β remains away from zero and the case β > 0< 0) will be applied to u as a subsolution supersolution) only. Lemma..6 implies v W, 0 ) with v 0, and therefore v is a valid test function. We claim that ū ψ ū β C ψ + ψ )ū β+,.5) 6

18 which is 8.5) in [GT0]. To see this, we test with v and get Lu, v) = + A u ū βū β ψ + A u ψ ψū β M β ) + u B ψ ū β M β )) C u ψ ū β M β ) Duψ ū β M β ) = I + + V 0 for u a subsolution and M = M 0 for u a supersolution and M = m. Using ellipticity and the sign of β, we get λ β ū ū β ψ II + III + IV + V. The claim.5) is proven once we have shown that each term II,, V can be bounded by C+ β ) ψ + ψ )ū β+ +ε+c ε ε )+ β ) u ū β ψ for ε and ε small. Roughly speaking, this follows from Lemma..), ū β M β ū β and the fact that u can be replaced by ū in every term. To make it precise, let us start with II: II M ū ū β β+ ψ) ψ ū ) ε ū ū β ψ + C ε ψ ū β+, therefore II is done. For IV we use Young s inequality with ε + β ) and Lemma.. to get ) IV ū ū β ψ C ψū β M ) β )ū β ε + β ) ū ū β ψ + C εε ψ[ū β + β M β ]ū β ) ε + β ) ū ū β ψ + C εε ψ ū β+ + β + C εε ψ ū β ū β ) )ū β ) + β + C εε ψ ū β + β M β ) u β ū ) β ) ) + β )ε + C ε ε ) ū ū β ψ + C εε ψ ū β+, + β which completes term IV. For V, we write V = β+ ū ψ)dψū β M β )ū β ). So V can be bounded in the same way as IV. For III, we have III = ū B ψ ū β M β )) = B ūψ ū β M β )) B ū ψ ū β M β ) = III A + III B. 7

19 For III B, one proceeds as for IV. For III A, we get III A C ūψ ū β M β )) +C ū ψū β β+ )ψū ) + C ψ ū β+ β+ )ψū ) + C β β β+ ū ψū )β ψū ) ε + β ) ū ū β ψ + C ε + β ) ψ + ψ )ū β+, which completes claim.5). Observe that a smaller ε allows a smaller β. Let w = ū β+, β and w = log ū for β =. Then.5) with γ = β + where γ has to stay away from ) says {Cγ ψ w ψ + ψ )w for β C ψ + ψ.6) for β = From the Poincaré inequality, we get the following for χ = n n : ψw L χ C ψ w) L + ψ w ) L C + γ ) ψ + ψ )w L. Thus for ψ on B r Q) and ψ 0 on Br c Q), with r < r 3 and ψ have C r r, we w L χ B r ) C + γ r r w L B r )..7) For p 0, define Φp, r) = B ū p rq) ) p, then by Lemma A.0.5 in the Appendix we get lim p Φp, r) = sup BrQ) ū and lim p Φp, r) = inf BrQ) ū. By the definition of w and.7), we have Φγχ, r ) γ C+ γ ) r r Φγ, r ) γ, hence C + γ ) Φχγ, r ) r r C + γ ) Φγ, r ) r r ) γ Φγ, r ) for γ > 0.8) ) γ Φχγ, r ) for γ < 0.9) We can start the iteration on k with γ = pχ m k and r k = + m k m, k = 0,, m to get C + pχ Φχ m+ p, r 0 m ) χ ) m p ) Φχ m p, r ) k=0 m+0) m C + pχ m k ) ) χ m k p Φp, r m ). m k) The term Φp, r m ) is bounded by Φp, ). The product of terms is bounded by a constant, since for example m k=0 m k) χ m k p = p m k=0 kχ k C Thus we see that Φχ m p, ) CΦp, ). Sending m, we get the desired result for subsolutions, whereby the closer p gets to, the closer γ gets to, i.e. the smaller ε has to be. 8

20 For u a supersolution, we can use the same method of iteration to show that for fixed 0 < p 0 < p < χ we get Φp, ) CΦp 0, 3) by.8) for 0 < γ <, Φ p 0, 3) CΦ, ) by.9). It remains necessary to show that there exists 0 < p 0 < min{χ, p} such that Φp 0, 3) Φ p 0, 3). Using.6) for β = we get with Ψ on B r Q), Ψ 0 on B r Q) and Ψ C/r that B r w Cr n. Theorem A.0.7 in the Appendix implies the existence of a p 0 > 0 such that e p0 w w B Q) 3 C. B 3Q) Hence B B ep0w e p0w 3Q) 3Q) Ce p0w B 3 Q) e p0w B 3 Q) = C and therefore, since w = log ū, we get Φp 0, 3) CΦ p, 3) and so the supersolution result is proven. For the result about weak solutions, let ū = u + M and v = ψūβ M β ) for β 0. Then, as before,.5) holds for a constant C uniformly in β, if β stays away from zero. Therefore.8) holds for γ > 0, whereby γ has to stay away from. One can apply the same iteration as for subsolutions. In the case that there exists an m N such that qχ m =, one has to choose a slightly smaller q. Moreover, the smaller q is the smaller ε has to be, since sup 0<q0<q inf m q 0 χ m 0 as q 0. With Theorem..7 proven one can follow [GT0] to show that the continuous Dirichlet problem is solvable. Since we did not prove Theorem..7 in such a generality as D. Gilbarg and N.S. Trudinger in [GT0], we have to work a little bit more carefully. Theorem..8. Assume that u is a weak solution in the Lipschitz domain for L O, then for any 0 < R < δ min{r 0, β}, we have [ R ) α osc TR Q)u C sup u + σ ] Rδ), δ T δ Q) where σr) = osc R Q)u and osc TR Q)u = sup TR Q) u inf TR Q) u and osc R Q) accordingly. Proof. We follow the proof of Theorem 8.7 in [GT0] and include a few lines since we did not prove Theorem..7 in such a generality as in [GT0]. We define the following numbers: M = sup TR Q) u, M 4 = sup T4R Q) u, M = sup B4R Q) u, m = inf TR Q) u, m 4 = inf T4R Q) u, m = inf B4R Q) u. We will consider first the case if M 4 0 and m 4 0. By the assumptions on L O, we have that positive constants are supersolutions. Thus the functions M 4 u and u m 4 are supersolutions. As in Theorem..7, we define { u m 4 ) inf{u m 4, m m 4 } x m m 4 = m m 4 x /, and similarly for M 4 u) M 4 M. Using this definition for the first inequality in each of the following two lines and then the result about supersolutions in Theorem..7 for the second inequality, we get M 4 M) B RQ)\ R n R n m m 4 ) B RQ)\ R n R n B R Q) B R Q) M 4 u) M 4 M CM 4 M ), u m 4 ) m m 4 Cm m 4 ). 9

21 Since is a Lipschitz Domain, it satisfies a uniform exterior cone condition and so B R Q)\ CR n for a uniform constant C = C). Thus the two inequalities above imply Adding them together, we are left with which implies that M 4 M CM 4 M ), m m 4 Cm m 4 ). M m C )M 4 m 4 ) + M m), C osc TR Q)u γ osc T4R Q)u + C osc 4R Q)u for some 0 < γ <. The theorem in the case that M 4 0 and m 4 0 then follows from Lemma A.0.6 in the Appendix. In the other case we can assume without losing generality that 0 < m 4 M 4. As before we then consider M 4 u and u m 4. In order to repeat the previous argument it remains to show that the result about supersolutions from the previous theorem can be applied to u m 4, which is a subsolution. For this it is enough to show that.5) holds for β < 0 Therefore we define v and M as in the previous proof for supersolutions, then Lu m 4, v) = I + II + III + IV = m 4 B ψ ū β M β )) + m 4 Dψ ū β M β ). } {{ } V As in the previous proof we use ellipticity to get λ β ū ū β ψ II + + V I. } {{ } V I Since u m 4 u the terms II, III, IV are treated as the terms II, III, IV. For V and V I observe that m 4 ū and so for example m 4 ψ ū β M β )) ū ψ ū β M β )). Therefore, one can proceed as for the terms II, III, IV to bound the terms IV and V I. Thus.5) is proven for u m 4 and so one can apply the supersolution result of Theorem..7 for the subsolution u m 4. One proceeds as in the case that m 4 0 and M 4 0 to finish the proof. Remark..9. The previous theorem tells us that if σr) 0 as R 0, then lim ux) = uq) X Q is well defined, i.e. we can extend u to in a continuous way. Lemma..0. Let L O and be a Lipschitz domain. Assume that u is a weak solution and osc R Q)u 0 as R 0 for all Q, then u is uniformly continuous on. Proof. The proof is obvious since continuous functions on a compact set are always uniformly continuous. Nevertheless, we will formulate a proof, which will provide a further result. Fix ε > 0. Since is a Lipschitz domain, satisfies a uniform cone condition at every Q. Let M = sup u. By Theorem..8, we see that there exists β > 0 depending on M and the constants C and α in Theorem..8 such that osc Tβ Q)u ε for all Q, i.e. u is uniformly continuous on ) β/. It remains to be shown that osc BγX)u ε for some γ < β / and all X β/. Weak solutions 0

22 are Hölder continuous on β/. Thus, there exists α and C such that ux) uy ) C X Y α. For X Y < γ with γ small enough, depending on C, β and α we get ux) uy ) < ε. Hence osc Bγ X) u ε for all X, where γ can be chosen in a way that it depends on ε and the constants in the definition of the operator class O, but is independent of u. Remark... Following the proof of Lemma..0, we see that a uniformly bounded sequence of weak solutions u k corresponding to a sequence of elliptic operators L k O is equicontinuous on. Finally, we can show that the continuous Dirichlet Problem for elliptic operators in O is solvable: Theorem... Let L O and be a Lipschitz domain. For every g C 0 ), there exists a unique u W, loc ) C0 ) such that Lu = 0 in and u g on. Proof. Choose {ϕ m } C ), which converge uniformly to g. Let u m be the weak solution to Lu m = 0 in and u m = ϕ m on existence is guaranteed by Theorem..6). Define α mk = sup ϕ m ϕ k we also denote the weak solution with the constant boundary value α mk by α mk ), then lim X Q u m u k α mk 0 for all Q, hence by the maximum principle u m u k α mk. The same holds for u k u m and therefore sup u m u k sup ϕ m ϕ k 0. So u m converges uniformly to some u C 0 ) with u ϕ on. In addition, by the interior Cacciopoli estimate we get u m u k ) 0 for all compact. The uniqueness of limits implies u W, loc ), and since the coefficients are in L ), one sees that u is a weak solution. Uniqueness follows from the maximum principle. We will finish this section with a result about subsolutions from [Sta65] for our type of elliptic operators. The proof given in [Sta65] works equally well for L O. For completeness, we will include this proof from [Sta65], which is based on the following Hilbert space result: Corollary..3 Corollary. in [Sta65]). Let B, ) be a bilinear form on W, ) W, ) such that for fixed g W, ), Bg, ) is continuous on W, 0 ) and B, ) is coercive on W, 0 ) W, 0 ). Let U W, ) be a convex and closed subset with Tru) = Trg) for all u U. Then, for a fixed f W, )), there exists a unique u U such that Bu, v) f, v for all v V u = {v W, ) : it exists ε > 0 such that u + εv U}. Theorem..4. Let be a Lipschitz domain and L O. Assume that u and v are subsolutions, then w = max{u, v} is a subsolution. Proof. We follow the proof of Theorem 3.5 in [Sta65]. Let U = {ϑ W, ) : T rϑ) = T rw), ϑ w}. Then U is closed and convex. Let L σ be the bilinear form corresponding to L σ, σ > 0, which was defined in the proof of Theorem..5. Then L σ is coercive on W, 0 ) W, 0 ) and so by Corollary..3, there exists a unique η U such that L σ η, ϕ) σ wϕ.0) for all ϕ V η. Since all non-positive ψ C0 ) are in V η we get for all non-negative ϕ C0 ) since η w by the definition of U that Lη, ϕ) σw η)ϕ 0.

23 This says that η is a subsolution. We claim that w η which clearly finishes the proof, since it implies η = w. To see this, let ζ = max{u, η}, then ζ U. We will show that ζ = η. Since ζ U, we get ζ η V η choose ε = ) and so, by.0), L σ η, ζ η) σ wζ η). Since u is a subsolution and ζ η = 0 on the set where η u, we get Therefore, L σ ζ, ζ η) σ ζζ η). Thus Lζ, ζ η) = Lu, ζ η) 0. L σ ζ η, ζ η) σ ζ w)ζ η) 0, since ζ η 0 by the definition of ζ and ζ w 0 by the definition of U. Therefore, ζ = η and so u η. The same reasoning shows that v η and so w η..3 Behaviour of weak solutions at the Boundary In this section, we will study the behaviour of weak solutions, which vanish at a part of the boundary. The first result we get is that weak solutions of that kind are viable test functions for that part of the boundary. Lemma.3.. Let L O and be a Lipschitz domain. Assume that u W, loc T RQ)) CT R Q)) is a weak solution or a non-negative subsolution for L O on T R Q). In addition, assume that u vanishes on R Q). Then u W, T R Q)). Proof. We modify the proof for weak solutions to elliptic operators of the form diva u in the remark. of [CFMS8] for our type of elliptic operators. Without losing generality, we can assume that R min{r 0, β}. For s > 0 let ψ = [u s] + φ, where φ C R n ) with φ on B R Q) and 0 outside of B 3R/ Q) we mutually extend u to be 0 outside of T R Q)). Then ψ is non-negative and in W, 0 T R Q)). Hence it is a viable test function and so 0 Lu, ψ) = A u [u s] + φ ) + u B [u s] + φ ) C u [u s] + φ Du[u s] + φ. Using the product rule in the term involving A, ellipticity and Young s inequality with ε, we get that u φ C φ u + C u B [u s] + φ ) {u>s} {u>s} {u>s} + C C u [u s] + φ + C Du[u s] + φ. {u>s} {u>s} We claim that the terms involving B, C, D are bounded by C T R Q) φ u + 0 {u>s} u φ.