Generalizations of a Result of Lewis and Vogel

Transcription

1 Generalizations of a Result of Lewis and Vogel Kris Kissel A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 007 Program Authorized to Offer Degree: Mathematics

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3 University of Washington Graduate School This is to certify that I have examined this copy of a doctoral dissertation by Kris Kissel and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. Chair of the Supervisory Committee: Tatiana Toro Reading Committee: Zhen-Quin Chen Steffen Rohde Tatiana Toro Date:

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5 In presenting this dissertation in partial fulfillment of the requirements for the doctoral degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of this dissertation is allowable only for scholarly purposes, consistent with fair use as prescribed in the U.S. Copyright Law. Requests for copying or reproduction of this dissertation may be referred to Proquest Information and Learning, 300 North Zeeb Road, Ann Arbor, MI , , to whom the author has granted the right to reproduce and sell (a) copies of the manuscript in microform and/or (b) printed copies of the manuscript made from microform. Signature Date

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7 University of Washington Abstract Generalizations of a Result of Lewis and Vogel Kris Kissel Chair of the Supervisory Committee: Professor Tatiana Toro Mathematics We discuss two generalizations of the fact that a bounded domain with a wellbehaved harmonic measure and a constant Poisson kernel is a ball. One generalization studies the case when the domain is unbounded and the Poisson kernel is close to a constant in a pointwise sense. The second generalization studies the bounded situation when the Poisson kernel is close to constant in the sense that it has small BMO-seminorm. A prioiri regularity assumptions are Ahlfors regularity and nontangential acessibility.

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9 TABLE OF CONTENTS Page List of Figures ii List of Symbols iii Chapter : Introduction Free Boundary Problems History Preliminaries Chapter : An Unbounded Generalization of Lewis and Vogel Definitions Lemmas Main Theorem Chapter 3: A Bounded Setting With a BMO Condition Crude Estimates A Finer Estimate on the Gradient of the Green s Function The Inside Ball The Outside Ball Bibliography i

10 LIST OF FIGURES Figure Number Page. The boundary Ω sits between the two spheres with radii R and R An M-non-tangential ball with center P, radius r and distance d B from the boundary Some non-tangential balls in an interior corkscrew, and a Harnack Chain Not NTA Domains Not Unbounded NTA In the first graphic, Θ(Q, r) = ǫ. In the second graphic, G F(σ + ; σ ; τ) The darkened arcs on the figure at right represent the image under the projection P of the ellipse in the left figure ii

11 LIST OF SYMBOLS R R n Ω B(x, r) or B r (x) B(x, r) or B r (x) S L n H k α(n) the real numbers n-dimensional euclidean space an open set in R n the open ball of radius r centered at x in R n the closed ball of radius r centered at x in R n the topological boundary of the set S R n n-dimensional Lebesgue measure k-dimensional Hausdorff measure the Lebesgue volume of the unit ball in R n σ n the H n (surface) measure of the unit sphere in R n+ X f dx the average value of f over the set X Df or f D f of f the gradient of the function f (this may be a classical or weak derivative, depending on context) the hessian matrix of the function f (classical or weak second derivative) C k (Ω) the space of k-times continuously differentiable functions on Ω for k C(Ω) C k c (Ω) L p (Ω) L p loc (Ω) the space of continuous functions on Ω the space of functions in Ck (Ω) with compact support the space of functions f on Ω such that Ω f p dx < the space of functions f on Ω such that K f p dx < for each compact set K Ω iii

12 α, β or α β the inner product n i= α iβ i on R n diam(e) the diameter of the set E, given by sup x,y E x y spt(f) the support of the function f iv

13 ACKNOWLEDGMENTS The author wishes to thank the Mathematics Department of the University of Washington for its generous support throughout his time as a graduate student there, and in particular his advisor, Tatiana Toro, whose patience and guidance have made this dissertation possible. v

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15 Chapter INTRODUCTION. Free Boundary Problems A typical problem in the beginning study of partial differential examples is a boundary-value problem, and the Dirichlet problem for Laplace s equation provides one of the simplest examples: Does there exist a function u satisfying u = 0 in D. (.) u = g on D If so, what are it s regularity properties? Here, the domain D and the boundary data g are given. A free-boundary problem turns this question inside-out. Assume that the domain D is unknown, but that a solution u of (.) is known to exist, and that it has certain regularity properties. What then can be said about the regularity or the geometry of the domain D? The known properties of u can come in a variety of forms. A typical setting occurs when u arises from a minimization problem in the calculus of variations as in the following example. Given a domain Ω R n, let J be the functional J(u) = u + Q(x) dx, (.) Ω {u>0} where Q is a given function and we have the side condition u 0 on D. Here the domain of integration Ω {u > 0} is variable. Taking the first variation for

16 this functional tells us that a minimizer will satisfy u = 0 in Ω {u > 0}, u = 0 and u = Q on Ω {u > 0}. Problems like this in the calculus of variations arise in many physical situations. For example, in [] the authors study the problem of optimizing heat flow in a steady state through a surface Ω with a given finite volume of insulation on it s interior. This question may be cast in terms similar to (.). The existence and regularity properties of u, and of the boundary {u > 0}, for (.) were studied by Alt and Caffarelli in []. Although our point of view will be very different from this calculus-of-variations one, we mention this example in particular because the paper [] is the source of an important technique, called non-homogeneous blow-up, of which we will make use later on. This technique also played a major role in some of the articles on which this dissertation is based. Our approach will be to study the geometry and regularity of a domain as determined by certain properties of the harmonic measure on its boundary. The harmonic measure for the ball B r (P) with pole at P is a constant multiple of the surface measure (n-dimensional Hausdorff measure) on B r (P). In [5], Lewis and Vogel proved the converse result (with an additional hypothesis): If ω P is harmonic measure for a bounded domain Ω with pole P Ω satisfying ω P = ch n Ω, and such that ω P (B ρ (Q)) Lρ n for all Q Ω and ρ > 0, then Ω is a ball centered at P. The additional hypothesis is needed to get a crude lower bound on the radius of a ball contained in Ω and to make careful estimates of certain integrals that describe the rate of decay of positive harmonic functions at the boundary Ω.

17 3 There are several directions in which one could try to generalize this result: What if Ω is unbounded? What if the harmonic measure is given by hh n Ω, where the function h is close to a constant. The second question here also requires that we specify what we mean by a function being close to a constant, and there are multiple ways one might define this. Kenig and Toro [3] answered the first question. Preiss and Toro [6] answered the second question with the assumption that h was close to a constant in a point-wise sense. In this dissertation, we will combine these ideas to address what happens when h is close to constant in a point-wise sense and Ω is unbounded. We will also consider the bounded case with the point-wise assumption replaced by one that says log h has small mean oscillation.. History We begin with a tour of the earlier results upon which this work is based. Readers unfamiliar with the terminology may wish to look through Section.3 first. In 00, Lewis and Vogel published in [5] that, if Ω R n+ is a bounded domain, regular for the Dirichlet problem, containing the origin, such that the harmonic measure with pole at 0 satisfies ω 0 (B r (X)) Ω) Lr n for all X Ω and 0 r r 0 and ω 0 = ah n on Ω

18 4 for a positive constant a, then Ω is a ball with center at 0. It is then immediate that the radius of the ball is also determined by the constant a, since we must have ah n = ω 0 ( Ω) =, so that H n ( Ω) = a. If we now write Ω = B r (0), we get Ω σ n r n = a, where σ n = H n (B (0)). Therefore r = (σ n a) n. This result was a generalization of an earlier proof (from 99) by the same authors that had required an additional hypothesis regarding regularity of the boundary of Ω. The following is a rough sketch of the 00 result. The first step is to obtain a crude estimate of v near Ω, where v is the Green s function for Ω with pole at 0. (When we say crude in this dissertation, we mean that it is not the best estimate we will obtain just a starting point that will lead to more refined results later.) This is done using the Riesz Representation Formula for Subharmonic Functions. The idea is that, once you have an estimate of the form v N near Ω, with N = N(L), the comparison principle for harmonic functions allows you to obtain a lower bound on the radius of the largest ball centered at the origin that is contained in Ω. Let R = sup{r > 0; B r (0) Ω}, and let G R denote the Green s function for B R (0) with pole at 0. Then if Q Ω B R (0), the comparison principle gives G R (0) v(q) N(L);

19 5 but we also know G R (0) = σ n R n, so we can conclude R (σ n N(L)) n. That is to say, R is bounded below in terms of L. The next step is to define M = lim sup X Ω v and to prove, via contradiction, that M a. Achieving this allows one to repeat the calculations above to get R (σ n a) n ; the isoperimetric inequality then implies Ω = BR (0). Having a crude upper bound on v near Ω is an important element of the indirect proof. Preiss and Toro [6] generalize this result as follows: Suppose that Ω R n+ is a bounded domain containing the origin that satisfies sup sup 0<r< Q Ω H n (B r (Q) Ω) r n <. Then given ǫ > 0 small enough, if the Poisson kernel h for Ω with pole at 0 exists and satisfies then with sup log h < ǫ, Ω B R (0) Ω B R (0) e ǫ σ n R n σ n R n e ǫ. One may think of this as a stability result for the theorem of Lewis and Vogel: a small perturbation of the Poisson kernel from constant results in only a small geometric perturbation of Ω from a ball. The paper [6] actually goes further than this geometric result. The authors also prove a regularity result for the boundary Ω: if δ > 0 is sufficiently small

20 6 R R Figure.: The boundary Ω sits between the two spheres with radii R and R. there exists ǫ > 0 so that, under the same conditions as above, Ω is δ-reifenberg flat. (See Definition (9) below.) That argument is based in large part on the techniques in []. The idea there is to define flatness of the boundary at a point Q Ω in terms of the lineargrowth behavior of the Green s function and the associated Poisson kernel near Q. Having done this, one can use the theory of partial differential equation to improve estimates, and thereby improve the measure of flatness at Q in successively smaller neighborhoods. Alt and Caffarelli employ this argument to show that, if the function Q used in the functional (.) is Holder continuous, then the free-boundary {u > 0} Ω of the minimizer u is in fact smooth. That technique is modified slightly in [6]. The setup there does not allow the estimates to improve dramatically as one looks at successively smaller neighborhoods of Q. But the estimates do persist as one looks at successively smaller neighborhoods of Q, so the measure of flatness at a larger scale can be

21 7 duplicated at all smaller scales. Heuristically, the idea in [] is that flatness at one scale implies greater flatness at a smaller scale ; in [6] the idea is that flatness at one scale implies similar flatness at all smaller scales. Kenig and Toro [3] develop a generalization of the results in [] and, in the process, obtain a generalization of Lewis and Vogel along different lines. Instead of considering a bounded domain, the authors asked what happens when Ω is unbounded. They proved that there exists δ n > 0 such that if Ω R n+ is an unbounded δ-reifenberg flat chord arc domain (for δ (0, δ n )), the Green s function with pole at, v, and the corresponding Poisson kernel, h, satisfy sup v(x) and h(q) for H n a.e. Q Ω, X Ω then Ω is a half space, and in suitable coordinates, v(x, x n+ ) = x n+. To summarize, given very loose assumptions about the growth properties of harmonic measure on Euclidean balls, Lewis and Vogel proved that constant Poisson kernels correspond to balls for bounded domains, while Kenig and Toro showed that they correspond to half spaces for unbounded domains. Again, a key idea was to use the flat at one scale implies greater flatness at smaller scales argument described in []. The generalization sought in Chapter of this dissertation combines these ideas. We assume that Ω R n+ is an unbounded domain, and that it s Poisson kernel is not much less than, while the Green s function maintains a linear growth near the boundary; and we assume that the boundary is flat at very large scales. We also require that Ω be Ahlfors regular and that the harmonic measure be well-behaved: ω(b r (Q)) Lr n for all balls B r (Q) centered on Ω. We use procedures similar to those in [3] to prove that the boundary is also flat at all small scales. This result is a quantitative version of [3].

22 8 Chapter 3 of this dissertation takes a very different approach to showing stability of the result of Lewis and Vogel. We start with a bounded domain. However, instead of perturbing the Poisson kernel h from constant in a pointwise sense, as in [6], we perturb it in an average sense by assuming that log h is a function of bounded mean oscillation (BMO) with small BMO-seminorm. Our argument will begin as did Lewis and Vogel s, with a crude estimate on the gradient of a Green s function near Ω. However, to improve the gradient estimate, the processes in [5] and [6] make explicit use of point-wise properties of h which we will not have assumed. Instead, we modify gradient estimates used in [4] when studying Poisson kernels of bounded mean oscillation. As above, a comparison principle argument allows us to turn a gradient estimate for the Green s function into a lower bound for the radius of the largest ball B R (0) contained in Ω. Moving from there to an estimate on the radius of a ball containing Ω again requires a different approach than is used in any of these other papers. In [5] and [6], the authors were able to make use of their knowledge of the pointwise behavior of h to estimate the total surface measure H n ( Ω); then another comparison principle argument in [6] or an application of the isoperimetric inequality in [5] completes the argument. With only a hypothesis about the average behavior of h, we will not be able to use either approach. Instead, we use a kind of piecewise projection of Ω onto concentric balls around the origin to make an estimate of how much of the boundary is far from the inner ball already discovered. That quantity cannot be too large, and then using the assumption that Ω is Ahlfors regular, we manage to conclude that no point of Ω can be very far from that inner ball. The process that goes from flatness at a large scale to flatness at smaller

23 9 scales does not seem to work here, however, because it requires some knowledge of the point-wise behavior of h. Therefore the results in Chapter 3 are purely geometric (Ω is close to being a ball ), and we do not discuss regularity..3 Preliminaries Because we will typically be concerned with the boundary of a domain, it will be most convenient for us to consider the boundary to be n-dimensional and therefore that our domains be open subsets of R n+..3. Geometric Measure Theory Definition. The Hausdorff distance between two nonempty sets A, B R n+ is defined to be D[A, B] = sup a A dist(a, B) + sup dist(b, A). b B Notice that the Hausdorff distance between two closed sets A and B is zero if and only if A = B. This notion of distance provides a metric on the class C R of nonempty compact subsets of B R (0) R n+ ; in fact, (C R, D) is a compact metric space. In particular, Cauchy sequences have limits: if A k is an infinite sequence in C and for all ǫ > 0, D[A l, A m ] < ǫ for l, m N(ǫ), then there is a compact set A C such that D[A k, A] 0 as k. The proof of this fact is a standard exercise in the theory of metric spaces. Definition. The k-dimensional Hausdorff measure of a set E in Euclidean space is { } H k (E) = lim inf c k (diam(e i )) s ; E E i and 0 diam(e i ) < ǫ, ǫց0 i= i=

24 0 where the constants c k are chosen so that H k agrees with k-dimensional Lebesgue measure on R k : H k (B r (0)) = r k B (0) dx for B (0) R k. (We will have no need here for Hausdorff measure of fractional dimension.) We used here the diameter of a set E R n+ : diam(e) = sup X Y. X,Y E The notation H k S means the measure is restricted to a set S: (H k S)(E) = H k (S E). Note that H n Ω is surface measure on Ω for a smooth domain in R n+. Definition 3. A domain Ω R n+ is said to have finite perimeter if { } sup div φ dx; φ Cc (Rn+ ; R n+ ), φ <. Ω When Ω is smooth, this supremum coincides with surface measure because div φ dx = φ ν dh n Ω Ω dh n (since φ ) Ω = H n ( Ω), and we obtain equality by choosing φ to agree with the outward unit normal vector field ν along Ω. An advantage of this definition is that it makes sense for any Ω R n+ no a priori regularity need be assumed. Definition 4. A set Ω R n+ is said to have locally finite perimeter if for each open set V R n+ with compact closure, we have { } sup div φ dx; φ Cc(V ; R n+ ), φ <. V Ω

25 As a consequence of the Riesz Representation Theorem, if Ω has locally finite perimeter, there is a Radon measure µ on R n+ and a µ-measurable function ν : Ω R n+ such that ν(x) = for µ a.e.x Ω, and div φ dx = φ ν dµ for all φ Cc(R n+ ; R n+ ). Ω R n+ Note that this appears to be a generalization of the divergence theorem, except that we do not yet have much information about the measure µ or its support. Definition 5. If Ω is a set of locally finite perimeter, we say that X Ω, the reduced boundary of Ω, if. µ(b r (X)) > 0 for all r > 0,. lim r 0 Br(X) σdµ = σ(x), and 3. ν(x) =, with ν and µ as above. By the Lebesgue-Besicovitch Differentiation Theorem (section.7. in [5], µ( Ω Ω) = 0. Blow-ups of the reduced boundary lead to half spaces: If X Ω, define Ω r = {Y R n ; r(y X) + X Ω} and Then H (X) = {Y R n+ ; σ(x) (Y X) 0}. χ Ωr χ H (X) in L loc (Rn+ ) as r ց 0.

26 (See Section 5.7. of [5].) Furthermore, ν = H n Ω. This implies H n ( Ω K) < for each compact set K R n+. Definition 6. Let X R n+. We say that X Ω, the measure theoretic boundary of Ω, if and lim sup rց0 lim sup rց0 L n+ (B r (X) Ω) r n+ > 0 L n+ (B r (X) \ Ω) r n+ > 0. Lemma in section 5.8 of [5] shows that Ω Ω and H n ( Ω \ Ω) = 0. Theorem in the same section then gives the full generalization of the divergence theorem: If Ω R n+ has locally finite perimeter, then H n ( Ω K) < for each compact K R n+ ; and, for H n a.e. X Ω, there is a unique measure-theoretic unit outer normal vector ν Ω (X) such that div φ dx = φ ν Ω dh n for all φ C c(r n+ ; R n+ ). E Ω Definition 7. A domain Ω R n+ is said to be Ahlfors regular if there is a constant A such that, for all Q Ω and all r (0, diam(ω)), r n A Hn ( Ω B r (Q)) Ar n. Observe that an Ahlfors regular domain has locally finite perimeter. Definition 8. A domain Ω R n+ is said to have the separation property if for each compact set K R n+ there exists R > 0 such that for Q Ω K and

27 3 r (0, R] there exists an n-dimensional plane L(Q, r) containing Q and a choice of unit normal vector to L(Q, r), n Q,r, satisfying { T + (Q, r) = X = x + tn Q,r B r (Q); x L(Q, r) and t > } 4 r Ω and T (Q, r) = { X = x + tn Q,r B r (Q); x L(Q, r) and t < } 4 r Ω c. Moreover, if Ω is an unbounded domain we also require that Ω divide R n+ into two distinct connected components Ω and Ω c. Definition 9. Let δ > 0 be small, and let Ω R n+ be a set of locally finite perimeter. We say that Ω is a δ-reifenberg flat chord arc domain or a Reifenberg flat chord arc domain if. Ω has the separation property.. For each compact set K R n+, there exists R K > 0 such that for every Q K Ω and every r (0, R K ], inf L { } r D[ Ω B r(q), L B r (Q)] δ, where the infemum is taken over all n-planes through Q. Moreover if Ω is unbounded we require R K =. 3. Ω is Ahlfors regular..3. Harmonic Functions Definition 0. A bounded domain Ω R n+ is said to be regular for the Dirichlet problem if, for every continuous function g C( Ω), there is a solu-

28 4 tion u of the boundary-value problem u = 0 u = g u C (Ω) C(Ω) in Ω on Ω (.3) The class of regular domains is very general (see chapter of [6]). Here, is the Laplace operator: u = u u solutions of u = 0 are called x xn+; harmonic functions. Notice that, if u and u solve (.3) for the boundary data g = g and g = g, respectively, then au +bu solves (.3) for g = ag +bg (where a and b are any constants). Harmonic functions satisfy the weak maximum principle: sup u = sup u. X Ω X Ω They also satisfy the strong maximum principle: if u(p) = sup Ω u for some P Ω, then u is constant on the connected component of Ω containing P. We will refer to either of these in this dissertation as a maximum principle. In particular, they imply that, for a connected domain Ω, solutions of (.3) are unique. Therefore, for any P Ω, the mapping is linear, and so for fixed P Ω, g C( Ω) u g C( Ω) u(p) (.4) defines a linear functional on C(Ω). Furthermore, if we equip C( Ω) with the uniform norm, g = sup Ω g, then we see that (.4) is actually a bounded linear functional because the strong maximum principle gives us u(p) g. Consequently, the Riesz Representation Theorem (see Theorem 6.9 in [7]) tells us that there is a probability measure ω P defined on Ω such that u(p) = g dω P. (.5)

29 5 Definition. The measure defined by (.5) is called the harmonic measure for Ω with pole P. Definition. For n, the function Φ(X) = is called the fundamental solution of. (n )σ n X n (.6) The fundamental solution satisfies Φ = δ 0 in the sense of distributions, where δ Y is the point mass at the point Y. That is to say, for any η C c (R n+ ) we have R n+ η(y )Φ(Y ) dy = η(0). This last equality can be verified directly by an argument using the divergence theorem and integration-by-parts. (See Chapter of [4].) (When n =, the function F(X) = ln X provides a fundamental solution; but in this dissertation we will only be concerned with the case n.) If Ω is regular for the Dirichlet problem and P Ω, then there is a solution u P of the boundary-value problem u P = 0 u P (X) = F(X P) u P C (Ω) C(Ω) in Ω for X Ω Then the function G P defined on Ω \ {P } by G(X) = F(X P) u P (X) satisfies G P = 0 on Ω, G P > 0 on Ω \ {P } and G P = δ P. Definition 3. G P is called the Green s function for Ω with pole at P. We typically extend G to be a continuous function on R n+ \ {P } by setting G = 0 on Ω c.

30 6 If Ω is a C domain (i.e. the boundary Ω is locally the graph of a continuously differentiable function), then it turns out that the normal derivative G P ν, where ν is the inward unit normal vector along Ω, provides us with a means to write down the harmonic measure for Ω with pole at P ; then dω P = G P ν dhn Ω. (.7) The proof of this fact is an application of the divergence theorem. It is important to note here that Ω need not be a C domain for the Green s function and the harmonic measure to exist. But as long as Ω is a domain for which a general divergence theorem holds, then something like (.7) will hold. (See the notion of domains of locally-finite perimeter, defined above.) Suppose now that Ω has locally finite perimeter, so that differentiation with respect to H n Ω makes sense. If it turns out that if ω P is absolutely continuous with respect to H n on Ω, then the Lebesgue-Radon-Nikodym derivative h P = dωp dh n exists, is nonnegative, is H n -measurable, is unique up to a set of H n -measure zero, and satisfies dω P = h P dh n. Definition 4. The function h P is called the Poisson kernel for Ω with pole at P. It satisfies u(p) = gh P dh n for u and g as in (.3). Because ω P is a probability measure and h P 0 h P dh n = dω P =, so we also have h P L ( Ω; H n ). Ω Ω Ω

31 7 EXAMPLE: Let Ω = B R (0) R n+. Then G(X) = (n )σ n X n (n )σ n R n is a Green s function for Ω with pole at the origin. Harmonic measure with pole at the origin turns out to be a multiple of surface measure: ω 0 (E) = H n (E) H n ( B R (0)). (This is a result of the mean value theorem for harmonic functions: see Chapter of [4].) The corresponding Poisson kernel is therefore constant: h P (Q) = H n ( B R (0)) for all Q Ω..3.3 Non-tangentially Accessible Domains The notion of a (bounded) non-tangentially accessible (NTA) domain was introduced in the 98 article [9] by Jerison and Kenig. In that paper, the authors generalize the classical theory of boundary behavior of harmonic functions known previously for Lipschitz domains (see the 970 article [8]). NTA domains are much more general than Lipschitz domains, but they maintain many of the classical properties of Lipschitz domains. For example, it was shown by Calderon in 950 (in [3]) that if u is a harmonic function in R n+ + which is non-tangentially bounded at every point of a measurable set E R n +, then u has a non-tangential limit at almost every x in E. In a 96 article [8], Stein posed the question of extending these results (and others) to the most general domains for which non-tangential behavior is meaningful. Hunt and Wheeden [8] took up the challenge for Lipschitz domains, and Jerison and Kenig extended the results to their even more general NTA domains.

32 8 Definition 5. A ball B r (X) Ω is called M-non-tangential if M + r dist(x, Ω) (M + )r. M This is equivalent to saying that M d r the ball B r (X) from the boundary Ω. M, where d is the distance of Ω P r d B Figure.: An M-non-tangential ball with center P, radius r and distance d B from the boundary. Suppose that B r (P ) and B r (P ) are M-non-tangential balls with B r (P ) B r (P ). Then M + M r d(p, Ω) P P + d(p, Ω) r + r + (M + )r, and thus r M(M + )r. Switching the roles of r and r yields M r r Mr for M = M(M + ). (.8) That is to say, M-non-tangential balls that meet have comparable radii, with comparison constant M. Definition 6. If X, X Ω, a Harnack chain from X to X is a collection of M-non-tangential balls B,..., B k such that X B, X B k and B i B i+

33 9 for all i =,..., k. The length of the chain, k, is the number of balls in the collection. Definition 7. A bounded domain Ω R N is said to be non-tangentially accessible (or NTA for short) if there exist M > 0 and r 0 > 0 such that the following conditions are satisfied:. (Interior Corkscrew Condition) For all Q Ω and every r (0, r 0 ) there exists a point A r (Q) Ω satisfying r M dist(a r(q), Ω) A r (Q) Q r.. (Exterior Corkscrew Condition) For all Q Ω and every r (0, r 0 ) there exists a point Ãr(Q) Ω c satisfying r M dist(ãr(q), Ω) Ãr(Q) Q r. 3. (Harnack Chain Condition) For every X, X Ω, if ǫ min{dist(x, Ω), dist(x, Ω)} and X X k ǫ, then there is a Harnack Chain in Ω from X to X of length at most Mk. Unbounded NTA domains will be defined below. Remarks: The definition here for bounded domains is slightly stronger than the one given in [9]. In that article, the authors do not require that the length of the Harnack chain be bounded by Mk, only that the length depends on k but not ǫ. However, in the intervening years, the definition given here has become the working definition in the field. See, for example, [0]. The points

34 0 A r (Q) and Ãr(Q) are called M-non-tangential points relative to Q, and the constants M and r 0 are referred to as the NTA constants of Ω Observe that the balls B r M (A r(q)) are 3M-non-tangential. Ω Q X X Ω Figure.3: Some non-tangential balls in an interior corkscrew, and a Harnack Chain. The Exterior Corkscrew Condition allows one to construct barriers whose existence prove that these domains are regular for the Dirichlet problem. The other two conditions are essential to producing certain estimates about the harmonic measures supported on the boundaries of NTA domains. In essence, these conditions generalize the main properties of smooth and Lipschitz domains which lead to doubling properties of the harmonic measure. Indeed, the following types of bounded domains are all subsets of the class of NTA domains: Smooth Domains Lipschitz Domains Zygmund Domains Quasispheres A quasisphere in R n+ is the image of B (0) R n+ under a quasi-conformal mapping R n+ R n+. A Zygmund domain is a domain whose boundary is

35 Ω X X P Q Ω Ω 3 Figure.4: Not NTA Domains locally the graph of a Zygmund-class function, i.e. a function in the family { } Λ (R n ) = φ : R n φ(x + z) + φ(x z) φ(x) R; sup <. x,z R n z Obviously C functions are Zygmund-class, but so are some nowhere-differentiable functions, including the Weierstrass function cos (3 n x) φ(x) =. n n= In order to better understand which domains are NTA, we illustrate in Figure.4 some domains which fail these hypotheses. The domain Ω fails the Interior Corkscrew Condition at Q; Ω fails the Exterior Corkscrew Condition at P ; and Ω 3 fails the Harnack Chain Condition, which we see by letting X and X get closer to each other on opposite sides of the vertex. Non-tangentially accessible domains were developed in [9] to generalize what was know about the behavior of harmonic functions on Lipschitz domains due to Hunt and Wheeden [8]. We collect some of those important facts here as the following three lemmas. Lemma. Harmonic measure on an NTA domain is a doubling measure: If Ω is NTA, X Ω and ω X is the harmonic measure for Ω with pole at X, then ω X (B r (Q)) C X ω X (B r (Q)).

36 This is proved as Lemma 4.9 in [9]. The next fact is Lemma 4. in [9]. Lemma. Positive harmonic functions that vanish continuously on the boundary of an NTA domain do so in a Hölder continuous fashion: If Ω is NTA with constants M and r 0, there exists β > 0 such that for all Q Ω, r < r 0 and every positive harmonic function u in Ω, if u vanishes continuously on Ω B r (Q), then for X Ω B r (Q), u(x) M( X Q r ) β C(u), where C(u) = sup{u(y ); Y B r (Q) Ω}. The main estimate we will need when working with NTA domains is a relationship between the harmonic measure of a ball and the value of the Green s function nearby (Lemma 4.8 in [9]): Lemma 3. There exists C = C(M) > such that C < ωy (B δ (x)(q)) δ(x) n G X (Y ) < C, where ω Y is harmonic measure with pole at Y, M is the NTA constant of Ω, G X is the Green s function for Ω with pole at X, and δ(x) = inf{ X P ; P Ω}..3.4 Unbounded Domains As mentioned previously, bounded NTA domains were introduced the the 98 paper [9], but unbounded NTA domains were not introduced until 999 in [] by Kenig and Toro. The definition is essentially the same, except we drop the constant r 0 and require the interior and exterior corkscrew conditions to hold for all distances r from the boundary. The modification is necessary in order to ensure a desired doubling property for the harmonic measure of unbounded Ω and a Harnack inequality.

37 3 Q Ω 4 Figure.5: Not Unbounded NTA Definition 8. An unbounded domain Ω R N is said to be non-tangentially accessible (or NTA for short) if there exist M > 0 such that the following conditions are satisfied:. (Interior Corkscrew Condition) For all Q Ω and every r > 0 there exists a point A r (Q) Ω satisfying r M dist(a r(q), Ω) A r (Q) Q r.. (Exterior Corkscrew Condition) For all Q Ω and every r > 0 there exists a point Ãr(Q) Ω c satisfying r M dist(ãr(q), Ω) Ãr(Q) Q r. 3. (Harnack Chain Condition) For every X, X Ω, if ǫ min{dist(x, Ω), dist(x, Ω)} and X X k ǫ, then there is a Harnack Chain in Ω from X to X of length at most Mk. The unbounded smooth domain Ω 4 in Figure.5 is not an NTA domain because it fails the interior corkscrew condition: non-tangential balls do not have enough room to grow proportionally to their distance from Q as that distance

38 4 increases without bound.the complement of Ω 4 would also not be NTA because it would fail the exterior corkscrew condition. We can also discuss Green s functions, harmonic measures and Poisson kernels for unbounded domains, except that these ideas become more complicated since solutions of the Dirichlet problem need not be unique on unbounded domains (because the maximum principle does not apply). In particular, the map (.4) is not well-defined, so we must be more direct. Definition 9. Let Ω R n+ be an unbounded domain. We say that a continuous function G : Ω R is a Green s function with pole at for Ω if G > 0 G = 0 G = 0 in Ω on Ω in Ω Definition 0. If Ω has locally finite perimeter, an associated Poisson kernel for Ω with pole at is a function h on Ω such that for all φ C c (Rn+ ) we have G φ dx = R n+ Ω φh dh n. Example: Let Ω = {(x, x n+ ) R n+ ; x n+ > 0}; let G(x, x n+ ) = c max{x n+, 0} for any positive constant c; and set h = c on Ω. Then G is a Green s function for Ω with pole at and h is an associated Poisson kernel, as the following application of the divergence theorem shows:

39 5 G φ dx = R n+ = = = 0 = = = {x n+ >0} {x n+ >0} {x n+ >0} cx n+ div ( φ) dx div (cx n+ φ) dx cx n+ φ, ν dh n div (φ cx n+ ) dx + φ cx n+, ν dx + {x n+ >0} {x n+ >0} {x n+ >0} {x n+ >0} {x n+ >0} {x n+ >0} cx n+, φ dx {x n+ >0} {x n+ >0} φ cx n+, e n+ dx + 0 φh dh n. c x n+, φ dx φdiv (c x n+ ) dx φc x n+ dx In the surface integrals above, ν represents an outward-pointing unit normal vector on the stated boundary. Unbounded NTA domains always admit Green s functions with pole at they are constructed as scaled limits of Green s functions with finite poles that tend to infinity (see []). As the previous example shows, unlike Green s functions on bounded domains, Green s functions with pole at are not unique; however, on unbounded NTA domains they differ only by scalar multiplication: G and G are Green s functions with pole at for an unbounded NTA domain Ω if and only if G = c G for some c > 0. Moreover, the Poisson kernel is unique up to a set of H n - measure zero for a given Green s function whenever it exists, as it does when Ω is Ahlfors regular. For these reasons, authors sometimes refer to these objects as the Green s function and the Poisson kernel. See [] for full details of the existence and uniqueness up to scalar multiplication for unbounded, Ahlfors regular NTA domains.

40 6 Chapter AN UNBOUNDED GENERALIZATION OF LEWIS AND VOGEL In this chapter, we begin with a hypothesis that says the boundary of a domain is flat at one point at all large scales, and we then prove that that boundary is flat everywhere at large scales. Then the domain is assumed to be unbounded and NTA and to admit a positive harmonic function that vanishes linearly at the boundary, and we show that the boundary is also flat at all small scales.. Definitions We define the following quantity to measure how much the boundary of a domain Ω differs from a plane near a point Q: Definition. Let Θ(Q, r) = r inf L Q D[ Ω B r(q), L B r (Q)], the infemum being taken over all planes L containing Q. When Θ(Q, r) is small, we think of Ω as being flat at Q at the scale r. There is another notion of flatness that will allow us to use the theory of partial differential equations to make estimates. It requires us to use properties of a Green s function for the domain Ω. The idea is due to Alt and Caffarelli (see []).

41 7 Definition. For Ω R n+, Q 0 Ω, ρ > 0 and σ +, σ, τ > 0, we say that a Green s function G on Ω satisfies G F(σ + ; σ ; τ) in B ρ (Q 0 ) in the direction ν if G(X) = 0 for X B ρ (Q) with X Q 0, ν σ + ρ, (.) G(X) h(q 0 )[ X Q 0, ν +σ ρ] for X B ρ (Q) with X Q 0 ; ν σ ρ, (.) and lim sup G(X) ( + τ) and h(q) > ( τ) for Q Ω B ρ (Q 0 ) (.3) X Q Ω B ρ(q 0 ) for an associated Poisson kernel h. We will refer to this notion later as F-flatness when we wish to distinguish it from the concept in definition, which we will call Θ-flatness. It is immediate that, if G F(σ + ; σ ; τ) in the ball B ρ (Q 0 ) in the direction ν, then G F(σ + ; σ ; τ) in the ball B ρ (Q 0) in the direction ν. The content of several of the lemmas later in this discussion will be refinements of this statement when G has certain properties. The relationship between the two notions of flatness discussed here is illustrated in Figure. If L is the plane through Q normal to ν and G F(σ + ; σ ; τ) in B ρ (Q) in the direction ν, then Θ(Q, ρ) max{σ + ; σ }. Going from Θ-flatness to F-flatness however requires an additional separation property. Definition 3. Let Ω R n+ be an unbounded domain. We say that Ω has the exterior separation property at large scales if there exists R > 0 such that for each r > R and Q Ω there is n-dimensional plane L(Q, r) containing Q and a choice of unit vector e(q, r), normal to L(Q, r), satisfying {X = (x, t) = x + te(q, r) B(Q, r) : x L(Q, r), t > 4 r } Ω c. (.4)

42 8 B (Q) r B (Q) ρ L (Q) r L (Q) r G=0 ε r Ω c σ r Ω c σ + r Q Q Ω Ω G > h(q)[<x Q, ν >+ σ + ρ ] Figure.: In the first graphic, Θ(Q, r) = ǫ. F(σ + ; σ ; τ). In the second graphic, G The coefficient in this definition is not special: any coefficient in the open 4 interval (0, ) would serve our purposes, but we fix it here for convenience. Proposition. Let Ω R n+ satisfy the exterior separation property at large scales. Let v be a Green s function on Ω with lim sup X Q Ω v + τ, and suppose the associated Poisson kernel satisfies h > τ. Assume also that there is a hyperplane L through Q Ω satisfying D[L B r (Q), Ω B r (Q)] < δr for some δ < and r sufficiently large. Then v F(δ; ; τ) in B 4 r(q) in the direction ν, where ν is perpendicular to L. Proof: Let ν be a unit vector perpendicular to L. Because D[L B r (Q), Ω B r (Q)] < δr, we see that Ω B r (Q) {X B r (Q); δr X Q, ν δr}. Let L(Q, r) be the plane and e(q, r) the unit vector guaranteed by the exterior separation property. Then we see that {X B r (Q); X Q, ν δr}

43 9 and {X = x + te(q, r) B(r, Q); x L(Q, r) and t 4 r } have a nonempty intersection. Let Y be a point in this intersection, and if it turns out that Y Q, ν < 0, replace ν by ν; thus by connectivity we have {X B r (Q); X Q, ν δr} Ω c. Consequently, the Green s Function for Ω is zero on this set, which tells us that (.) is satisfied with σ + = δ. The condition (.) is vacuously true when σ =. The other hypotheses of this proposition guarantee the conditions in (.3) are met, so we are finished.. Lemmas Lemma 4. Suppose that 0 Ω and lim inf r Θ(0, r) < δ for some sufficiently small δ > 0. Then for all Q Ω we have lim inf r Θ(Q, r) < 6δ. Proof: The hypothesis tells us that there is an increasing sequence r j ր such that Θ(0, r j ) < δ. Fix Q Ω and select J N such that r j > Q for j J. Let L j be a plane through 0 with unit normal n j and such that Let r j D[ Ω B rj (0); L j B rj (0)] < δ. L Q j = L j + e Q

44 30 where e Q = Q, n j n j. Note that L j + e Q = L j + Q and e Q < δr j. We will show that this plane satisfies the required estimates involving Hausdorff distance. There are two steps to that argument. Step : First we show that L Q j B r j /(Q) is not too far from Ω B rj /(Q). Let X L Q j B rj /(Q). Then there exists X L Q j B( 4δ)r j X X < δr j. Set X = X e Q, so that X L j. Also, (Q) such that X X Q + Q X X + X Q + Q < δr j + ( 4δ)r j < r j. + r j Thus X L B rj (0). Hence there exists Y Ω B rj (0) such that X Y < δr j. Therefore Y X Y X + X X + X X < δr j + δr j + δr j = 4δr j, and Y Q Y X + X X + X Q < δr j + δr j + ( 4δ)r j = r j.

45 3 That is to say, Y Ω B rj /(Q) and Y X < 4δr j. This proves that sup dist(x, Ω B rj /(Q)) < 4δr j. X L Q j B r j /(Q) Step : Next we show that Ω B rj /(Q) is not too far from L Q j B r j /(Q). Let X Ω B rj /(Q). Then there exists Y L B rj (0) such that X Y < δr j. Let Y = Y + e Q. Then Y L Q j B r j (0) and X Y X Y + Y Y < δr j + δr j = δr j. Note that Y Q Y X + X Q < δr j + r j. Then there exists Y L Q j B r j /(Q) such that and Y Q < r j Y Y < δr j.

46 3 Hence Y X Y Y + Y X < δr j + δr j = 4δr j. and That is to say, Y L Q j B r j /(Q) Y X < 4δr j. This proves that sup dist(x, L Q j B r j /(Q)) < 4δr j, X Ω B rj /(Q) completing Step. Putting together the results of Step and Step yields Consequently, D[ Ω B rj /(Q); L Q j B r j /(Q)] < 8δr j. (r j /) D[ Ω B r j /(Q); L Q j B r j /(Q)] < 6δ. This holds for all j such that r j > Q, and Q Ω was chosen arbitrarily. Therefore, for all Q Ω, we have lim inf r Θ(Q, r) < 6δ. Lemma 5. Let Ω R n+ be unbounded and NTA, let v be a Green s function for Ω with pole at, and suppose that the corresponding Poisson kernel h satisfies h > τ. Let Z Ω and assume there exists a ball B Ω c so that Z Ω B. Then v(x) lim sup X Z,X Ω dist(x, B) τ.

47 33 Proof: Let v(x) l = lim sup X Z,X Ω dist(x, B). There exists a sequence Y k Ω such that Y k Z and v(y k ) dist(y k,b) l as k. let d k = dist(y k, B). Then there exists X k Ω such that d k = Y k X k. Define for X B (0). Define Z k = Y k X k. v k (X) = v(d kx + X k ) d k By passing to a subsequence, we may assume that as k we have Z k e with e = ; v k v in C 0,β loc (Rn+ ); V k v weakly star in L loc(r n+ ) and weakly in L loc(r n+ ); d k ( Ω X k ) = {v k > 0} {v > 0} (in the Hausdorff distance sense, uniformly on compact sets); and χ {vk >0} χ {v >0} in L loc (Rn+ ). Note that v k (Z k ) = v(y k) d k, so v k (Z k ) l as k. Also, because v k converges uniformly to v on B (0), we obtain v (e) = l. Our goal is to show that Ω = {v > 0} is a half space, and that v is linear. Let r be the radius of the ball B. Let L k be the tangent plane to B through X k, and let α k = D[ B dk (X k ) B, L k B]. Observe that α k = d k r. Fix { P k P B dk (X k ); P X k, Y k X k d k < α k }. If Q k = P k X k d k, then Q k { } Q B (0); Q, Z k < d k r, and vk (Q k ) 0. Passing to the limit as k, we conclude that if Y B (0) and Y, e 0 then v (Y ) = 0.

48 34 Let Y B (0) satisfy Y, Z k > 0; then either d k Y + X k Ω c and V k (Y ) = 0 or d k Y + X k Ω and, given ǫ > 0, there exists k 0 N such that for k k 0, and which implies v(d k Y + X k ) dist(d k Y + X k, B) l + ǫ v(d k Y + X k ) (l + ǫ)dist(d k Y + X k, B) { (l + ǫ) d k Y, Y } k X k + d k d k r { (l + ǫ)d k Y, Z k + d } k, r v k (Y ) = v { k(d k Y + X k ) (l + ǫ) Y, Z k + d } k. d k r Letting k, we see, for Y B (0) with Y, e 0, that v (Y ) (l+ǫ) Y, e for every ǫ > 0; thus v (Y ) l Y, e. Moreover, v (e) = l. The maximum principle guarantees that v (Y ) = l max { Y, e, 0} for all Y B(0, ). If h k (X) = h(d k X + X k ), for η Cc (B (0)), η > 0, then as k we have ηh k dh n = v k η v η = lηdh n, {v k >0} R n+ R n+ { Y,e =0} thus lim ηh k dh n = lηdh n. (.5) k {v k >0} { Y,e =0} On the other hand, the divergence theorem gives us {v k >0} As k, we get ηdh n {v k >0} {v k >0} div(ηe) = = ηe ν k dh n = {v >0} {v >0} { Y,e =0} {v k >0} div(ηe) ηdh n ηdh n. div(ηe).

49 35 Therefore lim inf k {v k >0} ηdh n { Y,e =0} ηdh n. Then because the Poisson kernel h is at least τ for a.e. Q Ω, we obtain lim h k ηdh n ( τ) lim ηdh k {v k >0} k {v n, k >0} and together with (.5) this implies for any η C c (B (0)). Therefore l ηdh n ( τ) ηdh n { Y,e =0} { Y,e =0} l τ. Lemma 6. Let Ω R n+ be unbounded and NTA; let v be a Green s function for Ω with pole at. There exist δ n > 0 and τ n depending only on n so that for δ (0, δ n ) and τ (0, τ n ), if v F(σ; ; τ) in B ρ (Q 0 ) in the direction ν, then v F(σ; Cσ; τ) in B ρ(q 0) in the direction ν. The constant C here depends only on n. Proof: Without loss of generality, assume Q 0 = 0 Ω, ρ = and ν = e n+. Define η : R n R by exp η(y) = ( 9 y 9 y ) for y < 3 0 otherwise (The precise choice of test function here isn t important we just need to fix one since some constants will depend upon it.) Let. D = {X B (0); x n+ < σ sη(x)},

50 36 where X = (x, x n+ ) R n+. Choose s 0 to be the maximum s so that B (0) {v 0} D. Since 0 Ω = {v > 0}, we see that σ s 0 0, so s 0 σ. Since v F(σ; ; τ) in B (0) in the direction e n+, there exists Z D Ω B(0). Note that D B (0) is smooth. Let B D c be a tangent ball to D at 3 Z; because of our chose of η and the fact s 0 σ σ n, which is small, we may take the radius of B to be a constant C n. Define a function V by V = 0 in D D = V = 0 on D B (0). V = σ x n+ on D \ B (0) By the maximum principle, V > 0 in D. Also, we see that v V on D because v F(σ; ) in B (0) in the direction e n+. Thus the maximum principle also tells us v V in D, since v is subharmonic. We also have v(x) lim sup X Z,X Ω dist(x, B) V (Z), (.6) n where n denotes the inward unit normal vector to D. For X D define F(X) = (σ x n+ ) V (X). Then F is harmonic on D, continuous on D and 0 F s 0 on D; hence the maximum principle says 0 F s 0 on D. Since Z is a smooth point of D, standard boundary regularity arguments (see section 6. of [6]) ensure that Therefore sup X D F(X) C sup F Cs 0 Cσ. D V (Z) = F (Z) ( + Cσ). x n+ x n+

51 37 Consequently, V (Z) = V (Z), n n = V (Z), n + e n+ V x n+ V (Z) n + e n+ + ( + Cσ) ( + Cσ) n + e n+ + ( + Cσ). Near Z, D is a graph of a vertical translation of η, so we can calculate the inward normal vector there: ( n(z) = s η(x) + s η(x), ). + s η(x) This yields n + e n+ Cσ, with C = C(n). Putting this together with (.6), we have Together with Lemma 5, this gives us v(x) lim sup X Z,X Ω dist(x, B) V (Z) + Cσ. n v(x) τ lim sup X Z,X Ω dist(x, B) + Cσ. Next let ξ B3(0) {x n+ < }, and let ω 4 ξ satisfy the equation ω ξ = 0 in D \ B(ξ) 8 ω ξ = 0 on D. ω ξ = x n+ on B(ξ) 8 The Hopf boundary point lemma (see Lemma 3.4 in [6]) ensures that for some C = C(n), ω ξ n (Z) C > 0. Suppose d > 0 and that for every X B (ξ) we get 8 v(x) V (X) + σdx n+.

52 38 Then by the maximum principle, we have v(x) V (X) σdω ξ (X) on D \ B(ξ). 8 Consequently, τ V n (Z) σd ω ξ (Z) + Cσ Cσd. n Therefore τ Cσ Cσd, so or Cσd Cσ + τ, d C C + τ σ. That is to say, if d > C C + τ, then there exists X σ ξ B (ξ) for which 8 v(x ξ ) V (X ξ ) + σd (X ξ ) n+. Let X B(X ξ ), and recall that V (X) x n+. Then 4 v(x) v(x ξ ) sup v X X ξ B 4 (ξ) V (X ξ ) + σd(x ξ ) n+ ( + τ) 4 (X ξ ) n+ + σd(x ξ ) n+ 4 τ σd 4 τ 4 = σd τ ( σ C ) + τ τ 8 C n 4. Thus if σ and τ are sufficiently small, we have v(x) > 6 > 0.

53 39 This tells us that v is harmonic on B(X ξ ), and so is V v. Furthermore, we 4 (ξ), so Harnack s inequality yields have V v > 0 on B 4 (X ξ ) B 8 (V v)(ξ) C n (V v)(x ξ ) Cσd(X ξ ) n+ Cσ, and v(ξ) V (ξ) Cσ ξ n+ Cσ. For X D B(0) there is a ξ B3(0) and a t > 0 such that X = ξ + te n+. 4 Then v(x) = v(ξ + te n+ ) v(ξ) t (ξ n+ + t) Cσ. Since v F(σ; ; τ) in B (0) in the direction e n+, the last inequality proves v F(σ; Cσ; τ) in B(0) in the direction e n+. Notation: For y R n, define B r (y) = {(x Rn ; x y < r}. In particular, B r = B r (0). (The point is that our usual ambient space is Rn+, and we wish to distinguish balls in that space from balls in R n.) Lemma 7. Let Ω R n+ be unbounded and NTA, and let v be a Green s function for Ω with pole at. Suppose that Q j Ω with Q j Q and σ j ց 0, and that v F(σ j ; σ j ; τ) in B ρj (Q j ) in the direction ν j. Let R j be the rotation that maps {(x, x n+ ) R n+ ; x n+ 0} to {X + tν j R n+ ; X, ν j = 0 and t 0}. Define v j (X) = ρ j v(ρ j R j X + Q j ) (.7)

54 40 and for y B f + j (y) = sup{h; (y, σ jh) {v j > 0}}, f j (y) = inf{h; (y, σ jh) {v j > 0}}. Then there exists a subsequence of indices k j such that lim sup f + k j (z) = lim inf f j, z y j, z y k j (z). (.8) Let f(y) denote the function defined by (.8). Then f is continuous in B, it satisfies f(0) = 0, and f + k j and f k j converge uniformly to f on compact subsets of B. Proof: Let D j = {(y, a) R n+ ; (y, σ j a) {v j > 0} B }. Note that 0 D j. Also, because v j F(σ j ; σ j ; τ) in B in the direction e n+, we see that the scalars a in the definition of D j are in the interval [, ], so D j B. Furthermore, we see that B {x ; (x, x n+ ) D j }. We can pass to a subsequence such that D j converges to a set D in the Hausdorff distance sense, and D B. by For y B, let A y = { {y j } j= B ; lim j y j = y }. Define a function f on B f(y) = sup lim sup f + j (y j). {y j } A y j We will show that f(y) is the quantity on both sides of equation (.8). Step : f is upper-semicontinuous. Let z l B satisfy z l z B. We want to show that lim sup l f(z l ) f(z). W.L.O.G. we may pass to a subsequence for which lim l f(z l ) exists and equals the limit superior of the original sequence. Fix ǫ > 0 and choose {z k l } k= A z l such that f(z l ) ǫ < lim sup f + k (zk l ). k

55 4 There is a diagonal-like subsequence {z k i l i } i= such that Therefore z k i l i z and f + k i (z k i l i ) f(z li ) ǫ. lim inf i f k i (z k i l i ) f(z). Thus f(z) > lim sup i f(z li ) ǫ = lim i f(z li ) ǫ, and letting ǫ 0 gives us the desired inequality. Step : Upper-semicontinuity implies some flatness. Fix y B, and choose y k B so that lim k y k = y and lim k f + k (y k) = f(y). For x B, (x, f(x)) D, so for ǫ > 0 there exists δ ǫ > 0 so that, for δ (0, δ ǫ ), D {(x, x n+ ); x B (y, δ) and x n+ > f(y) + ǫ} =. Then because {D k } converges to D in the Hausdorff-distance sense on compact subsets of B, we have that for sufficiently large k, D k {(x, x n+ ); x B δ(y) and x n+ > f + k (y k) + ǫ} =. That is to say, if (x, x n+ ) B δ (y k) [σ k f + k (y k)+σ k ǫ, ) then v(x, x n+ ) = 0. This implies that, for some τ, v k F( σ kǫ δ ; ; τ) in B δ(y k, σ k f + k (y k)) in the direction e n+. Now by Lemma 6 we get for k large that v k F( σ kǫ δ ; C σ kǫ δ ; τ) in B (y δ k, σ k f + k (y k)) in the direction e n+. (.9) Step 3: Flatness lets us control lim inf f k. Now we see that if z B δ (y k ) and σ k h < σ k f + k (y k) C ǫσ k 4 δ δ = σ k f + k (y k) Cǫσ k then f k (z) f+ k (y k) Cǫ for z B δ(y k ). (.0) 4