Boundary Behavior of Solutions of a Class of Genuinely Nonlinear Hyperbolic Systems by Julian Gevirtz

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1 Boundary Behavior of Solutions of a Class of Genuinely Nonlinear Hyperbolic Systems by Julian Gevirtz ABSTRACT We study the set of boundary singularities of arbitrary classical solutions of genuinely nonlinear planar hyperbolic systems of the form H5V5 œ, here H5 3) denotes differentiation in the direction / 5 ÐV ßV Ñ, 5 œ,, and here the defining functions ) 5 satisfy ( i ) ) œ ) 1, and ) ( ii) E 5 l` ÐVÑ `V l F, for all V, 5 œ, 5 for some positive constants E, F We sho that for any system of this kind there is a 7 such that for any locally Lipschitz solution V in a smoothly bounded domain K, the set of points of `K at hich V fails to have a nontangential limit has Hausdorff dimension at most 7, and, on the other hand, for any such system for hich the ) 5 G Ð Ñ, e construct a G solution V on a half-plane for hich the set of points of ` at hich V fails to have a nontangential limit has positive Hausdorff dimension These results are immediately applicable to constant principal strain mappings, hich are defined in terms of a system of this kind for hich is a linear function of V and V ) 1 I ntroduction For hyperbolic systems in to independent variables B and >, most often associated ith space and time, one usually studies the Cauchy problem in hich one seeks a solution?ðbß>ñ, > for hich?ðbßñ coincides ith a given? ÐBÑ, the questions considered including ell-posedness, global existence, blo-up and behavior of solutions as > p _ In the nonlinear case discussion is often limited to initial data ith a small range and even for such data, generalized solutions must be considered In this paper e concern ourselves ith the folloing inverse question for a certain family of genuinely nonlinear hyperbolic systems: What can be said about the boundary values of an arbitrary classical solution in a plane domain K? Here classical can be taken to mean G _, although the treatment e give ill be valid for locally Lipschitz solutions In the first place, e are interested in systems hose formulation imposes no a priori limit on the range of characteristic directions, that is, systems such that for a characteristic given parametrically by DÐ=Ñ, argöd Ð=Ñ can potentially cover all of, in contrast to hat is implicitly the case in the standard spacetime context Secondly, e are interested in statements valid for all solutions rather than ones knon to arise from some form of initial value problem Because of this generality, even in geometrically simple domains such as disks or half-planes characteristics can be quite contorted curves Although the specific focus of this paper is the size of the set of boundary points at hich arbitrary solutions can fail to have nontangential limits, it ould be reasonable to investigate other aspects of their behavior and that of the associated characteristics In any event, given the nonstandard nature of the boundary value question and of several of the issues that arise in dealing ith it, e shall begin ith a somehat detailed discussion of a system for hich it is physically meaningful, namely the system hich describes smooth planar mappings ith constant principal stretches Ðcps-mappings Ñ, about hich e have previously ritten ([ChG],[G1]-[G5]) It is in fact the study of the boundary behavior of such mappings that is the main goal of this 1

2 paper, and e have only chosen to ork in a ider context because it is possible to do so ith little additional effort, and because this broader approach suggests some interesting questions A cps-mapping ith principal stretch factors 7 Á 7 is a mapping 0 À K p ith locally Lipschitz continuous Jacobian N 0 œ XÐ 9Ñ5 Ð7 ß7 ÑXÐ) Ñ, here cos ) sin ) 7 XÐ) Ñ œ and 5Ð7 ß7 Ñ œ ) ) sin cos 7 As is explained in the cited references, apart from regularity considerations, functions ) and 9 ill correspond to such a mapping on a simply connected domain K if and only if they satisfy the autonomous quasilinear hyperbolic system (11) H Ð7 ) 7 9 Ñ œ à H Ð7 ) 7 9Ñ œ, here H? œ Ðcos ) Ñ? B Ðsin ) Ñ? C and H? œ Ð sin ) Ñ? B Ðcos ) Ñ? C 3) 3) The characteristics of a solution are the integral curves of the fields / and 3/ It turns out that a net a made up of to mutually orthogonal families of curves covering a simply connected K is the net of characteristics of a cps-mapping if and only if for any to curves G ß G belonging to one of the families of a, the change in the inclination of the tangent is the same along all subarcs of curves of the other family hich join G to G Nets ith this property are knon as Hencky-Prandtl nets (See [CS], [G3], [Hem], [Hen], [Hi], [Pr]) The theory of cps-mappings e have developed is based on direct application of (11) together ith this Hencky-Prandtl (HP) property and the equations (12) H H ) œ ÒH ) Ó and H H ) œ ÒH Ó 2 ), hich are also effectively equivalent to (11) and hich are very special cases of equations derived by Lax [L] in the context of considerably more general genuinely nonlinear hyperbolic systems in the plane and used by him in connection ith the inevitability of singularity formation The blo-up equations (12) imply that if a characteristic G has curvature, at : then the orthogonal characteristic arc emanating from : toards the concave side of G can have length at most,, that is, that the boundary of K must be encountered after moving at most a distance of, along this orthogonal characteristic A characteristic length bound of this kind is a sine qua non for the theory e are developing and plays a fundamental role in all that is to follo When regarded as deformations ith constant principal strains, cps-mappings are of concrete interest as models in a number of physically interesting contexts (see [Y]) Consider, for example, a thin liquid film on a plane surface hich upon solidification takes on a rectangular cryptocrystalline structure, that is, at each point a suitably oriented minute square of the original liquid becomes a rectangular crystal hose side lengths are constant multiples of the side length of the square In this light global geometric results for cps-mappings tell one about the extent to hich the shape of the original film can change as a result of such solidification and about ho matter is moved around in the process, and statements about the existence of boundary limits of ) (and, in light of (11), of 9, and consequently of the Jacobian of the mapping) tell one to hat extent the cryptocrystalline structure is present at the very edge of the solidified lamina Applied to the system (11) the main result of this paper says that there is some number 7 such that if K is smoothly bounded, then ) and 9 can fail to have nontangential limits on a set W `K of Hausdorff dimension at most 7 On the other hand, the construction of 2

3 Section 5 shos that the set of boundary points at hich ) does not have nontangential limits can in fact have positive Hausdorff dimension Beyond their immediate physical significance, cps-mappings constitute a particularly important and tractable class of planar quasi-isometries, for hich e believe they ill ultimately be shon to display extremal behavior for many of the as yet unsolved distortion questions (see [J1], [J2]) In this direction a very significant inroad as made a fe years ago by Gutlyanskii and Martio [GM], ho shoed that the spiral mappings given by log< 3< 3Ð< < Ñ KÐ</ Ñ œ </ log 3, are extremal for the problem of determining for given 3 and <, the smallest 7 ratio 7, such that there is a quasi-isometry 0 of the annulus ldl 3 onto itself ith stretching bounds 7, 7 hich satisfies the boundary conditions 0ÐDÑ œ D and 0Ð3DÑ œ 3/ 3< D, for ldl œ It is in fact not hard to see that K is indeed a cps-mapping of the entire punctured plane ÏÖ ith È+ + È+ + 7 œ and 7 œ, here + œ < log 3 We call these spiral mappings because the corresponding inclination functions are of the form 3< (13) ) Ð</ Ñ œ <, here tan œ 7, hich is to say that the characteristics form to mutually orthogonal families of logarithmic spirals all members of each of hich are rotations of each other We no describe the class of systems for hich e treat the boundary singularity question Let ) 5 œ ) 5 ÐV ßV Ñ, 5 œ ß For given V ÐBßCÑ, V ÐBßCÑ and any? œ?ðbßcñ e rite ` (14) H 5? œ 5 ÐV ÐBßCÑßV ÐBßCÑÑ? ` cos ) sin ) 5 ÐV ÐBßCÑßV ÐBßCÑÑ? `B `C It is ell-knon that in general an autonomous quasi-linear homogeneous hyperbolic system for unknon functions 0 and 1 is formally equivalent to a system of the form H5V5 œ, 5 œ ß, ith appropriate inclination functions ) 5 ÐV ßV Ñ The relationship beteen the Riemann invariants V, V and 0, 1 is of the form V5( BßCÑ œ J5Ð0( BßCÑß1( BßCÑÑ Henceforth _ e œ Ð) ß) Ñ and use the term system to mean a G À p, although the smoothness requirement could be eakened substantially Obviously, a solution of the in a domain K of the plane is then a pair of functions V ÐBßCÑ, V ÐBßCÑ for hich V5 is constant on each integral curve (henceforth referred to as a 5-3) characteristic) of the field / 5 ÐV ÐBßCÑßV ÐBßCÑÑ, 5 œ ß A system is said to be genuinely `) nonlinear if the derivatives 5 `V, 5 œ, 2, never vanish It is clear that the system (11) 5 for the ) and 9 associated ith cps-mappings is already in Riemann invariant form ith V3 œ 73) 749, so that in this case to inclination functions are given by 7 V 7 V 1 (15) ) œ ) œ and ) œ ) 7 7 (Here e have used the convention, in force throughout this paper, to the effect that Ö3ß 4 œ Öß ) This system is obviously genuinely nonlinear and in fact is the simplest possible such system in that the to families of characteristics are mutually orthogonal 3

4 is a linear function of V œ ÐV ßV Ñ We no define the family of systems ith hich e ork Definition 11 A normal system is a for hich ( i) ) œ ) 1 and ( ii) There are constants E, F such that E l ` ) 5 `V l F for all V 5 The only hyperbolic systems ith hich e deal ill be normal systems, for hich e use the symbol ) to denote ) We shall sho that for any normal there is a 7 œ 7 Ð@ Ñ such that for any smoothly bounded K and any solution V on K the set of points of `K at hich V does not have a nontangential limit has Hausdorff dimension at most 7 This ill follo as an immediate consequence (Corollary 42) of our principal result, Main Theorem 22, hich deals ith boundary singularities of a class of functions effectively more general than the class of solutions of normal systems An outline of the proof, hich is actually made simpler by this somehat greater generality, is given early in Section 2, just after the statement of the main theorem Furthermore, in _ Section 5 e shall sho that for any there is a G solution V in the upper halfplane for hich the set of points of ` at hich V does not have nontangential limits has positive Hausdorff dimension 2 Quasi-HP Functions, the Characteristic Length Bound and Related Matters Let K be a domain If ) is a locally Lipschitz function on K, the integral 3) ÐDÑ 3) ÐDÑ curves of the fields / and 3/ ill be called the - and -characteristics of ), respectively The term full characteristic ill refer to the complete integral curve and e ill use the term half-characteristic to refer to either of the to arcs into hich a full characteristic is divided by one of its points (Characteristics hich are closed curves ill not arise in this paper) As indicated in the Introduction e ill use the convention Ö3ß4 œ Öß throughout Arcs of 5-characteristics ill be called 5-arcs, or less specifically, characteristic arcs With reference to a given ), a characteristic arc joining points +ß, H ill be denoted by +, and e shall use the abbreviation?) Ð+,Ñ œ ) Ð,Ñ ) Ð+Ñ A domain U K ill be said to be a positively ( negatively) oriented characteristic quadrilateral of ), and e rite U œ +,-, if `U is a Jordan curve lying in K containing four points +,,,, - occurring in that order hen `U is traversed in the positive (negative) sense and such that +, and - are 3 -arcs and +- and, are 4-arcs We say that +, and - are translates of each other ith respect to or along any 4-characteristic passing through +, For a curve parametrized by D œ DÐ=Ñ, =, e use the terms to the right of G, to the left of G in the obvious sense, so that, for example, if G is a characteristic arc and : G, e describe an orthogonal characteristic arc or a halfcharacteristic as emanating from : to the right or left of G We shall denote the -dimensional measure of \ by Ð\Ñ and the - dimensional measure of a set E by -ÐEÑ The parameter = ill alays refer to arc length We use the notation RÐ+ß<Ñ œ ÖD À ld +l < and denote the line segment joining + to, by +, The overline ill also be used to denote closure, but this should cause no confusion For \, ], dist Ð\ß] ) œ infölc Bl À B \, C ], and for +, distð+ß\ñ œ dist ÐÖ+ ß\Ñ 4

5 Definition 21 Let O A locally Lipschitz function ) on a domain K ill be said to be a O-quasi-HP function or to have the O-quasi-HP property if for any characteristic quadrilateral +,- K there holds (21) Ol?) Ð+-Ñl Ÿ l?) Ð,Ñl Ÿ Ol?) Ð+-Ñl The net consisting of all of the full - and -characteristics of a O-quasi-HP function ill be called a O-quasi-HP net A simple continuity argument shos that this definition implies that the?)ð+-ñ and?)ð,ñ in (21) must in fact have the same sign (unless both vanish) It is also obvious that a -quasi-hp net is an HP-net Note that hile the HP-property is a local condition that implies its global counterpart, this is not the case for the O-quasi-HP property hen O The net of characteristics of any locally Lipschitz solution to a normal is a O -quasi-hp net, here O œ OÐ@ Ñ Indeed, if E and F are as in Definition 11, then it is clear that e can take OÐ@ Ñ œ FÎE We can no state our Main Theorem 22 There is a number 7 œ 7 ÐOÑ ith the property that if ) is any O-quasi-HP function on a Jordan domain K ith G boundary, then the set of points of `K at hich ) fails to have a nontangential limit has Hausdorff dimension at most 7 Because the proof of this theorem, to be given in Section 4, is quite involved and depends on the prior development of a considerable amount of machinery in this and the folloing section, e shall briefly explain here ho it proceeds As e shall sho (see Corollary 329) for any point : `K at hich ) does not have a nontangential limit there is either a nontrivial fan of characteristics emanating from :, or for 5 œ or, : is not the endpoint of a 5 -characteristic; in the latter case e say that : is a 5-singularity Since, as ill be apparent, any quasi-hp function can have at most a countable number of fans, e only need to sho that the Hausdorff dimension, dimðwñ, of the set W of 5- singularities satisfies dimðwñ Ÿ 7 For this to be the case it is enough that there be some $ œ $ ÐOÑ such that any almost straight arc E `K has a subarc of length at least $- ÐEÑ hich has at most a countable set of 5-singularities of ) If for some ) there ere no such $, then for any R there ould have to be an almost straight arc E `K such that there is a set WR of 5-singularities of ) hich are essentially uniformly distributed along E It is not too hard to sho that any 5 -singularity : is surrounded by arbitrary small 5 -characteristics hich join boundary points on either side of : (see Proposition 319) For sufficiently large R, starting ith a set of R small 5- characteristics, one surrounding each of the points of WR, e sho that there must be a 5- characteristic G (here the term 5-characteristic is used in an appropriate sense - see the discussion of extended characteristics beteen Propositions 313 and 314) hose endpoints lie on E and are at least $ - ÐEÑ apart (here $ depends solely on O) and hich is tangent to E at a point 7 E, here 7 is appropriately bounded aay from the endpoints of G We then use this to obtain a subarc E of E hich contains 7, hose length is bounded belo by $ - ÐEÑ, here $, like $, depends only on O, and on hich ) can have at most a countable number of 5-singularities (see Proposition 333), thereby arriving at the desired contradiction A good measure of the complexity of the proof lies in establishing the existence of the extended characteristic G, hich is carried out in Section 4, but hich depends on properties of the net of extended characteristics of quasi- HP-functions developed in Section 3 5

6 We begin ith the folloing proposition, hich is immediate Proposition 23 (Invariance of O ) Let ) be a O-quasi-HP function on K, and +ß, ith + Á Then ) Ð+D,Ñ arg + is a O-quasi-HP function on + ÐK,Ñ The next proposition is a special case of [G1, Lemma 2]; its proof is included for the sake of completeness A function ) is said to be a locally P-Lipschitz function on K if each point of K has a neighborhood in hich ) satisfies a Lipschitz condition ith constant P Proposition 24 (Length Change Estimate) Let K be a simply connected domain and let ) be a locally P-Lipschitz function on K Let U œ +,- K be a positively oriented characteristic quadrilateral such that l, +l œ 6, l- +l œ Ÿ 6, and distðuß`kñ œ ( There is some 6 œ 6ÐPß( Ñ such that (22) l -l œ 6?) Ð+,Ñ SÐ 6 $ Ñ 0or 6 Ÿ 6, here the constant implied by the big- S depends only on P and ( Proof In hat follos, hen e say that some quantity is big- S of some expression, e mean that this is so for all 6 less than some positive number hich depends only on P and (, and that the constant corresponding to the big- S depends only on P and ( Let +, and - be 3 -arcs Without loss of generality e can assume that + œ and, œ 6 Let 1 1 and be the inclinations of the tangents to the 4 -characteristics at + and,, 1 1 respectively, and let œ argö- +, œ argö, Clearly,,, and are all SÐ6Ñ and œ ) It easily follos from the Lipschitz condition that 3 3 œ SÐ Ñ We have œ 3/ >/, here > œ l -l œ SÐ6Ñ and œ SÐ6Ñ We also have œ 6 =3/ 3, here = œ SÐ6Ñ Just as œ SÐ Ñ, one sees that œ SÐ=Ñ From the to expressions for e have that / >/ œ 6 =3/, so that considering real and imaginary parts e have (23) sin > cos œ 6 = sin and cos > sin œ = cos The latter equation implies that =Ð SÐ6 ÑÑ œ Ð SÐ6 ÑÑ SÐ6 Ñ, so that = œ SÐ6 Ñ Thus, since œ SÐ=Ñ, e have (24) œ SÐ 6 Ñ From (23) it no follos that $ $ >Ð SÐ6 ÑÑ œ 6 sin = sin œ 6 Ð SÐ6 ÑÑ Ð SÐ6 ÑÑÐ SÐ6 ÑÑ, so that from the fact that œ SÐ Ñ and (24) it follos that $ l -l œ > œ 6 Ð Ñ SÐ6 Ñ $ œ 6 Ð SÐ ÑÑ Ð SÐ 6 ÑÑ SÐ6 Ñ œ 6 Ð Ñ SÐ 6 $ Ñ Since œ?)ð+,ñ, e are done è Let G be a characteristic arc parametrized by D œ DÐ=Ñ, = Let H Ð) ßGÑ ) Ð= Ñ ) Ð= Ñ denote the infimum of all 7 such that = = Ÿ 7 for all = ß= Ð ß Ñ Note that H Ð) ßGÑ depends on the orientation of G implicit in the parametrization Proposition 25 (Characteristic Length Bound) Let ) be a O-quasi-HP function on K and let G K be an open 3-arc ith H Ð) ßGÑ œ, Then for any ( there is 6

7 some point : G such that the 4 -half-characteristic emanating from : to the left of G O has length at most, ( Proof Let $ be a small positive number Let G be parametrized by D œ DÐ=Ñ $ Clearly there is a point : œ DÐ= Ñ such that ) is differentiable at : and ) Ð= ), Ð Ñ Without loss of generality e can take = œ Let N be the 4-half-characteristic emanating from : to the left of G Let G be any compact arc of N hose initial point is : It is clearly sufficient to sho that O (25) - ÐG Ñ Ÿ $, Ð $ Ñ $ Let > $ be so small that the entire characteristic quadrilateral U for hich an 3-side is DÐÒß> ÓÑ and a 4-side is G is contained in K Let ( œ dist ÐUß`KÑ Let P be such that ) is an P-Lipschitz function in the (-neighborhood of U By decreasing the size of > if necessary e can assume that ) ÐDÐ>ÑÑ ) Ð:Ñ, Ð $ Ñ, for > Ÿ > ldð>ñ :l and that > Ÿ 6ÐPß( ), here 6ÐPß( ) is as in the preceding proposition Let ) Ð>Ñ œ ) ÐDÐ>ÑÑ ) Ð:Ñ Clearly, by taking a smaller positive value for > e can assume ) that for all > Ðß> 0 Ó and all ) Ð>Ñ $Î O, if 6 Ÿ ldð>ñ :l and œ 6, then $ (26) 6 ) SÐ 6 Ñ Ÿ 6 ) Ð $ Ñ, here the big- S is that of the previous proposition Let G be parametrized by A œ AÐ=Ñ, Ÿ = Ÿ - We take > Ÿ > such that if +, +, â, + 8 is an ordered sequence of points on G, the distance beteen any successive to of hich is at most >, then 8 (27) l+ + l Ð $ Ñ-Ð+ + Ñ 5œ Let 6 œ ldð>ñ :l, and let + œ :,, œ DÐ>Ñ Let G be the other 4-side of U Let $Î œ dist ÐG ßG Ñ Let + G to be the (first) point for hich l+ + l œ œ 6 Let G be the 3 -half-characteristic emanating from + to the right of N and let, be the point common to G and G From Proposition 24 and (26) it follos that 6 œ l, + Ÿ 6 ) Ð>ÑÐ $ Ñ Clearly, e can continue this process until e come to an + 8 G ithin 6 of AÐ- Ñ Taking into account that ) is a O-quasi-HP function and (27) e see that for this 8 e have that 8 ) Ð>Ñ ) Ð>Ñ Ÿ l, + l Ÿ 6 Ð $ Ñ l+ + l Ÿ 6 Ð $ Ñ -Ð+ + Ñ 8 8 O 5 5 O 8 5œ ) Ð>Ñ ) Ð>Ñ Ÿ 6 O Ð $ Ñ Ð- ÐG Ñ 6 Ñ Ÿ 6 O Ð $ Ñ Ð- ÐG Ñ $ Ñ, 6 O O ) Ð>Ñ Ð Ñ ) Ð>Ñ Ð Ñ Ð Ñ$ $ $, $ so that - ÐG Ñ $ œ Ÿ Thus indeed (25) holds è O 6 If D œ DÐ=Ñ, =, is a parametrization of an 3-arc G, then, 3ÐDÐ=ÑÑ œ ) ÐDÐ=ÑÑÎ= exists almost everyhere on Ð ß Ñ and gives the curvature of G at DÐ=Ñ It follos immediately from the preceding proposition that if G is an 3-arc and, 3 Ð:Ñ exists, then the 4 -half-characteristic emanating from : toards the concave side of G (that is, to the left or right of G according as, 3Ð:Ñ or, 3Ð:Ñ ) has length at O most, 3 Ð:Ñ Let 9 be a real-valued function defined on Ð ß Ñ, and let = Ð ß Ñ We denote 9Ð=Ñ 9 Ð= Ñ by H 9Ð= Ñ and H 9Ð= Ñ the loer and upper limits of as = p = and by = = 7

8 HV9Ð= Ñ and HP9Ð= Ñ ( HV9Ð= Ñ and HP9Ð= Ñ) the corresponding right and left loer (upper) limits We have the folloing simple corollary of the characteristic length bound Proposition 26 (Curvature Bound) Let D œ DÐ=Ñ, = be an 3-characteristic of a O-quasi-HP function ) on K and let N be the 4-half-characteristic emanating from DÐ= Ñ O to the left of G and joining it to `K Then H ÐDÐ= ÑÑ Ÿ ) -ÐNÑ Proposition 27 (Length Monotonicity) Let +,- be a positively oriented characteristic quadrilateral of a quasi-hp function ) on K Let D œ DÐ=Ñ, Ÿ = Ÿ, DÐ Ñ œ +, parametrize +, Let I be a --measurable subset of +, such that ) Ð=Ñ Ÿ at all points of I (That is, +, is nonconcave toards the inside of +,- at all points of I) If I is the set of points of - hich are joined to points of I by translates of +-, then -ÐI Ñ -ÐIÑ Proof This is a simple consequence of the length change estimate, the fact that ) is a quasi-hp function on K and elementary measure theory è In Section 4 e shall make use of the folloing to loer bounds for the area of certain regions made up of families of characteristic arcs of a O -quasi-hp function ) They follo as simple corollaries of Proposition 26; the constants, ( and ( hich appear in them depend solely on O Proposition 28 (Area Bound 1) If G is the interior of an 3-arc of a O-quasi-HP function ) of length - 3, and for each A G, G ÐAÑ is a 4-arc of G containing A and of length at least - 4 such that Y œ ÖG ÐAÑ À A G is open, then ÐYÑ (-3-4 Proposition 29 (Area Bound 2) If G is the interior of an 3-arc of a O-quasi-HP function ), and for each A G, G ÐAÑ is a 4-arc of G containing A and of length at least - such that Y œ ÖG ÐAÑ À A G is open, then ÐYÑ ( - l?) ÐGÑl Extended Characteristics, Regular and Singular Boundary Behavior Our approach to boundary behavior requires the examination of curves hich are in effect characteristics hose interiors (ie, sets of nonendpoints) contain boundary points; the subtleties that arise in this connection require careful discussion Hereafter the symbol Z ill denote the family of all Jordan domains K for hich `K is a G curve and ZÐ3Ñ Z ill denote the family of all K Z such that for each : `K the interior of one of the circles of radius 3 tangent to `K at : is contained in K and that of the other such circle is contained in ÏK Obviously, for K ZÐ3Ñ the unsigned curvature of `K is everyhere bounded by 3, so that Z œ ÖZÐ3Ñ À 3 Furthermore, HP ÐKß OÑ ill denote the family of O-quasi-HP functions on K Although sometimes the hypotheses of the propositions of this section do not state so explicitly, they alays deal ith K Z and ) HP ÐKßOÑ (Although the results of this paper apply to unbounded and multiply connected domains as ell, the proof of the main theorem itself ill entail only consideration of Jordan domains, and in fact e ill be able to ork largely ith Jordan domains of the kind e call characteristic subdomains, as defined belo) By an arc G e shall henceforth mean a continuous one-to-one mapping D œ DÐ>Ñ of an interval Ò ß Ó into the closure K of K and DÐÐ ß ÑÑ ill be referred to as the interior of G As in Section 2, hen an arc is considered to be oriented, e use the term to (toards) the right (left) of G to refer to the part of G (immediately) to the right (left) of G, and the term full characteristic ill refer to a complete integral 8

9 1 curve of ) or ) in K The folloing proposition as proved in [G5, Proposition 28] for HP-nets Although the proof is virtually the same in the present more general context e include it here for the sake of completeness Proposition 31 Let G be a full characteristic of a quasi-hp function in K parametrized by D œ DÐ=Ñ, = Ð ß Ñ Then limdð=ñ exists and belongs to `K =Ä Proof Clearly the conclusion holds if Á _, so that e assume œ _ First of all, e sho that dist ÐDÐ=Ñß`KÑ p as = p _ If this ere not true, then there ould be a D K and an such that for some sequence Ö= 3 tending to _, DÐ= 3 Ñ p D, but DÐÒ= 3ß= 3 ÓÑ `RÐD ß Ñ Á g But from this it ould follo that some orthogonal characteristic crosses G tice, an impossible occurrence in light of the simple connectivity of K We can no sho that, in fact, DÐ=Ñ p, `K as = p _ If this is not so, the foregoing then implies that there is an arc I of `K, -ÐIÑ, each point of hich is an accumulation point of G œ ÖDÐ=Ñ À = for each Ð ß_Ñ Since K is bounded and œ _, G cannot be a straight line, so that from the characteristic length bound it follos that there is an orthogonal half-characteristic G of finite length hich joins some DÐ5Ñ to a point / `K Since G cannot cross G tice in K, G 5 KÏG Let D, D be distinct points of IÏÖ/ For each $, G5 has a subarc :: RÐ`Kß$ ÑÏG, ith :ß: RÐD ß$ Ñ and a point : :: RÐD ß$ Ñ For obvious topological reasons, for each sufficiently small $, there must be a point ; on :: hich is joined to a point in RÐD ß$ Ñ by an orthogonal characteristic arc F of length at least ld D l $ such that the curvature of G at ; exists and tends to infinity as $ p and G is concave toards the side from hich F emanates But this clearly violates the characteristic length bound (Proposition 25), as indicated in the paragraph immediately folloing its proofè Definition 32 An arc G for hich DÐ Ñ, DÐ Ñ `K and for hich DÐÐ ß ÑÑ K and is a full 3-characteristic ill be called an elementary 3-characteristic In other ords, an elementary characteristic is a full characteristic together ith its endpoints, hich are ell defined by the preceding proposition Note that for each : `K, Ö: is a trivial elementary 3-characteristic An 3- characteristic arc (also called an 3-arc) ill be a subarc of an elementary 3-characteristic Lemma 33 Let K Z Then there exists a number F œ F ÐKÑ ÐßÓ ith the folloing property Let +,, `K Let G and G be the to closed arcs into hich `K is divided by + and, (If + œ,, G œ Ö+ and G œ `K) Then distðdßg Ñ distðdßg Ñ F distðdßö+ß, Ñ, for all D K Proof This is self-evidentè Proposition 34 (Bounded Length of Characteristics) Let K Z There is some Q œ QÐKßOÑ such that -ÐGÑ Ÿ Q for all elementary characteristics G of any ) HPÐKßOÑ Proof Let ) be a O-quasi-HP function on K and let G be an elementary 3-characteristic of ) We regard G as being oriented and let E œ G `K (that is, E is the set of endpoints of G ) Let œ diam ÐKÑ, and let œ supödistðdßeñ À D G Ÿ Þ Clearly, (31) ÐÖD K À dist ÐDßEÑ Ÿ )< Ñ Ÿ F<, here F œ ) 1 For 5, let K œ ÖD K À Ÿ distðdßeñ Ÿ

10 and G5 œ G K5, so that G K œ ÖG5 À 5 For each nonendpoint : of G, let NÐ:Ñ denote the elementary 4 -characteristic containing : Obviously, NÐ:Ñ NÐ: Ñ K œ g for : Á : For 5 and : G5 let 6Ð:Ñ and <Ð:Ñ be the first points encountered hen moving along NÐ:Ñ from : to the left and right of G, respectively, hich are not in the interior of K5 K5 K 5 Each ; Ö6Ð:Ñß<Ð:Ñ is either on `K or is in K and satisfies one of dist Ð;ßEÑ œ 5 or dist Ð;ßEÑ œ 5 Say ;  `K and dist Ð;ßEÑ œ 5 Then l: ;l 5, since otherise distð:ßeñ Ÿ l: ;l dist Ð;ßEÑ 5, hich contradicts the fact that : G 5 If ;  `K and dist Ð;ßEÑ œ 5, then l: ;l 5 since otherise distð:ßeñ dist( ;ßEÑ l: ;l, hich is inconsistent ith : G5 Thus l: ;l if ;  `K Hence, if at least one 5 of 6Ð:Ñ, <Ð:Ñ is not in `K e have for the open subarc N Ð:Ñ of N Ð:Ñ ith endpoints 6Ð:Ñ, <Ð:Ñ that - ÐN Ð:ÑÑ If, on the contrary, Ö6Ð:Ñß<Ð:Ñ `K, then it follos from 5 the preceding lemma that F F -ÐN Ð:ÑÑ l6ð:ñ :l l: <Ð:Ñl 5 5, so that this bound holds in all cases for : G5 since F Ÿ Since N Ð:ÑÏÖ6Ð:Ñß<Ð:Ñ K5 K5 K 5 for : G5, it follos from (31) that Fˆ ÐÖD K À Ÿ ÐDß`KÑ Ÿ Ñ œ ÐK K K Ñ 5 5 dist Ð ÖN Ð:Ñ À : G 5 Ñ (-ÐG5Ñ F 5, F by Proposition 28 From this e have that -ÐG Ñ Ÿ But since 5 ( F 5 F F F ( F ( G œ ÖG5 À 5, e conclude that -ÐGÑ Ÿ Ÿ Since F œ ) 1 and F and depend only on K, and ( depends only on O, e are done è Definition 35 An elementary characteristic G one of hose endpoints is : `K said to exit K at : If G is parametrized by D œ DÐ=Ñ, Ÿ = Ÿ P and lim ÐDÐ=ÑÑ exists, then G =Ä ) is said to exit regularly at DÐÑ, otherise it is said to exit singularly Proposition 36 Let K Z and ) HPÐKßOÑ Let G œ +, K be a closed 3-arc of ) and let I be an arc joining + and, in K such that I G œ Ö+ß, Then diamðgñ Ÿ &- ÐIÑ Proof Let H be the interior of the simple closed curve G I Obviously H K Let : G and ; I satisfy l: ;l œ sup ÖlD Al À D G and A I Let G be the elementary 4 -characteristic containing : Since a 4-characteristic can have at most one point in common ith an 3 -characteristic, there is a point ; I such that G contains a subarc N œ :; hose interior lies in H Let diam ÐGÑ œ ld D l, here D, D G Then diam ÐGÑ œ ld D l Ÿ ld +l l+,l l, D l Ÿ l: ;l -ÐIÑ, since ld +l, l, D l Ÿ l: ;l But l: ;l Ÿ l: ; l l; ;l Ÿ -ÐNÑ - ÐIÑ, so that diam ÐGÑ Ÿ -ÐNÑ $-ÐIÑ 10

11 But it follos by a simple argument based on Proposition 27 (Length Monotonicity) that -ÐNÑ Ÿ -ÐIÑ, since the elementary 3-characteristic through each interior point D of N contains a subarc joining to distinct points of I Thus diam ÐGÑ Ÿ &-ÐIÑ as claimed è Proposition 37 Let G œ :; and G œ :; be distinct elementary 3-characteristics of a O-quasi-HP function on K ith an endpoint : `K in common Let N K be a 4- characteristic arc joining a point of - of G K to a point - of G K, so that G, G and N form the three sides of a characteristic triangle X Let T N denote the set of points at hich N is not strictly concave toards the inside of X Then -ÐTÑ œ Proof Suppose not Then since the elementary 3-characteristic passing through any point of N must exit at :, after replacing the original N by an appropriate subarc and changing G and G accordingly e can assume that - ÐT NÑ œ, - Ð) ÐNÑ) O and -ÐRÑ ), here R œ NÏT Let G5 be parametrized by D5Ð=Ñ, Ÿ = Ÿ -5, ith DÐÑ œ : Let Ö5ß6 œ Öß Let N5Ð=Ñ denote the 4-arc joining D5Ð=Ñ to a point A5Ð=Ñ G6 K Note that e only kno that N5Ð=Ñ is defined for = Ÿ -5 sufficiently near - 5 It follos from length monotonicity (Proposition 27) that -ÐN5Ð=ÑÑ - ÐT5Ð=Ñ), here T5Ð=Ñ is the set of points of N5Ð=Ñ joined to points of T by an 3-arc By the quasi-hp property - Ð) ÐN5Ð=ÑÑÑ, so that N5Ð=Ñ is almost straight and in particular the distance beteen its endpoints is at least -ÐN5Ð=ÑÑ -ÐT5Ð=ÑÑ Let 05 be the infimum of all = for hich N5Ð=Ñ is defined Since the distance beteen the endpoints of N5 Ð=Ñ is at least, it is clear that at least one of 0, 0 must be positive, and for definiteness e assume that 0 For 5 Ð0 ß- Ó let N Ð5 Ñ be parametrized by ' Ð=ß5Ñ, Ÿ = Ÿ -ÐN Ð5 ÑÑ, ith ' Ðß5Ñ G It is clear that there are $, X such that (32) dist Ð' Ð=ß5 ÑßG G Ñ X=, for = Ðß$ Ñ, 5 Ð0 ß- Ó From the fact that a 4-arc can intersect an 3-arc at most once in K it easily follos that for each point D G K there is a $ œ $ ÐDÑ such that (33) ld ' Ð=ß5 Ñl $, for = Ò$ ß-ÐN Ð5ÑÑÓ, 5 Ð0 ß- Ó From (32) and (33) together ith the fact that for 5 Ð0 ß- Ó, N Ð5Ñ œ N Ð5 Ñ for some 5 Ð0 ß- Ó it follos that as 5 p0, N Ð5Ñ tends to an arc N hich contains :, hich joins D Ð0 Ñ to the point D Ð0 Ñ of G and the distance beteen hose endpoints is at least Furthermore, either N consists of a 4-arc joining D Ð0 Ñ to : or (in the case that 0 Ñ it consists of such an arc together ith another 4 -arc joining : to D Ð0 Ñ One of the endpoints, hich e henceforth call ;, is at a distance of at least from : For definiteness or by renaming e assume that ; œ D Ð0 Ñ G Let N be the subarc of N hich joins ; to : Let E denote the arc :; of G Then N and E form the to sides of a characteristic bilateral F Let T be the subset of points of N hich correspond to (ie, are joined by 3-arcs to) points of T, (that is, T is the set of points at hich N is nonconcave toards the inside of F), and let R N be the subset corresponding to R Since by length monotonicity, -ÐR Ñ Ÿ -ÐRÑ Ÿ ), it follos that (34) -ÐT Ñ -ÐN Ñ -ÐR Ñ ) œ ) Let N be parametrized by D Ð=Ñ, Ÿ = Ÿ -ÐN Ñ, ith D ÐÑ œ :, and for = Ðß-ÐN ÑÑ let EÐ=Ñ be the part of the elementary 3-characteristic through D Ð=Ñ in F, so that EÐ=Ñ joins D Ð=Ñ to : (since its interior can cross neither E nor N ) It follos from Proposition 36 that diam ÐEÐ=ÑÑ tends to as = p Let ;: be an initial arc of E for hich 11

12 - Ð) Ð;: ÑÑ O Since - Ð) Ð[ÑÑ for any translate [ of either N or ;: by the quasi-hp-property, it follos that e are able to translate ;: in F all the ay don N ÏÖ: from ; ithout meeting the boundary of F, since for simple geometric reasons all these translates, being essentially perpendicular to the virtually straight arc N must stay aay from : If E Ð=Ñ denotes the translate of ;: ith initial point D Ð=Ñ, then E Ð=Ñ EÐ=Ñ, and therefore diam ÐE Ð=ÑÑ p as = p -ÐN Ñ This means that each point D ;: is joined to : in F by a 4-arc N ÐDÑ such that if T ÐDÑ is the set of points of N ÐDÑ joined to points of T by 3 arcs in X, then by (34) and length monotonicity -ÐT ÐDÑÑ ) and by the quasi-hp-property - Ð) ÐN ÐDÑÑÑ, so that (35) ld :l ' We can no repreat this process starting ith N Ð: Ñ instead of N œ N Ð;Ñ, and continue doing so to obtain in the end N ÐDÑ for all D EÏÖ: Hoever, by the argument e just gave e no have (35) for all D E, hich is absurd since : is an endpoint of E Therefore -ÐTÑ œ è Proposition 38 The to endpoints of a nontrivial elementary characteristic of a quasi- HP function on K must be different Proof This follos easily from Proposition 36è Proposition 39 To elementary 3-characteristics of a quasi-hp function on G ith the same endpoints must be identical Proof Assume to the contrary that points : Á : of `K are joined by distinct elementary 3-characteristics G and G It follos from Proposition 38 that the elementary 3-characteristic through any point of the simply connected domain H bounded by G G must also have endpoints : and : But then it follos easily from Proposition 37 that all 4-characteristic arcs in H are straight line segments But this contradicts Proposition 37è Proposition 310 To different elementary characteristics of a quasi-hp function on K hich both exit K regularly at : K cannot be tangent to each other there Proof This is an easy consequence of Proposition 37 and the quasi-hp propertyè Definition 311 Let ) be a quasi-hp function on K and let G be an elementary 3- characteristic of ) ith endpoints +ß, `K Let F be one of the boundary arcs of `K ith endpoints +,, Then the subdomain H of K for hich `H œ G F ill be called an 3-characteristic subdomain When e ish to indicate the elementary 3-characteristic involved, e ill denote the 3-characteristic subdomain by ÐHßG Ñ The arc F œ `H G `K, called the bottom of ÐHßG Ñ and denoted by bot ÐHÑ, ill be considered to have the order corresponding to the positive orientation of `H We shall freely use interval notation as ell as the terms, to the right of, to the left of, beteen, etc hen dealing ith bot ÐHÑ Furthermore if +, is an elementary 3 -characteristic joining points +,, of bot ÐHÑ, it ill be understood that + Ÿ,, unless otherise indicated When dealing ith a characteristic subdomain ÐHßG Ñ, e shall ork ith the class \ ÐHÑ of nontrivial elementary 3-characteristics G hich join points :, ; of bot ÐHÑ Note that G \ ÐHÑ If G œ :; \ ÐHÑ, then above G refers to the part of H not in the closed region bounded by G Ò:ß;Ó If J H is a compact set for hich J bot ÐHÑ Á g, e say that an elementary 3-characteristic I œ :; \ ÐHÑ envelopes J, and rite J I, if J is contained the closed set bounded by the simple closed curve Ò:ß;Ó :; We only deal 12

13 ith sets J for hich each component of J has points in bot ÐHÑ For to such sets J, J H e say that J precedes J and rite J Ÿ J if for all, 0, 0 ith 05 J5 bot ÐHÑ, 5 œ ß there holds 0 Ÿ 0 It is clear that if G, G \ ÐHÑ, then one of the folloing is true: G G, G Ÿ G or G Ÿ G The first of these possibilities includes the case G œ G In hat follos PÐKß:ßÑ ill denote the segment :; of length hich is orthogonal to `K at :, hich emanates from : into K and hich is oriented from : to ; Proposition 312 There are an absolute constant = œ = and a constant = œ = ÐOß3Ñ ith the folloing properties Let K ZÐ3 Ñ and ) HPÐKßOÑ, and let G, parametrized by DÐ=Ñ, Ÿ = Ÿ -ÐGÑ, be an arc of an 3-characteristic the distance beteen hose endpoints is at least 3, for hich (36) G PÐKß:ß= 3Ñ œ ÖDÐÑ, hich emanates to the right (left) of PÐKß:ß= 3Ñ and for hich each 4-half- characteristic emanating to the left (right) of G has length at least 3 Let `K be parametrized by AÐ=Ñ, Ÿ = - Ð`KÑ, respectively ith DÐÑ, AÐÑ PÐKß:ß= 3Ñ and largöd ÐÑ argöa ÐÑ l Ÿ 1 Then argöd ÐÑ argöa ÐÑ = ÈlDÐÑ :l, if G emanates to the right of PÐKß:ß0 Ñ, and argöd ÐÑ argöa ÐÑ = È ldðñ :l, if G emanates to the left of PÐKß:ß= Ñ Proof Clearly it is enough to handle the case in hich G emanates to the right of PÐKß:ß3 Ñ Without loss of generality e can assume that : œ œ argöa ÐÑ, so that in particular H œ RÐ 3 3ßÑ ÏK and `RÐ 33ßÑ is tangent to `K at Let 3 V œ Ö3 >/ À >, œ ÐÑ Ðß 1 Ñ be the ray emanating to the right of PÐKßß3Ñ from the point 3 PÐKßß3Ñ hich is tangent to `H; and let the point of tangency be D Then 3 cos œ 3 œ 3 SÐ 3 Ñ, so that there is some E such (37) É É, for Ÿ E Let X be the curvilinear triangle bounded by PÐKßßÑ, the line segment Ò3ßD Ó and the (shorter) arc of `H ith endpoints, D Then it is easy to see that Ð38 Ñ diamðx Ñ Ÿ 3 Ÿ 23É œ È3, for Ÿ E 3, 3 E by a smaller value, if necessary) so that (after replacing (39) diam ÐX Ñ 3, for Ÿ E 3 No assume that G is as in the hypothesis ith DÐÑ œ 3, here Ÿ E 3 and that œ argöd ÐÑ Ÿ R Here R is some (large) number yet to be determined Then, since by (39) diamðgñ 3 diam ÐX Ñ and because obvious angle considerations imply that G enters X at 3, in light of assumption (36), G must exit X at a point of the segment Ò3ß D Ó Say = is the smallest value of = for hich DÐ=Ñ Ò3ßD Ó Then obviously 13

14 ) ÐÒß= ÓÑ Ò ß Ó, so that Òß= Ó has a subinterval Ò= ß= Ó for hich 1 ) ÐÒ= ß= ÓÑ œ [, Ó Since Ÿ it follos that = = œ -ÐDÐÒ= ß= ÓÑ Ÿ ldð= Ñ DÐ= Ñl Ÿ diam ÐX Ñ Ÿ Ð È3 Ñ, by (38) Thus by the mean value theorem there must be some = Ò= ß= Ó for hich ÐR Ñ (310) H ) ÐDÐ= ÑÑ Ð È3 Ñ But by the curvature bound (Proposition 26) and the hypothesis regarding the 4- O characteristics H ) ÐDÐ= ÑÑ Ÿ 3, so that in light of (310) and the loer bound in (37) e have 3ÐR ÑÉ Î (8È3 4 Ñ Ÿ O 3 É R 3 ) È E 3 But the left hand side of this inequality is ÐR ÑÎÐ) 4 Ñ for Ÿ E, so that e have a contradiction for R œ Ð) È E ÑO Thus if Ÿ E 3 e must have Ð) ÈE ÑO argöd ÐÑ 2 È, in light the upper bound in (37) Thus e have proved the proposition ith Ð) ÈE ÑO = œ È3 2 è È3 = œ E Proposition 313 There are positive constants 0 œ 0 ÐOß3Ñ and 0 œ 0 ÐOß3Ñ ith the folloing properties Let K ZÐ3 Ñ, ) HPÐKßOÑ, and let G œ +, be an elementary 3- characteristic, ith corresponding characteristic subdomain ÐHß GÑ Let G and E œ bot ÐHÑ be given by DÐ=Ñ, Ÿ = Ÿ -ÐGÑ and AÐ=Ñ, Ÿ = Ÿ -ÐEÑ, respectively, ith DÐÑ œ AÐÑ œ +, + being the leftmost point of botðhñ Then for each : œ AÐ5 Ñ E for hich distð:ß `KÏEÑ 3, the segment PÐKß:ß0 Ñ contains at most one point of G, and if there is such a point DÐ=Ñ, then there holds (311) largöd Ð=Ñ argöa Ð5 Ñ l Ÿ 0 ÈlDÐ=Ñ :l Proof Without loss of generality e may assume that : œ œ argöa Ð5 Ñ Let = and = be as in the preceding proposition Let PÐ Ñ œ PÐKß:ß Ñ We assume for the 3 moment that DÐ=Ñ PÐ= Ñ is the only point of G on this segment Since immediately to the left of G there are points in the complement of H and the interior of the segment PÐlDÐ=ÑlÑ lies in H, it is clear that if e rite argöd Ð= Ñ œ / 3 7, ith 1 7 Ÿ 1, then 1 l7l Ÿ Let I be one of the to subarcs of G one of hose endpoints is DÐ=Ñ and the other of hich is a point ; K for hich ldð=ñ ;l œ 3 It follos from the preceding 3 proposition that if = ÐOß ÑÈ 1 3 ldð=ñl Ÿ, then G cannot be tangent to PÐ= Ñ Thus, of 3 the to arcs I one moves to the right of PÐ= Ñ as e move along it aay from DÐ=Ñ and the other moves to the left But then by the preceding proposition e have (311) ith (312) 0 œ = ÐOß 3 Ñ for any DÐ=Ñ for hich = ÐOß3 ÎÑ = ÐOß3ÎÑ O (313) ldð=ñl Ÿ minö= ß Š Ÿ minö= ߊ ß œ 0 Assume Ÿ 0 and that PÐ Ñ contains at least to points of G Then there are =, = Ðß- ÐGÑÑ, = =, such that the interior of PÐminÖlDÐ= ÑlßlDÐ= Ñl Ñ contains no point of G (and is therefore contained in H) and DÐ= ÑDÐ= Ñ PÐ Ñ œ ÖDÐ= ÑßDÐ= Ñ By 3 Proposition 36, diam ÐDÐ= ÑDÐ= ÑÑ Ÿ & = There are the folloing to cases and 14

15 ( i) ldð= Ñl ldð= Ñl In this case, simple topological arguments sho that DÐ= ÑDÐ= Ñ must lie to the right of PÐ Ñ Also, by the foregoing and our definition of 0, one easily has largöd Ð= Ñ argöa Ð5 Ñ l Ÿ 1 But then since G crosses PÐ Ñ again at DÐ= Ñ, e must have that for some > > in Ð= ß= Ñ, argöd Ð> Ñ argöd Ð> Ñ œ, so that by mean value considerations as in the proof of the preceding proposition together ith the curvature bound e see that there must be a point 5 Ð> ß> Ñ at such that O argödð=ñ 1Î 1 1, 3 Î = =œ5 diamðdð= ÑDÐ= ÑÑ O 0 Ÿ O hich implies that, a contradiction, since ( ii) ldð= Ñl ldð= Ñl In this case, simple topological arguments sho that DÐ= ÑDÐ= Ñ must lie to the left of PÐ Ñ and e proceed analogously to the ay e did in case ( i) This completes the proof of the proposition ith 0 and 0 as defined in (312) and (313), respectivelyè We no extend of the notion of characteristic to include certain arcs hose interiors contain points of `K Although hat follos does not give the most exhaustive extension possible, it is sufficient for our present needs Let ÐH ßG Ñ be an 3- characteristic subdomain of K and consider a monotone decreasing sequence ÖG5 œ + 5, 5 À 5 in \ ÐH Ñ In other ords, + 5 Ÿ + 5, 5 Ÿ, 5, 5, so that the arcs E5 œ Ò+ 5ß, 5 Ó of bot ÐH Ñ œ E are nested Let H 5 H be the 3-characteristic subdomain bounded by G5 E5, so that H5 H5, 5 We regard G5 as being oriented from + 5 to, 5 Let + 5 p + and, 5 p, We define G to be the set of limits of sequences ÖD 6, here D6 G5, 56 p _ Any such G ill be called an extended 6 characteristic ith endpoints + and, An extended characteristic consisting of a single point : `K ill be called trivial The folloing proposition, in hich the notation is the same as that of the immediately preceding sentences, contains the basic properties of extended characteristics Note that QÐKßOÑ is, as in Proposition 34, an upper bound on the length of elementary characteristics of O-quasi-HP functions on K Proposition 314 Let G be an extended characteristic ith endpoints + and, If + œ, then G œ Ö+ Otherise, G is a simple arc joining + to, and -ÐGÑ Ÿ QÐKßOÑ If G is parametrized by DÐ=Ñ, Ÿ = Ÿ -ÐGÑ ith DÐÑ œ +, then G `K Ò+ß,Ó and for points of G Ò+ß,Ó the order ith respect to bot ÐH Ñ coincides ith the order ith respect to = Furthermore, the function D is continuously differentiable on Ðß-ÐGÑÑ and for > Ðß-ÐGÑÑ, DÐ>Ñ is joined to a point of `KÏÐ+ß,Ñ by a unique 4-characteristic arc NÐ>Ñ emanating to the left of G and O (314) H ÖD Ð>Ñ Ÿ arg -ÐN Ð>ÑÑ Proof We begin by observing that (315) E G Ò+ß,Ó To see this, note that for each A E ÏÒ+ß,Ó there is an 8 for hich A  E8 Since G8 E œ Ö+ 8ß, 8, A is not in G8 either, and therefore, A  H8 But then, by the monotonicity of ÖH 5, distðaßg5ñ distðaßh5ñ dist ÐAßH8Ñ for 5 8, from hich the desired conclusion follos at once From (315) it follos immediately that G `K Ò+ß,Ó 15

16 Next e note that if + œ,, then Proposition 36 implies that diam ÐG5Ñ p, so that G œ Ö+ For the remainder of the proof e therefore assume that +, By this assumption and Proposition 34 on the boundedness of the lengths of characteristics there exist 6 and 6 such that 6 Ÿ -ÐG5 Ñ Ÿ 6 _ Let G5 be parametrized by D œ D5Ð=Ñ, Ÿ = Ÿ -ÐG5 Ñ Let 8 be such that l+ 5 +l, l, 5,l l, +lî$ for 5 8 Let l, +lî$, and for 5 8 let 5 œ supö= À D5Ð=Ñ `RÐ+ß Ñ and 5 œ infö= À D5Ð=Ñ `RÐ,ß Ñ, and let I5 œ I5Ð Ñ œ D5ÐÒ 5ß 5ÓÑ Then (316) $ œ infödist ÐI5ß`KÏE5 Ñ À 5 8, since otherise there ould be a point of G in E ÏÒ+ß,Ó, in contradiction of (315) For 5 8 and = Ò 5ß 5Ó, let N5Ð=Ñ denote the 4-half-characteristic emanating to the left of G5 and joining D5Ð=Ñ to a point A5Ð=Ñ of `KÏE5 In light of (316), (317) - ÐN5Ð=ÑÑ dist ÐD5Ð=Ñß`KÏE5 Ñ $ for all 5 8, = Ò 5ß 5Ó By the curvature bound this means that for each e have an upper bound on the curvature to the left of I 5 More precisely, e have that (318) H ) O O O ÐD Ð=ÑÑ Ÿ Ÿ Ÿ, for 5 8, = Ò ß Ó 5 - ÐN Ð=ÑÑ distðd Ð=Ñß`KÏE Ñ $ The curvature bound trivially implies that O (319) H ÐD Ð=ÑÑ ) 5 distðd5ð=ñß`kñ There is a neighborhood Y of `K such that for each D Y there is a unique :ÐDÑ `K for hich ld :ÐDÑl œ min Ödist ÐDß' Ñ À ' `K and for hich : is continuous For ; in 39Ð;Ñ 39Ð:ÐDÑÑ `K, let / be the positively oriented unit tangent to `K at ;, so that in Y, / is continuous It follos from (318) and Proposition 310 that there is an A œ A Ð Ñ such that 39Ð:ÐD5Ð=ÑÑÑ (320) ld5ð=ñ / l Ÿ AÈlD5Ð=Ñ :ÐD5Ð=ÑÑl Bounds (318), (319), (320) and the imply that the family ÖD5Ð=Ñ is uniformly bounded and equicontinuous (that is, has a uniform modulus of continuity valid for all 5 8 on the Ò 5ß 5ÓÑ From this it follos that there are, and a sequence Ö5 6 such that 5 p and 5 p and D 6 6 5Ð=Ñ p D Ð=Ñ uniformly (in the obvious sense), here D is 6 continuously differentiable and parametrizes an arc GÐ Ñ hich joins point of `RÐ+ß Ñ to point of `RÐ,ß Ñ in the part of H lying outside of both these circles The arc GÐ Ñ is simple since by Proposition 313 any sufficiently small neighborhood of a point of Ð+ß,Ñ can contain at most a single arc of G5 Clearly, - ÐGÐ ÑÑ Ÿ QÐKßOÑ The nested nature of the H 5 then implies that (a least for sufficiently small ) the entire sequence converges to GÐ Ñ Also, it is clear that GÐ Ñ is an extension of GÐ Ñ, for We have that G œ ÖGÐ Ñ À, since diam ÐD5ÐÒß 5ÓÑÑ and diam ÐD5ÐÒ 5ß-ÐG5ÑÓÑÑ tend to uniformly in 5 as p by Proposition 36 It follos from G œ ÖGÐ Ñ À that - ÐGÐ ÑÑ Ÿ QÐKßOÑ and that G is a simple arc ith endpoints +,, and furthermore that G is parametrized by a function DÐ=Ñ hich is continuously differentiable on Ðß-ÐGÑÑ That the to possible orderings of the points of G Ò+ß,Ó coincide follos from the fact that G is a simple arc and (315) The existence and uniqueness of NÐ>Ñ is trivial hen DÐ>Ñ K When DÐ>Ñ Ð+ß,Ñ the existence of NÐ>Ñ follos from a straightforard compactness argument In light of (320), for such >, any corresponding N Ð>Ñ must be orthogonal to `K at DÐ>Ñ, so 16

17 that the uniqueness of NÐ>Ñ follos from Proposition 310 Bound (314) follos from (318) è We ill refer to an extended characteristic G joining +,, `K as +, ; points of +, `K ill be called contact points and contact points other than + and, ill be called proper contact points It is clear that ) can be continuously extended to K Ð+,ÏÖ+ß, Ñ For hat is to follo it is important to understand that if ÐH ßG Ñ is an 3-characteristic subdomain, then in addition to the extended characteristics constructed above, H might contain extended characteristics G joining points +, arising from a sequence of 3- characteristic subdomains ÖÐH5ßG5Ñ, here G5 œ + 5, 5 ith + 5 Ÿ + 5, 5 Ÿ, 5 (here the order is ith respect to E œ bot ÐH Ñ, as above) Here the G5 are contained in H but the other part of the boundary of H5 is `KÏÐ+ 5ß, 5Ñ (here here again the interval notation refers to the order on E œ bot ÐH Ñ) Note that if for such an extended characteristic G, G `K has points other than + and, (that is, if G is not simply an elementary characteristic), then the contact points ill not occur monotonically ith respect to the order on E hen G is traversed from + to, With respect to the characteristic subdomain H the extended characteristics G constructed originally ill be referred to as monotone, and this other kind of extrended characteristic G ith proper contact points ill be said to be nonmonotone We consider that an extended characteristic +, exits K at all of its contact points, and e use the terms exits regularly and exits singularly at + or, in the obvious fashion Clearly, +, exits regularly at its proper contact points Proposition 315 Let G be an extended 4 -characteristic hich exits at : and is parametrized by D œ DÐ=Ñ, Ÿ = Ÿ P, ith DÐÑ œ : Let 9Ð=Ñ œ argöd Ð=Ñ and assume that lim9ð=ñ does not exist For = ÐßPÑ let IÐ=Ñ be the elementary 3-characteristic =Ä containing DÐ=Ñ and for, X let TÐ ßXÑ œ Ö= À 9 Ð=Ñ X and RÐ ßXÑ œ Ö= À 9 Ð=Ñ Ÿ X Then for all, X, TÐ ßXÑ and RÐ ßXÑ have positive measure and diamðið=ññ p as = p Proof To see that - ÐTÐ ßXÑÑ, assume to the contrary for some, X, - ÐTÐ ßX ÑÑ œ, so that 9 Ð=Ñ Ÿ X ae on Ðß Ó Then 9 œ 9 9 on Ðß Ó, here 9 is Lipschitz continuous and nondecreasing and 9 is continuous and nonincreasing From this in turn it follos that either 9 has a finite limit as = p or it tends to _ as = p, so that in fact the latter is the case since G exits singularly But this means that G spirals around :, hich is clearly impossible Thus TÐ ßXÑ must have positive measure for all ßX One sees similarly that RÐ ßXÑ has positive measure That diam ÐIÐ=ÑÑ p follos immediately from the characteristic length bound è Let ÐHßG Ñ be a characteristic subdomain of K If J H bot ÐHÑ is a compact set for hich J bot ÐHÑ Á g, then the family V ÐJÑ œ ÖG \ ÐHÑ À J G is clearly linearly ordered ith respect to the relation From this it easily follos that there is a unique monotone extended 3-characteristic I œ +, for hich J bot ÐHÑ Ò+ß,Ó and I G for all G VÐJÑ Notation 316 This monotone extended 3-characteristic I ill be denoted by minhðjñ In the statement and proof of the folloing proposition all order relations are ith respect to ÐHßG Ñ 17