Partial regularity and smooth topology-preserving approximations of rough domains
PPartial regularity and smooth topology-preservingapproximations of rough domains
John M. Ball ∗ and Arghir Zarnescu †‡§ June 4, 2018
In memoriam
J. Bryce McLeod
Abstract
For a bounded domain Ω ⊂ R m , m ≥ , of class C , the properties are studied of fields of‘good directions’, that is the directions with respect to which ∂ Ω can be locally represented asthe graph of a continuous function. For any such domain there is a canonical smooth field ofgood directions defined in a suitable neighbourhood of ∂ Ω, in terms of which a correspondingflow can be defined. Using this flow it is shown that Ω can be approximated from the insideand the outside by diffeomorphic domains of class C ∞ . Whether or not the image of a generalcontinuous field of good directions (pseudonormals) defined on ∂ Ω is the whole of S m − is shownto depend on the topology of Ω. These considerations are used to prove that if m = 2 ,
3, or if Ωhas nonzero Euler characteristic, there is a point P ∈ ∂ Ω in the neighbourhood of which ∂ Ω isLipschitz. The results provide new information even for more regular domains, with Lipschitzor smooth boundaries.
In this paper we study bounded domains Ω ⊂ R m , m >
1, of class C , showing that they can beapproximated from the inside and outside by bounded domains Ω ε of class C ∞ that are diffeomorphicto Ω, and such that ¯Ω ε are homeomorphic to ¯Ω. Thus the approximating smooth domains preservetopological properties of the rough domain; in particular, for instance, a simply-connected boundeddomain of class C can be approximated from the inside and outside by smooth simply-connecteddomains. The method of approximation uses a construction of smooth fields of ‘good directions’,that is directions with respect to which the boundary ∂ Ω can be locally represented as the graph ofa continuous function, and a corresponding flow. We analyze topological properties of fields of gooddirections, and exploit them to study partial regularity of the boundary of such domains.Domains of class C represent one of the largest class of domains relevant to the analysis of partialdifferential equations and their numerical approximation, and have been widely studied, see for ∗ Oxford Centre for Nonlinear PDE, Mathematical Institute, University of Oxford, Andrew Wiles Building, RadcliffeObservatory Quarter, Woodstock Road, Oxford OX2 6GG, U.K. † IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Bizkaia, Spain ‡ BCAM, Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Basque Country, Spain § “Simion Stoilow” Institute of the Romanian Academy, 21 Calea Grivit¸ei Street, 010702 Bucharest, Romania a r X i v : . [ m a t h . C A ] F e b nstance [50, 51, 18, 42]. They are in particular relevant for physical applications to non-Lipschitzdomains arising in optimal design, free-boundary problems (e.g. for two-phase flow) and fracture(see for instance [9, 10, 17, 20, 48]). Such rough domains also appear in the theoretical study ofparabolic equations through the use of H¨older-continuous space-time domains [11, 40] or ellipticproblems in domains with point singularities [35]. Other applications include problems in whichthe domain is an unknown (such as optimal design), for which the flow of good directions might beused to construct domain variations, and various problems in partial differential equations [26] andpotential theory [57]. Nevertheless there is a a lack of tools for treating general domains of class C and they are often treated on a case-by-case basis, under various additional simplifying assumptions(see for instance [35, 43]). One aim of our study is to provide a set of versatile tools for dealing withsuch domains, without the need of ad hoc methods adjusted to specific cases.We recall that Ω ⊂ R m , m > a domain of class C (respectively of class C r , r = 1 , , . . . , ∞ ,Lipschitz) if Ω is a connected open set such that for any point P belonging to the boundary ∂ Ω thereexist a δ > Y def = ( y (cid:48) , y m ) = ( y , y , . . . , y m ) with origin at P , together with a continuous (respectively C r , Lipschitz) function f : R m − → R , such thatΩ ∩ B ( P, δ ) = { y ∈ R m : y m > f ( y (cid:48) ) , | y | < δ } (1.1)from which it follows that ∂ Ω ∩ B ( P, δ ) = { y ∈ R m : y m = f ( y (cid:48) ) , | y | < δ } and f (0) = 0. Wecall the unit vector n ( P ) = e m ( P ) in the y m − direction for this coordinate system a pseudonormal at P . More generally, if P ∈ R m is not necessarily a boundary point, but is such that for thecoordinate system Y with origin at P we have that (1.1) holds with ∂ Ω ∩ B ( P, δ ) nonempty, wecall the corresponding unit vector n ( P ) a good direction at P . We show (Lemma 2.1) that the setof good directions at a point P is a (geodesically) convex subset of the unit sphere S m − . Using apartition of unity we deduce (Proposition 2.1) that for a bounded domain of class C there exists afield of good directions G ( P ), that we call canonical , depending smoothly on P .Although good directions and pseudonormals are defined locally, their properties depend on thetopology of the domain. If Ω is a bounded domain of class C then the negative Gauss map ν ∂ Ω : ∂ Ω → S m − , defined by ν ∂ Ω ( P ) = the inward normal to ∂ Ω at P , is surjective. We show(Theorem 6.1) that the same is true for an arbitrary continuous field of pseudonormals if m = 2 orif m ≥ ⊂ R is a standard solid torus in R then, using an observation of Lackenby [37], we show (Proposition 6.1) that there is a continuousfield of pseudonormals with image contained in an arbitrarily small neighbourhood of the great circlein S perpendicular to the axis of cylindrical symmetry of the torus.In our approximation result (Theorem 5.1) the approximating domains are given byΩ ε def = { x ∈ R m : ρ ( x ) > ε } , < | ε | < ε , (1.2)where ρ is a regularized signed distance to the boundary. In general these domains would provide,for suitably small ε and almost any ε ∈ ( − ε , ε ), an interior and exterior approximation for an arbitrary open set Ω (see Remark 5.5). However if one requires that the approximating domains arein the same diffeomorphism (or homeomorphism up to the boundary) class as the initial domain oneneeds to impose some restrictions on Ω (see Examples 5.1, 5.2). The bounded domains of class C form a large class of domains for which this type of approximation is possible.(In fact Theorem 5.1holds for trivially for the larger class of domains that are the image of a bounded domain of class C under a suitable diffeomorphism; see Remark 5.2.)In our paper we use the regularized signed distance to the boundary defined by Lieberman [39]. Thisregularized distance has the property that ∇ ρ ( P ) · G ( P ) > G ( P ) in2 suitable neighbourhood of ∂ Ω. This enables us to use the flow of canonical good directions S ( t ) x defined as the solution of ˙ x ( t ) = G ( x ( t )) for t ∈ R ,x (0) = x , suitably extended so as to be globally defined, to provide the deformation showing that Ω and Ω ε are C ∞ -diffeomorphic and their closures are homeomorphic.A surprising by-product of our study of the topology of the set Ω and that of the good directions isthe fact that in R m for m = 2 , C must necessarily have portions ofthe boundary with better regularity, namely Lipschitz regularity (Theorems 7.2, 7.3). This is truefor arbitrary m if Ω has nonzero Euler characteristic (Theorem 7.1). However it is in general falsefor unbounded domains of class C (Remark 7.1).Let us now mention some related literature. Further results on domains of class C and theirproperties, in particular their geometric characterization as domains with the segment property ,are available in Fraenkel [15]. The relation between domains of class C and their closure is addressedin Grisvard [18, Theorem 1.2.1.5]. Also in [18, Corollary 1.2.2.3], it is noted that a bounded openconvex set necessarily has Lipschitz boundary. Somewhat related domains are the ‘cloudy manifolds’defined by Kleiner & Lott [33] which are subsets of an Euclidean space with the property that neareach point they look “coarsely close” to an affine subspace of the Euclidean space. It was shown in[33] that any cloudy k -manifold can be well interpolated by a smooth k -dimensional submanifold ofthe Euclidean space. Another strand of research that can be compared with our partial regularityresult for bounded domains of class C is that described by Jones, Katz & Vargas [30], in which theauthors prove and generalize a conjecture of Semmes [56] that for a bounded open set Ω ⊂ R m with H m − ( ∂ Ω) =
M < ∞ there exist ε > H m − (Γ ∩ ∂ Ω) ≥ ε .Close in spirit to our work is that of Verchota ([61, Appendix], [62, Proposition 1.12]), who studiesbounded Lipschitz domains Ω and constructs smooth approximating domains whose boundaries areshown to be homeomorphic with ∂ Ω using a smooth flow.In Hofmann, Mitrea & Taylor [28] a definition is given of a continuous vector field transversal to theboundary of an open set with locally finite perimeter, in which it is required that the inner productof the vector field with the normal is bounded away from zero. For the case of bounded domains ofclass C (which need not have finite perimeter) this is a similar but stronger requirement than beinga continuous field of good directions. A result [28, Proposition 2.2] analogous to our Proposition 2.1is then proved giving conditions under which the existence of a continuous locally tranversal fieldimplies the existence of a global smooth transversal field.The closest to our work seems to be that of Iwaniec & Onninen [29]. There they note, as we do inProposition 3.1, that the distance is a monotone function along a good direction, and they use atype of canonical field of good directions and its flow to construct approximating domains similarlyto our construction in Section 4.Finally, in a more general framework related types of questions were addressed in the papers of Cairns[6], Whitehead [64] and Pugh [53], which study conditions under which a topological manifold can besmoothed: that is when its (maximal) topological atlas contains a smooth subatlas. Indeed, it seemslikely that the techniques of smoothing of manifolds could lead to inner and outer approximations bydomains of class C ∞ for the wider class of bounded domains Ω ⊂ R m , m (cid:54) = 4 ,
5, having boundariesthat are locally flat (that is, ¯Ω and Ω c are topological manifolds with boundary), though without i.e. for each point P ∈ ∂ Ω there exists a neighbourhood U ( P ) in R m and a non-zero vector b ( P ) ∈ R m such that x + tb ∈ Ω , for all x ∈ Ω ∩ U ( P ) , < t < C can be found in [18], p.5 − C ∞ and simply-connected, for more general simply-connecteddomains (though recent work of Bedford [3] gives a way of proving the desired orientability withoutapproximation of the domain). Definition 2.1.
Let Ω ⊂ R m be a domain of class C . For a point P ∈ R m we define a gooddirection at P , with respect to a ball B ( P, δ ) , δ > , with B ( P, δ ) ∩ ∂ Ω (cid:54) = ∅ to be a vector n ∈ S m − such that there is an orthonormal coordinate system Y = ( y (cid:48) , y m ) = ( y , y , . . . , y m ) with origin atthe point P and such that n = e m is the unit vector in the y m direction, together with a continuousfunction f : R m − → R (depending on P and n and δ ), such that Ω ∩ B ( P, δ ) = { y ∈ R m : y m > f ( y (cid:48) ) , | y | < δ } . (2.1) We say that n is a good direction at P if it is a good direction with respect to some ball B ( P, δ ) with B ( P, δ ) ∩ ∂ Ω (cid:54) = 0 .If P ∈ ∂ Ω then a good direction n at P is called a pseudonormal at P . Remark 2.1.
A good direction at a point need not be unique but for a bounded domain of class C there always exists at least one for each point in a (small enough) neighbourhood of ∂ Ω. Note alsothat if both n and ¯ n are good directions at P then we can choose δ = min { δ ( P, n ) , δ ( P, ¯ n ) } so thatthe corresponding two representations (2.1) hold for δ . However a possible choice of δ ( P, n ) may notbe a possible choice of δ ( P, ¯ n ). Remark 2.2.
If one has for instance a domain in R such that part of its boundary can be locallyrepresented as { ( x, f ( x )) , x ∈ ( − , } with f ( x ) = (cid:112) | x | then there is only one good direction at thepoint (0 ,
0) namely (0 , ∈ S . This suggests that there exists a connection between the uniquenessof a good direction and the regularity of the boundary, and this topic will be explored in detail inthe last section.We refer to a subset S of a Riemannian manifold M as geodesically convex if given any two points P, Q ∈ S there is a unique shortest curve (minimal geodesic) in M joining P and Q , and this curvelies in S (there are several differing definitions in the literature). The following lemma asserts thatthe set of good directions at any given point is a geodesically convex subset of S m − . For a closelyrelated result see Wilson [63, Lemma 4.2]. Lemma 2.1.
Let Ω ⊂ R m be a ( possibly unbounded ) domain of class C . If p, q ∈ S m − are gooddirections at a point P ∈ R m with respect to the ball B ( P, δ ) then p (cid:54) = − q and for any λ ∈ (0 , thevector λp +(1 − λ ) q | λp +(1 − λ ) q | is a good direction at P with respect to the ball B ( P, δ ) . roof. Let p, q be good directions at P , and 0 < λ <
1. We can assume that P = 0. Let B def = B (0 , δ ).If p = q there is nothing to prove. So assume p (cid:54) = q . Then Definition 2.1 implies that p (cid:54) = − q , andso λp + (1 − λ ) q (cid:54) = 0. Let N = λp +(1 − λ ) q | λp +(1 − λ ) q | . We choose coordinates such that e m = N . Take any ξ ∈ B \ Ω. Then since p and q are good directions the intersections of B with the open half-lines { ξ + tp : t < } and { ξ + tq : t < } lie in R m \ Ω. Similarly, if ξ ∈ B ∩ Ω the intersections of B withthe open half-lines { ξ + tp : t > } and { ξ + tq : t > } lie in Ω.p q PN δe m = N = λp +(1 − λ ) q | λp +(1 − λ ) q | Figure 1: Geodesically convex combination of two good directionsGiven any ξ (cid:48) ∈ B m − (0 , δ ) = { z ∈ R m − : | z | < δ } define the line L ( ξ (cid:48) ) = { ( ξ (cid:48) , t ) : t ∈ R } ,and let S = { ξ (cid:48) ∈ B m − (0 , δ ) : L ( ξ (cid:48) ) ∩ ∂ Ω ∩ B is nonempty } . We claim that if ξ (cid:48) ∈ S then L ( ξ (cid:48) ) intersects B ∩ ∂ Ω in a unique point ( ξ (cid:48) , t ( ξ (cid:48) )), and that { ( ξ (cid:48) , t ) : t > t ( ξ (cid:48) ) } ∩ B ⊂ Ω and { ( ξ (cid:48) , t ) : t < t ( ξ (cid:48) ) } ∩ B ⊂ R m \ Ω. To prove the claim let ξ = ( ξ (cid:48) , t ( ξ (cid:48) )) ∈ B ∩ ∂ Ω, and suppose ξ − he m ∈ B for some h >
0. Then for some ε >
0, dist ( ξ, ∂B ) > ε, dist ( ξ − he m , ∂B ) > ε . Choosea positive integer k > hε | λp +(1 − λ ) q | , and divide the interval (0 , h ) into k subintervals of length h/k .Then ¯ ξ = ξ − hk · (1 − λ ) q | λp +(1 − λ ) q | ∈ B , and so ¯ ξ ∈ R m \ Ω. Thus ξ − hk e m = ¯ ξ − hk · λp | λp +(1 − λ ) q | ∈ R m \ Ω anddist ( ξ − hk e m , ∂ Ω) > ε . Proceeding inductively, after k steps we find that ξ − he m ∈ R m \ Ω, so that { ( ξ (cid:48) , t ) : t < t ( ξ (cid:48) ) } ∩ B ⊂ R m \ Ω. It follows similarly that { ( ξ (cid:48) , t ) : t > t ( ξ (cid:48) ) } ∩ B ⊂ Ω, establishingthe claim.Now define f ( ξ (cid:48) ) = | ξ (cid:48) | = δt ( ξ (cid:48) ) if ξ (cid:48) ∈ S − (cid:112) δ − | ξ (cid:48) | if L ( ξ (cid:48) ) ∩ B = L ( ξ (cid:48) ) ∩ B ∩ Ω (cid:112) δ − | ξ (cid:48) | if L ( ξ (cid:48) ) ∩ B = L ( ξ (cid:48) ) ∩ B ∩ ( R m \ Ω) . (2.2)5ote that ( ξ (cid:48) , f ( ξ (cid:48) )) ∈ ¯ B for all ξ (cid:48) ∈ B m − (0 , δ ). Clearly Ω ∩ B = { ( ξ (cid:48) , ξ m ) ∈ B : | ξ (cid:48) | < δ, ξ m > f ( ξ (cid:48) ) } ,and it remains to prove that f is continuous, since then we can extend f by zero for | ξ (cid:48) | > δ to get asuitable continuous f : R m − → R . Let ξ (cid:48) ( j ) → ξ (cid:48) in B m − (0 , δ ). If | ξ (cid:48) | = δ then | ( ξ (cid:48) ( j ) , f ( ξ (cid:48) ( j ) )) | ≤ δ implies | ξ (cid:48) ( j ) | + f ( ξ (cid:48) ( j ) ) ≤ δ and so f ( ξ (cid:48) ( j ) ) → f ( ξ (cid:48) ). If ξ (cid:48) ∈ S then ( ξ (cid:48) , f ( ξ (cid:48) )) ∈ B ∩ ∂ Ω andso for any sufficiently small ε > x + ε = ( ξ (cid:48) , f ( ξ (cid:48) ) + ε ) and x − ε = ( ξ (cid:48) , f ( ξ (cid:48) ) − ε ) belongto Ω ∩ B and to ( R m \ Ω) ∩ B respectively. Since Ω and R m \ Ω are open, there exists δ ∈ (0 , ε )such that B ( x + ε , δ ) ⊂ Ω ∩ B and B ( x − ε , δ ) ⊂ ( R m \ Ω) ∩ B . Hence for sufficiently large j , the line L ( ξ (cid:48) ( j ) ) has points in both B ( x + ε , δ ) and B ( x − ε , δ ) and thus intersects ∂ Ω in B at the unique point( ξ (cid:48) ( j ) , f ( ξ (cid:48) ( j ) )), where | f ( ξ (cid:48) ( j ) ) − f ( ξ (cid:48) ) | < ε . Since ε is arbitrarily small, f ( ξ (cid:48) ( j ) ) → f ( ξ (cid:48) ). Similarly, if L ( ξ (cid:48) ) ∩ B = L ( ξ (cid:48) ) ∩ Ω so that f ( ξ (cid:48) ) = − (cid:112) δ − | ξ (cid:48) | , then for all sufficiently small ε > δ ∈ (0 , ε ) such that the ball B (( ξ (cid:48) , f ( ξ (cid:48) ) + ε ) , δ ) ⊂ Ω ∩ B . Hence for all sufficiently large j we have f ( ξ (cid:48) ( j ) ) ≤ f ( ξ (cid:48) ) + ε , so that f ( ξ (cid:48) ( j ) ) → f ( ξ (cid:48) ). The case when L ( ξ (cid:48) ) ∩ B = L ( ξ (cid:48) ) ∩ B ∩ ( R m \ Ω) ishandled in a similar way.The above can be easily extended to the case of an arbitrary number of good directions:
Lemma 2.2.
Let k = 1 , , . . . . If n , n , . . . , n k ∈ S m − are good directions at a point P with respectto the ball B ( P, δ ) and < λ i < , i = 1 , , . . . , k , with Σ ki =1 λ i = 1 then Σ ki =1 λ i n i (cid:54) = 0 and Σ ki =1 λ i n i | Σ ki =1 λ i n i | is a good direction at P with respect to B ( P, δ ) .Proof. This follows easily from Lemma 2.1 by induction on k , noting that (cid:80) ki =1 λ i n i is parallel to µ (cid:80) k − i =1 λ i n i (cid:12)(cid:12)(cid:12)(cid:80) k − i =1 λ i n i (cid:12)(cid:12)(cid:12) + (1 − µ ) n k , where µ = (cid:12)(cid:12)(cid:12)(cid:80) k − i =1 λ i n i (cid:12)(cid:12)(cid:12) λ k + (cid:12)(cid:12)(cid:12)(cid:80) k − i =1 λ i n i (cid:12)(cid:12)(cid:12) . Despite the fact that the boundary is just of class C we can easily construct a smooth field of gooddirections in a neighbourhood of the boundary: Proposition 2.1.
Let Ω ⊂ R m be a bounded, open set with boundary of class C . There exists aneighbourhood U of ∂ Ω and a smooth function G : U → S m − so that for each P ∈ U the unit vector G ( P ) is a good direction.Proof. As Ω is of class C , for each point ¯ P ∈ ∂ Ω there is a good direction n ¯ P at ¯ P , with cor-responding δ = δ ( ¯ P ). Then n ¯ P is a good direction at any P ∈ B ( ¯ P , δ ( ¯ P )) since for such P we have ¯ P ∈ B ( P, δ ( ¯ P )) ⊂ B ( ¯ P , δ ( ¯ P )). As ∂ Ω is compact, there exist P i , i = 1 , . . . , k, suchthat ∂ Ω ⊂ U def = ∪ ki =1 B ( P i , δ ( P i )). Consider a partition of unity subordinate to the covering { B ( P i , δ ( P i )) } , i = 1 , . . . , k, of ¯ U , namely functions α i ∈ C ∞ ( R m , R + ) , i = 1 , , . . . , k with supp α i ⊂ B ( P i , δ ( P i )) and Σ ki =1 α i = 1 in ¯ U . If P ∈ U and i ∈ S P def = { j ∈ { , , . . . , k } : P ∈ B ( P j , δ ( P j )) } then n P i is a good direction at P with respect to the ball B ( P, ∆( P )) where ∆( P ) def = min i ∈ S p δ ( P i ).It then follows from Lemma 2.2 that G ( P ) def = Σ ki =1 α i ( P ) n P i | Σ ki =1 α i ( P ) n P i | , for all P ∈ U (2.3)has the required property. Definition 2.2.
We call a field of good directions, constructed by (2.3) , a canonical field of gooddirections . A proper generalized distance
For a bounded open set Ω define the signed distance d ( x ) to the boundary ∂ Ω by d ( x ) = (cid:26) inf y ∈ ∂ Ω | x − y | if x ∈ Ω − inf y ∈ ∂ Ω | x − y | if x (cid:54)∈ Ω . (3.1) Proposition 3.1.
Let Ω ⊂ R m be a bounded domain of class C . There exists a function ρ ∈ C ∞ ( R m \ ∂ Ω) ∩ C , ( R m ) such that ≤ ρ ( x ) d ( x ) ≤ , for all x ∈ R m \ ∂ Ω (3.2) and |∇ ρ ( x ) | (cid:54) = 0 for all x in a neighbourhood of ∂ Ω , x (cid:54)∈ ∂ Ω . (3.3) Proof.
We let ρ be a regularized distance function, as constructed by Lieberman [39] (followingrelated earlier work of Fraenkel [14]). To define it let ϕ ∈ C ∞ ( R m ) be a nonnegative function, whosesupport is the unit ball and is such that (cid:82) R m ϕ ( x ) dx = 1. For x ∈ R m , τ ∈ R , let G ( x, τ ) def = (cid:90) | z | < d (cid:16) x − τ z (cid:17) ϕ ( z ) dz. (3.4)Since d is 1-Lipschitz, | ∂G/∂τ | ≤ /
2, and so there is a unique solution ρ ( x ) of the equation ρ ( x ) = G ( x, ρ ( x )). Thus defined, ρ is a Lipschitz function, smooth outside ∂ Ω, that satisfies (3.2)but not necessarily (3.3) (see [39], Lemma 1 . x in a neighbourhood of the boundary and x ∈ Ω. To this endwe consider a point P ∈ ∂ Ω. Without loss of generality we can suppose that P = 0 and that in asuitable coordinate system there exist δ > f : R m − → R such that U δ def = Ω ∩ B (0 , δ ) = { y ∈ R m : y m > f ( y (cid:48) ) , | y | < δ } . (3.5)Denoting by e m the unit vector in the y m direction, let y, y + he m ∈ U δ/ for some h >
0. Then h < δ/
2. Since 0 ∈ ∂ Ω, d ( y ) ≤ | y | . If v ∈ B ( y, d ( y )) then | v | ≤ | v − y | + | y | ≤ d ( y ) + | y | ≤ | y | ≤ δ/ . Hence B ( y, d ( y )) ⊂ U δ . We claim that d ( y + he m ) > d ( y ) . (3.6)If not, there would exist w ∈ ∂ Ω with | w − y − he m | ≤ d ( y ). Thus w − he m ∈ B ( y, d ( y )) and so w − he m ∈ U δ and w m − h ≥ f ( w (cid:48) ) = w m , a contradiction. Since d is Lipschitz it follows that thederivative ∂d∂x m exists a.e. with strictly positive integral on every line segment in U δ/ parallel to e m .7y the definition of weak derivatives ∂G∂x m ( x, τ ) = ∂∂x m (cid:90) R m d (cid:16) x − τ z (cid:17) ϕ ( z ) dz = ∂∂x m (cid:90) R m d ( ζ ) ϕ (cid:18) τ ( x − ζ ) (cid:19) (cid:18) τ (cid:19) m dζ (3.7)= (cid:90) R m d (cid:16) x − τ z (cid:17) τ ϕ ,m ( z ) dz = (cid:90) {| z | < } ∂d∂x m (cid:16) x − τ z (cid:17) ϕ ( z ) dz. (3.8)Suppose now that x ∈ U δ/ . Then for | z | < x − ρ ( x )2 z ∈ Ω and | x − ρ ( x )2 z | ≤ δ + d ( x ) < δ ,where we have used (3.2). Hence, since the partial derivatives equal the weak derivatives almosteverywhere in Ω, by Fubini’s theorem ∂G∂x m ( x, ρ ( x )) >
0, and so differentiating ρ ( x ) = G ( x, ρ ( x )) weobtain ∂ρ∂x m ( x ) = ∂G∂x m − ∂G∂τ > . (3.9)Thus every point P ∈ ∂ Ω has a neighbourhood U ( P ) such that |∇ ρ ( x ) | (cid:54) = 0 for x ∈ U ( P ) ∩ Ω. Bycompactness this implies that there is a neighbourhood U of ∂ Ω such that |∇ ρ ( x ) | (cid:54) = 0 for x ∈ U ∩ Ω.The case when x is in a neighbourhood of the boundary ∂ Ω but x ∈ R m \ Ω is treated similarly.
Remark 3.1.
For Ω ⊂ R m bounded, of class C , the compactness of ∂ Ω implies that there exists aneighbourhood U of ∂ Ω such that
U ⊂ ∪ ki =1 B ( P i , δ i ) where k ≥ i = 1 , . . . , k , P i ∈ ∂ Ωand at P i there is a good direction n i ∈ S m − with respect to the ball B ( P i , δ i ). Then relation (3.9)in the previous proof shows that for any P ∈ U \ ∂ Ω we have ∂ρ∂n j ( P ) = n j · ∇ ρ ( P ) > j ∈ { , . . . , k } such that P ∈ B ( P j , δ j ).Moreover, for any n that is a convex combination of those good directions n j , j ∈ { , . . . , k } with P ∈ B ( P j , δ j ), we have ∂ρ∂n ( P ) > . Remark 3.2.
We claim now that for any R ∈ U (with U as in Remark 3.1) and any n that is aconvex combination of those good directions n j , j ∈ { , . . . , k } , such that R ∈ B ( P j , δ j ), we havethat n is also a good direction at R and there exists δ n > ρ ( R + sn ) < ρ ( R + tn )for all s, t ∈ ( − δ n , δ n ) with s < t .If R (cid:54)∈ ∂ Ω then Remark 3.1 suffices for obtaining the claim. If R ∈ ∂ Ω we consider the function h : [ − , → R defined by h ( τ ) = ρ ( R + τ n ). Then Remark 3.1 ensures that h (cid:48) ( τ ) > τ ∈ ( − δ n , ∪ (0 , δ n ) for some δ n >
0. This fact, together with h (0) = 0 and h ( τ ) τ > τ ∈ ( − δ n , ∪ (0 , δ n ) (since n is a pseudonormal at R ) suffices to obtain the claim in this case aswell. 8 The flow of canonical good directions
We continue working with Ω ⊂ R m a bounded domain of class C and ρ a proper regularizeddistance from the boundary ∂ Ω as described in Section 3. We take a function γ ∈ C ∞ ( R m , R + ) sothat supp γ ⊂ U (where U is as in Proposition 2.1 and U ⊂ U with U as in Remark 3.1) with γ ≡ W where W = { x ∈ R m : | ρ ( x ) | < ¯ ε } and ¯ ε > W ⊂ U , and γ ≤ R m \ W . Let G : U → S m − be the function from Proposition 2.1, so that G ( P ) is a good directionat P . Let S ( t ) x denote the solution at time t ∈ R of the system:˙ x ( t ) = (cid:26) γ ( x ( t )) G ( x ( t )) for t ∈ R , x ( t ) ∈ U t ∈ R , x ( t ) (cid:54)∈ U (4.1)with initial data x (0) = x .From now on we call the globally defined flow S ( · )( · ) : R × R m → R m the flow of canonical gooddirections .We first show that the regularized distance to the boundary increases along this flow. Lemma 4.1.
Let Ω ⊂ R m be a bounded domain of class C . Let P ∈ U , with γ ( P ) (cid:54) = 0 (where U and γ are defined at the beginning of the section). Then ρ ( S ( µ ) P ) < ρ ( S ( µ ) P ) , for all ≤ µ < µ . (4.2) Proof.
We proceed in two steps:
Step 1.
We claim that for any point P ∈ U with γ ( P ) (cid:54) = 0 we have ρ ( S ( µ ) P ) ≤ ρ ( S ( µ ) P ) for0 ≤ µ < µ . Then γ ( S ( t ) P ) (cid:54) = 0 for all t ≥ , µ + 1]. Thisis obtained by linearly interpolating between the points P k defined recursively by: P k +1 = P k + hγ ( P k ) G ( P k ) , h = µ + 1 l , k = 0 , . . . , l − , where P = P .Thus we have the approximate solution S l ( t ) = P k + ( t − k µ +1 l ) γ ( P k ) G ( P k ) for all t ∈ [ k µ +1 l , ( k +1) µ +1 l ] , k = 0 , . . . , l −
1. Note that by the convergence of the Euler approximation, for h smallenough γ ( P k ) (cid:54) = 0 for all k = 0 , . . . , l −
1. Using then Remark 3.2 we have that for h small enough ρ is an increasing function along S l ( t ) , t ∈ [0 , µ + 1]. Passing to the limit l → ∞ we have that ρ isa non-decreasing function along the limit function S ( t ) , t ∈ [0 , µ + 1] that is also a solution of thesystem (4.1). Step 2.
We claim now that for any µ < µ we have ρ ( S ( µ ) P ) < ρ ( S ( µ ) P ). To this end weclaim first that for any ε < (0 , µ − µ ) there exists an interval ( a, b ) ⊂ ( µ , µ + ε ) and a subset B ⊂ { , , . . . , k } so that α i ( S ( t ) P ) (cid:54) = 0 for all t ∈ ( a, b ) , i ∈ B and α i ( S ( t ) P ) = 0 for all t ∈ ( a, b ) , i ∈ { , , . . . , k } \ B (where the functions α i are those used in the definition of G in the proofof Proposition 2.1).In order to prove the claim let M i def = { t ∈ [ µ , µ + ε ] : α i ( S ( t ) P ) > } . Then each M i , ≤ i ≤ k, isrelatively open in [ µ , µ + ε ] and the M i cover [ µ , µ + ε ]. Each ∂M i is closed and nowhere dense.9ence by the Baire category theorem (for a finite number of sets) ∪ ki =1 ∂M i is closed and nowheredense. Let ( a, b ) ⊂ (cid:0) ∪ ki =1 ∂M i (cid:1) c . Then B ( t ) def = { i ∈ { , . . . , k } : t ∈ M i } is constant in ( a, b ) and wecan take B = B ( t ), thus finishing the proof of the claim.We consider now the function S ( t ) P for t ∈ ( a, b ) with ( a, b ) ⊂ ( µ , µ + ε ) as in the claim above.Let us denote R def = S ( a ) P and let B = { i , i , . . . , i j } . Then S ( a + s ) P = S ( s ) R and we have (for s < b − a ) that S ( a + s ) P = R + j (cid:88) r =1 n i r (cid:90) s γ ( S ( τ ) R ) α i r ( S ( τ ) R ) | (cid:80) jr =1 n i r α i r ( S ( τ ) R ) | dτ = R + j (cid:88) r =1 n i r ξ i r = R + (cid:32) j (cid:88) r =1 ξ i r (cid:33) j (cid:88) p =1 ξ i p (cid:16)(cid:80) jr =1 ξ i r (cid:17) n i p (4.3)where we denote ξ i r def = (cid:82) s γ ( S ( τ ) R ) α ir ( S ( τ ) R ) | (cid:80) jr =1 n ir α ir ( S ( τ ) R ) | dτ > , r = 1 , . . . , j . By our choice of the set of indices B , we have that n i p is a good direction at R for all i p ⊂ B, p = 1 , . . . , j, and thus their convexcombination (cid:18)(cid:80) jp =1 ξ ip ( (cid:80) jr =1 ξ ir ) n i p (cid:19) is also a good direction at R . Using then Remark 3.2 we havethat ρ ( S ( a ) P ) < ρ ( S ( a + s ) P ) for ε > s ∈ (0 , ε ), which combined withStep 1 completes the proof.We now show that in a neighbourhood of the boundary the flow of canonical good directions crossesthe boundary uniformly in time. Lemma 4.2.
Let Ω ⊂ R m be a bounded domain of class C . If ¯ ε is as defined at the beginning ofthe section and ≤ ε ≤ ¯ ε then (i) there exists t ε − < such that ρ ( S ( t ) P ) < − ε, for all t ≤ t ε − , P ∈ W , (4.4)(ii) there exists t ε + > such that ρ ( S ( t ) P ) > ε, for all t ≥ t ε + , P ∈ W , (4.5) where W ⊂ U is as defined at the beginning of the section.Proof. We consider case (i), the argument for case (ii) being similar. Let us denote by Z ⊂ U the α -limit set of the solution S ( t ) P . Since S ( R ) P is bounded (because all points outside a neighbourhoodof the boundary are stationary points), Z is a nonempty, compact, invariant set that attracts P alongthe flow S (see, for instance, [21, Lemma 3.1.1]).We show first that the conclusion is true if we let t ε − depend on P . We argue by contradiction andassume that the conclusion is false, so that there exists a sequence t k → −∞ such that ρ ( S ( t k ) P ) ≥− ε . As ρ ( S ( t ) P ) is increasing and bounded from below we have that lim t →−∞ ρ ( S ( t ) P ) = l ≥ − ε and ρ ( Q ) = l for all Q ∈ Z . In particular Z ⊂ ¯ W . As Z is invariant, for each Q ∈ Z we alsohave S ( t ) Q ∈ Z for all t ≤ ρ ( S ( t ) Q ) = l for all t ≤
0, which contradicts Lemma 4.1. Inorder to prove that t ε − can be chosen independent of P we assume for contradiction that this is not10ossible, so that there exist a sequence { P k } k ∈ N ⊂ W and a corresponding sequence of times { t k } k ∈ N so that ρ ( S ( t k ) P k ) = − ε and t k → −∞ . Using the compactness of W we can find a subsequence P k l → P ∈ W . But for P there exists a time t < ρ ( S ( t ) P ) < − ε and using thecontinuity with respect to the initial data for the solution of the system (4.1) together with the factthat ρ ( S ( t ) P k ) is increasing, we obtain a contradiction. C ∞ diffeomorphically equivalentapproximations of rough domains In this section we provide an application of the tools developed in the previous sections by showingthat one can approximate from the inside (and also from the outside) domains Ω of class C bysmooth domains Ω (cid:48) such that Ω and Ω (cid:48) are C ∞ -diffeomorphic and their closures are homeomorphic. Theorem 5.1.
Let Ω ⊂ R m , m ≥ be a bounded domain of class C . Let ρ be a regularized distanceas given in Proposition and for ε ∈ R define Ω ε = { x ∈ R m : ρ ( x ) > ε } . (5.1) There exists ε = ε (Ω) > such that if < | ε | < ε then Ω ε is a bounded domain of class C ∞ and (i) (cid:84) − ε <ε< Ω ε = Ω , (cid:83) ε >ε> Ω ε = Ω , ¯Ω ε ⊂ Ω ε (cid:48) if − ε < ε (cid:48) < ε < ε . (ii) For ≤ | ε | < ε there exists a homeomorphism f ( ε, · ) of R m onto R m , with inverse denoted f − ( ε, · ) , such that • f ( ε, ¯Ω) = Ω ε , f ( ε, ∂ Ω) = ∂ Ω ε , • f ( ε, x ) = x for | ρ ( x ) | > | ε | ( so that in particular f (0 , · ) = identity ) , • f ( ε, · ) : R m \ ∂ Ω → R m \ ∂ Ω ε is a C ∞ diffeomorphism.Furthermore f and f − are continuous functions of ( ε, x ) for ≤ | ε | < ε , x ∈ R m , and f ( resp. f − ) is a smooth function of ( ε, x ) for < | ε | < ε , x ∈ R m \ ∂ Ω ( resp. < | ε | < ε , x ∈ R m \ ∂ Ω ε ) . (iii) There exists a map ¯ f : (0 , ε ) × ( − ε , × R m → R m such that if < ε < ε , − ε < ε (cid:48) < then • ¯ f ( ε, ε (cid:48) , · ) is a C ∞ diffeomorphism of R m onto R m with inverse ¯ f − ( ε, ε (cid:48) , · ) : R m → R m , • ¯ f ( ε, ε (cid:48) , Ω ε ) = Ω ε (cid:48) , ¯ f ( ε, ε (cid:48) , ∂ Ω ε ) = ¯ f ( ε, ε (cid:48) , ∂ Ω ε (cid:48) ) , • ¯ f ( ε, ε (cid:48) , x ) = x if ρ ( x ) < ε (cid:48) or ρ ( x ) > ε .Furthermore ¯ f and ¯ f − are smooth functions of ( ε, ε (cid:48) , x ) ∈ (0 , ε ) × ( − ε , × R m .Proof. We choose ε > ε < ¯ ε , so that we can use in Ω ε all the constructionsfrom the previous section. Conclusion (i) is then immediate.In order to prove (ii) we begin by considering the problem of approximating Ω from the interior, andconstruct the desired function f first just on [0 , ε ) × Ω. Thus we assign to each x ∈ Ω and ε ∈ [0 , ε )11 value f ( ε, x ) ∈ Ω ε taken to be along the flow S ( · ) x defined in (4.1), starting at x . However, sincethe flow is defined to be non-stationary just in a neighbourhood of the boundary, we take f ( ε, x ) = x for x far enough from the boundary. Thus we define f ( ε, x ) = (cid:26) S ( t ( ε, x )) x, x ∈ Ω \ Ω ε ,x, x ∈ Ω ε , (5.2)for a t ( ε, x ) to be determined, where t (0 , x ) = 0 (so that f (0 , x ) = x ). In order to define t ( ε, x ) for ε ∈ (0 , ε ) let ˜ h : R + → [0 ,
1] be a smooth function such that ˜ h ≡ , h ≡ / , ∞ ) and − < h (cid:48) ( r ) ≤ r ≥
0. We take now h ( ε, r ) def = ε ˜ h ( rε ). Then h ( ε, · ) ≡ ε on [0 , ε ] and h ( ε, · ) ≡ ε/ , ∞ ) with − < ∂h∂r ( ε, r ) ≤ r ≥ , ε ∈ (0 , ε ). For x ∈ Ω \ Ω ε and ε ∈ (0 , ε ) define t ( ε, x ) to be the unique t ≥ ρ ( S ( t ) x ) = ρ ( x ) + h ( ε, ρ ( x )) . (5.3)We claim now that t ( ε, x ) is well defined. We denote g ( t ) def = ρ ( S ( t ) x ) − ρ ( x ) − h ( ε, ρ ( x )). Thenx S(t( ε ,x))x ∂ Ω ∂ Ω ε Figure 2: Defining the diffeomorphism along the flow g (0) ≤
0. But ρ ( S (¯ t ) x ) = 3 ε for some ¯ t ≥ g (¯ t ) = 3 ε − ρ ( x ) − h ( ε, ρ ( x )) ≥
0, since ρ ( S (¯ t ) x ) ≥ ρ ( x ) and h ( ε, ε ) = 0 togetherwith ∂∂ρ ( ρ + h ( ε, ρ )) >
0. Finally g ( t ) is strictly increasing by Lemma 4.1. This proves our claimregarding the definition of t ( ε, x ). Note that the properties of t imply that f ( ε, · ) : ∂ Ω → ∂ Ω ε and f ( ε, · ) : Ω → Ω ε .We continue by claiming that t is a smooth function of ( ε, x ) in (0 , ε ) × (cid:0) Ω \ ¯Ω ε (cid:1) . This followsfrom the implicit function theorem applied to F ( t, ε, x ) def = ρ ( S ( t ) x ) − ρ ( x ) − h ( ε, ρ ( x )) (5.4)(the non-degeneracy condition needed for applying the implicit function theorem is a consequenceof the relation ddt ρ ( S ( t ) x ) = ∇ ρ ( S ( t ) x ) · ddt S ( t ) x = [( ∇ ρ · G ) γ ] ( S ( t ) x ) > , (5.5)where for the equality we used the definition of the flow S ( · ) x and for the last inequality we usedthe definition of G together with Remark 3.1 with x = P ).Since S ( t ) x, ρ ( x ) , h ( ε, ρ ( x )) are smooth in t , x ∈ Ω and ε > f is smooth on(0 , ε ) × (cid:0) Ω \ Ω ε (cid:1) . Since if ρ ( x ) ∈ [5 ε/ , ε ] then t ( ε, x ) = 0, and thus f ( ε, x ) = x , it follows that f
12s smooth in (0 , ε ) × Ω. To show that f : [0 , ε ) × Ω → Ω ε is continuous it is enough to show that t ( · , · ) is continuous at points ( ε, ˜ x ) where ε ∈ [0 , ε ) and ˜ x ∈ ∂ Ω. Assume for contradiction that thisis not the case, so that there exist sequences x k → ˜ x , x k ∈ Ω and ε k → ε in [0 , ε ) such that t ( ε k , x k )does not converge to t ( ε, ˜ x ). By the uniformity in time in Lemma 4.2 we may assume that t ( ε k , x k ) → τ (cid:54) = t ( ε, ˜ x ). We may also suppose without loss of generality that ε k > k . Replacing x with x k in (5.3) and passing to the limit k → ∞ we obtain that ρ ( S ( τ )˜ x ) = ρ ( S ( t ( ε, ˜ x ))˜ x ), and hence τ = t ( ε, ˜ x ), a contradiction which proves our assertion that t is continuous up to the boundary.Next we check that f ( ε, · ) is one-to-one. Suppose f ( ε, x ) = f ( ε, y ) for x, y ∈ Ω. If ρ ( x ) > ε and ρ ( y ) > ε then f ( ε, x ) = x, f ( ε, y ) = y and so x = y . If ρ ( x ) ≤ ε and ρ ( y ) > ε then ρ ( f ( ε, x )) = ρ ( x ) + h ( ε, ρ ( x )) ≤ ε + h ( ε, ε ) = 3 ε < ρ ( y ) = ρ ( f ( ε, y )) so this case cannot occur. Finally ifboth ρ ( x ) and ρ ( y ) are in [0 , ε ] then we have S ( t ( ε, x )) x = S ( t ( ε, y )) y and hence ρ ( x ) = ρ ( y ). If t ( ε, x ) = t ( ε, y ) there is nothing to prove, so we assume without loss of generality that t ( ε, x ) > t ( ε, y ).Then S ( t ( ε, x ) − t ( ε, y )) x = y , hence ρ ( x ) < ρ ( y ) since ρ ( S ( t ) x ) is strictly increasing in t , giving acontradiction.We next show that f ( ε, · ) is onto. This is obvious if ε = 0 so we suppose ε ∈ (0 , ε ). To this endwe take an arbitrary z ∈ ¯Ω ε and seek x ∈ ¯Ω with f ( ε, x ) = z . If ρ ( z ) ≥ ε then f ( ε, z ) = z , so wesuppose ρ ( z ) < ε . First note that by Lemma 4.2 there exists α ( ε, z ) ≤ ρ ( S ( α ( ε, z )) z ) = 0.We look for x of the form x = S ( β ( ε, z )) z with α ( ε, z ) ≤ β ( ε, z ) ≤
0. Denoting¯ g ( ε, z, τ ) def = ρ ( z ) − ρ ( S ( τ ) z ) − h ( ε, ρ ( S ( τ ) z )) , (5.6)we have that ¯ g ( ε, z, ≤ , ¯ g ( ε, z, α ( ε, z )) = ρ ( z ) − ε ≥
0. Since ¯ g ( ε, z, τ ) is strictly decreasing in τ it follows that there exists β ( ε, z ) ∈ [ α ( ε, z ) ,
0] with ¯ g ( ε, z, β ( ε, z )) = 0, that is ρ ( S ( − β ( ε, z )) S ( β ( ε, z )) z ) = ρ ( S ( β ( ε, z )) z ) + h ( ε, ρ ( S ( β ( ε, z )) z )) . Also, since β ( ε, z ) ≥ α ( ε, z ) we have that ρ ( S ( β ( ε, z )) z ) ≥ ρ ( S ( α ( ε, z )) z ) = 0, so that S ( β ( ε, z )) z ∈ Ω. Hence t ( ε, S ( β ( ε, z )) z ) = − β ( ε, z ) and f ( ε, S ( β ( ε, z )) z ) = S ( − β ( ε, z )) S ( β ( ε, z )) z = z . Hence¯ f ( ε, · ) : ¯Ω → ¯Ω ε is onto, and so also ¯ f ( ε, · ) : Ω → Ω ε is onto. Furthermore, relation (5.5) impliesthat ∂ ¯ g∂τ ( ε, z, β ( ε, z )) <
0, so that by the implicit function theorem β is a smooth function of ( ε, z )for z ∈ Ω ε and ε ∈ (0 , ε ). Therefore f − ( ε, z ) = S ( β ( ε, z )) z is also a smooth function of ( ε, z ) ∈ (0 , ε ) × Ω ε . That f − : [0 , ε ) × ¯Ω ε → ¯Ω is continuous follows from the continuity of f .This completes the construction of the mapping f = f ( ε, x ) in (ii) for ε ∈ [0 , ε ) and x ∈ ¯Ω. Inparticular, since Ω is by hypothesis connected, so is Ω ε for ε ∈ [0 , ε ). To show that Ω ε is of class C ∞ for ε ∈ (0 , ε ) let ¯ x ∈ ∂ Ω ε , so that ρ (¯ x ) = ε . As |∇ ρ (¯ x ) | (cid:54) = 0 at least one of the partial derivatives ∂ρ∂x i (¯ x ) is nonzero. Without loss of generality we may assume that ∂ρ∂x m (¯ x ) >
0. By the implicitfunction theorem there exist δ > f ∈ C ∞ ( R m − ) such that { x ∈ B (¯ x, δ ) : ρ ( x ) = ε } = { ( x (cid:48) , x m ) ∈ B (¯ x, δ ) : x m = ˜ f ( x (cid:48) ) } . Since for δ > ∂ρ∂x m ( x ) > x ∈ B (¯ x, δ ) it follows that Ω ε ∩ B (¯ x, δ ) = { ( x (cid:48) , x m ) ∈ Ω ε : x m > ˜ f ( x (cid:48) ) } as required.Notice that the same argument proves that there is a continuous mapping F : ( − ε , × Ω c → R m such that if ε (cid:48) ∈ ( − ε ,
0] then F ( ε (cid:48) , · ) is a homeomorphism of Ω c onto Ω cε (cid:48) with inverse F − ( ε (cid:48) , · ) :Ω cε (cid:48) → Ω c , such that F ( ε (cid:48) , · ) is a C ∞ diffeomorphism of ( ¯Ω) c onto ( ¯Ω ε (cid:48) ) c if ε (cid:48) ∈ ( − ε , F ( ε (cid:48) , x ) = x if ρ ( x ) < ε (cid:48) . Furthermore F − : ( − ε , × Ω cε (cid:48) → Ω c is continuous, while F : ( − ε , × ( ¯Ω) c → ( ¯Ω ε (cid:48) ) c and F − : ( − ε , × ( ¯Ω ε (cid:48) ) c → ( ¯Ω) c are smooth. In fact we can deducethis by applying the above to each of the finite number of connected components of the bounded open13et Ω R = B (0 , R ) \ ¯Ω for a large R with ¯Ω ⊂ B (0 , R ), noting that each such component is a boundeddomain of class C , and redefining the mapping to be the identity in a suitable neighbourhood of ∂B (0 , R ). We will use this observation below when extending the definition of f to the whole of( − ε , ε ) × R m .In order to define f in (ii) on ( − ε , × ¯Ω we cannot proceed in exactly the same way as we did todefine f on [0 , ε ) × ¯Ω via an analogue of the definition (5.2), because we would then not be able toprove smoothness via the implicit function theorem at points on, or with images on, ∂ Ω (since ρ isnot smooth there). Instead we proceed by first proving (iii), from which the extension of f to thewhole of ( − ε , ε ) × R m will follow easily.We make use of the following lemma. Lemma 5.1.
There exists a smooth function h : (0 , × (0 , × R → R satisfying: h ( a, c, t ) = t for all a, c ∈ (0 , , t (cid:54)∈ [0 , ,∂h∂t ( a, c, t ) > for all a, c ∈ (0 , , t ∈ R ,h ( a, c, a ) = c. .Proof. Let ϕ ∈ C ∞ ( R ) , ϕ ≥ , ϕ ( t ) = ϕ ( − t ) for all t , supp ϕ ⊂ ( − , (cid:82) ∞−∞ ϕ ( t ) dt = 1. Define δ = δ ( a, c ) by δ ( a, c ) := 14 a (1 − a ) c (1 − c ) , and H ( a, c, t ) := t if t (cid:54)∈ [ δ, − δ ] ,δ + ( t − δ ) c − δa − δ if t ∈ [ δ, a − δ ] ,t + c − a if t ∈ [ a − δ, a + δ ] ,c + δ + ( t − a − δ ) − c − δ − a − δ if t ∈ [ a + δ, − δ ] . Note that min( a, − a, c, − c ) > δ , so that the continuous piecewise affine function H ( a, c, · ) iswell defined and has strictly positive slope. We claim that h ( a, c, t ) := (cid:90) ∞−∞ δ φ (cid:18) t − s ) δ (cid:19) H ( a, c, s ) ds = (cid:90) ∞−∞ ϕ ( σ ) H ( a, c, t + δσ dσ has the required properties. First, splitting the range of integration into the different parts in which H ( a, c, · ) is affine, we see that h ( a, c, t ) is the sum of five integrals each having the form (cid:90) q i ( a,c,t ) p i ( a,c,t ) ϕ ( σ ) θ i ( a, c, t, σ ) dσ, where, for each i = 1 , . . . , p i , q i are smooth functions of ( a, c, t ) in the set D := (0 , × R , and θ i is smooth on (0 , × R , so that h is smooth on D . Since h t ( a, c, t ) = (cid:90) ∞−∞ ϕ ( σ ) H t ( a, c, t + δσ dσ
14e have that h t ( a, c, t ) >
0. If t (cid:54)∈ ( δ , − δ ) then t + δ (cid:54)∈ ( δ, − δ ) for | σ | <
1, so that, since ϕ ( t ) iseven, h ( a, c, t ) = t . Similarly h ( a, c, a ) = (cid:90) − ϕ ( σ )( c + δσ dσ = c. Continuing with the proof of (iii), given any x ∈ R m with | ρ ( x ) | < ε and any τ ∈ ( − ε , ε ), thereexists a unique t = t ( τ, x ) such that ρ ( S ( t ) x ) = τ . Note that if also | ρ ( S ( σ ) x ) | < ε then t ( τ, S ( σ ) x ) = t ( τ, x ) − σ. (5.7)Furthermore t ( τ, x ) is smooth in τ (cid:54) = 0 and x , and ddτ t ( τ, x ) > τ (cid:54) = 0. For such x and ε ∈ (0 , ε ) , ε (cid:48) ∈ ( − ε , α ( ε, ε (cid:48) , x ) = 1 t (2 ε, x ) − t (2 ε (cid:48) , x ) , β ( ε, ε (cid:48) , x ) = − t (2 ε (cid:48) , x ) t (2 ε, x ) − t (2 ε (cid:48) , x ) . (5.8)Note that by (5.7), if y = S ( σ ) x and | ρ ( x ) | , | ρ ( y ) | < ε then α ( ε, ε (cid:48) , x ) = α ( ε, ε (cid:48) , y ) , β ( ε, ε (cid:48) , x ) = β ( ε, ε (cid:48) , y ) − σα ( ε, ε (cid:48) , y ) . (5.9)We define the desired ¯ f in (iii) by¯ f ( ε, ε (cid:48) , x ) = (cid:26) S ( η ( ε, ε (cid:48) , x )) x, if ρ ( x ) ∈ (3 ε (cid:48) , ε ) ,x, otherwise , (5.10)where η ( ε, ε (cid:48) , x ) = α ( ε, ε (cid:48) , x ) − [ h ( r ( ε, ε (cid:48) , x ) , s ( ε, ε (cid:48) , x ) , β ( ε, ε (cid:48) , x )) − β ( ε, ε (cid:48) , x )] , (5.11) h is as in Lemma 5.1 and r ( ε, ε (cid:48) , x ) = α ( ε, ε (cid:48) , x ) t ( ε, x ) + β ( ε, ε (cid:48) , x ) , (5.12) s ( ε, ε (cid:48) , x ) = α ( ε, ε (cid:48) , x ) t ( ε (cid:48) , x ) + β ( ε, ε (cid:48) , x ) . (5.13)Note that r ( ε, ε (cid:48) , x ) , s ( ε, ε (cid:48) , x ) ∈ (0 ,
1) so that ¯ f ( ε, ε (cid:48) , x ) is well defined. Also, from (5.9), if y = S ( σ ) x and | ρ ( x ) | , | ρ ( y ) | < ε then r ( ε, ε (cid:48) , x ) = r ( ε, ε (cid:48) , y ) , s ( ε, ε (cid:48) , x ) = s ( ε, ε (cid:48) , y ) . (5.14)Furthermore, if ρ ( x ) ≥ ε then t (2 ε, x ) ≤ β ( ε, ε (cid:48) , x ) ≥
1. From the properties of h we thushave η ( ε, ε (cid:48) , x ) = 0 and ¯ f ( ε, ε (cid:48) , x ) = x . Similarly η ( ε, ε (cid:48) , x ) = 0 and ¯ f ( ε, ε (cid:48) , x ) = x if ρ ( x ) ≤ ε (cid:48) .To prove that ¯ f ( ε, ε (cid:48) , · ) is one-to-one, suppose that ¯ f ( ε, ε (cid:48) , x ) = ¯ f ( ε, ε (cid:48) , y ). If both ρ ( x ) , ρ ( y ) (cid:54)∈ (2 ε (cid:48) , ε ) then clearly x = y . If ρ ( x ) (cid:54)∈ (2 ε (cid:48) , ε ) and ρ ( y ) ∈ (2 ε (cid:48) , ε ) then t ( ε, y ) ∈ ( t (2 ε (cid:48) , y ) , t (2 ε, y )).Hence β ( ε, ε (cid:48) , y ) ∈ (0 , α ( ε, ε (cid:48) , y ) η ( ε, ε (cid:48) , y ) + β ( ε, ε (cid:48) , y ) ∈ (0 , η ( ε, ε (cid:48) , y ) ∈ ( t (2 ε (cid:48) , y ) , t (2 ε, y )). Hence ρ ( ¯ f ( ε, ε (cid:48) , y )) ∈ (2 ε (cid:48) , ε ) and so ¯ f ( ε, ε (cid:48) , y ) (cid:54) = x = ¯ f ( ε, ε (cid:48) , x ). Sothis case cannot occur.If both ρ ( x ) , ρ ( y ) ∈ (2 ε (cid:48) , ε ) then y = S ( σ ) x , where σ = η ( ε, ε (cid:48) , x ) − η ( ε, ε (cid:48) , y ). Let γ = ρ ( ¯ f ( ε, ε (cid:48) , x )) = ρ ( ¯ f ( ε, ε (cid:48) , y )). Then η ( ε, ε (cid:48) , x ) = t ( γ, x ) , η ( ε, ε (cid:48) , y ) = t ( γ, y ). Thus by (5.7), (5.9) α ( ε, ε (cid:48) , x ) η ( ε, ε (cid:48) , x ) + β ( ε, ε (cid:48) , x ) = α ( ε, ε (cid:48) , y ) η ( ε, ε (cid:48) , y ) + β ( ε, ε (cid:48) , y ) . h ( r ( ε, ε (cid:48) , x ) , s ( ε, ε (cid:48) , x ) , β ( ε, ε (cid:48) , x )) = h ( r ( ε, ε (cid:48) , x ) , s ( ε, ε (cid:48) , x ) , β ( ε, ε (cid:48) , y )) , which implies by the strict monotonicity of h ( a, c, t ) in t that β ( ε, ε (cid:48) , x ) = β ( ε, ε (cid:48) , y ) and hence σ = 0and x = y .To prove that ¯ f ( ε, ε (cid:48) , · ) is onto it suffices to show that if ρ ( y ) ∈ (3 ε (cid:48) , ε ) then S ( η ( ε, ε (cid:48) , x )) x = y forsome x , and we claim that such an x with ρ ( x ) ∈ (3 ε (cid:48) , ε ) is given by x = S ( τ ( ε, ε (cid:48) , y )) y ; τ ( ε, ε (cid:48) , y ) = α ( ε, ε (cid:48) , y ) − [ h − ( r ( ε, ε (cid:48) , y ) , s ( ε, ε (cid:48) , y ) , β ( ε, ε (cid:48) , y )) − β ( ε, ε (cid:48) , y )] , (5.15)where h − ( a, c, · ) denotes the inverse function of h ( a, c, · ). To show that ρ ( x ) ∈ (3 ε (cid:48) , ε ) it suffices toprove that τ ( ε, ε (cid:48) , y ) ∈ ( t (3 ε (cid:48) , y ) , t (3 ε, y )). But this holds because α ( ε, ε (cid:48) , y ) t (3 ε, y ) + β ( ε, ε (cid:48) , y ) > α ( ε, ε (cid:48) , y ) t (3 ε (cid:48) , y ) + β ( ε, ε (cid:48) , y ) <
0, using h t ( a, c, t ) > h ( a, c, t ) = t for t (cid:54)∈ (0 , x given by (5.15) we deduce from (5.9) that β ( ε, ε (cid:48) , x ) = β ( ε, ε (cid:48) , y ) + τ ( ε, ε (cid:48) , y ) α ( ε, ε (cid:48) , y ) = h − ( r ( ε, ε (cid:48) , y ) , s ( ε, ε (cid:48) , y ) , β ( ε, ε (cid:48) , y )) , so that η ( ε, ε (cid:48) , x ) = α − ( ε, ε (cid:48) , y )[ β ( ε, ε (cid:48) , y ) − β ( ε, ε (cid:48) , x )] = − τ ( ε, ε (cid:48) , y )as required.Next we note that ρ ( x ) > ε if and only if t ( ε, x ) <
0, which holds if and only if h ( α ( ε, ε (cid:48) , x ) t ( ε, x )+ β ( ε, ε (cid:48) , x ) , α ( ε, ε (cid:48) , x ) t ( ε (cid:48) , x )+ β ( ε, ε (cid:48) , x ) , β ( ε, ε (cid:48) , x )) > α ( ε, ε (cid:48) , x ) t ( ε (cid:48) , x )+ β ( ε, ε (cid:48) , x ) , since h ( a, c, a ) = c and h t ( a, c, t ) >
0, thus if and only if ρ ( S ( η ( ε, ε (cid:48) , x )) x ) > ε (cid:48) . Thus ¯ f ( ε, ε (cid:48) , Ω ε ) =Ω ε (cid:48) and ¯ f ( ε, ε (cid:48) , ∂ Ω ε ) = ∂ Ω ε (cid:48) . That ¯ f , ¯ f − are smooth functions of ( ε, ε (cid:48) , x ) ∈ (0 , ε ) × ( − ε , × R m follows from the smoothness of α, β, r, s, h and h − (the latter by h t > f : [0 , ε ) × ¯Ω → R m constructed at the beginning of the proof by ˜ f . We need to extend ˜ f to a map f : ( − ε , ε ) × R m → R m satisfying (ii). We define f = f ( ε, x ) by f (0 , x ) = x and f ( ε, x ) = ˜ f ( ε, x ) if ε ∈ (0 , ε ) , x ∈ ¯Ω , ¯ f − ( ε, − ε, F ( − ε, x )) if ε ∈ (0 , ε ) , x ∈ Ω c , ¯ f ( − ε, ε, ˜ f ( − ε, x )) if ε ∈ ( − ε , , x ∈ ¯Ω ,F ( ε, x ) if ε ∈ ( − ε , , x ∈ Ω c . (5.16)Note that the domains of definition of f ( ε, x ) overlap for x ∈ ∂ Ω. However the definitions coincidethere because by construction in each case f ( ε, x ) lies on the intersection of the orbit of the flow ofgood directions through x with ∂ Ω ε , and this point is unique. The properties of ˜ f , ¯ f and F implythat f ( ε, · ) is a homeomorphism of R m onto R m with f ( ε, Ω) = Ω ε , f ( ε, ∂ Ω) = ∂ Ω ε , with inversegiven by f − (0 , x ) = x and f − ( ε, x ) = ˜ f − ( ε, x ) if ε ∈ (0 , ε o ) , x ∈ ¯Ω ε ,F − ( − ε, ¯ f ( ε, − ε, x )) if ε ∈ (0 , ε ) , x ∈ Ω cε , ˜ f − ( − ε, ¯ f − ( − ε, ε, x )) if ε ∈ ( − ε , , x ∈ ¯Ω ε ,F − ( ε, x ) if ε ∈ ( − ε , , x ∈ Ω cε . (5.17)16he continuity of f ( ε, x ) and f − ( ε, x ) for 0 ≤ | ε | < ε , x ∈ R m follows since, as is easily checked, f ( ε, − ε, x ) → x, f − ( ε, − ε, x ) → x uniformly as ε → f ( ε, x ) for 0 < | ε | <ε , x (cid:54)∈ ∂ Ω, and of f − ( ε, x ) for 0 < | ε | < ε , x (cid:54)∈ ∂ Ω ε , follows from the corresponding properties of˜ f , ¯ f and F . Finally, by construction f ( ε, x ) = x for | ρ ( x ) | > | ε | .This completes the proof. Remark 5.1.
The above proof uses part (iii) of the theorem to help prove part (ii). Conversely,given (ii), for − ε < ε (cid:48) < < ε < ε we can define f ( ε, ε (cid:48) , x ) = f ( ε (cid:48) , f − ( ε, x )), which is ahomeomorphism of R m onto R m such that f ( ε, ε (cid:48) , Ω ε ) = Ω ε (cid:48) , f ( ε, ε (cid:48) , ∂ Ω ε ) = ∂ Ω ε (cid:48) , f ( ε, ε (cid:48) , · ) : Ω ε → Ω ε (cid:48) is a diffeomorphism, and f ( ε, ε (cid:48) , x ) = x if | ρ ( x ) | > max( − ε (cid:48) , ε ). However (iii) gives extrainformation, in particular that f ( ε, ε (cid:48) , · ) is a diffeomorphism of ¯Ω ε onto ¯Ω ε (cid:48) . Remark 5.2.
As observed by Fraenkel [15, Section 5] the image under a diffeomorphism of abounded domain of class C need not be a domain of class C , since a cusp such as in Remark 2.2can be bent by the diffeomorphism so that the boundary is not locally a graph. However Theorem 5.1immediately implies a corresponding smooth approximation result for the larger class of boundeddomains Ω ⊂ R m which are the image under a C ∞ diffeomorphism ϕ : U → R m of a boundeddomain Ω (cid:48) ⊂ R m of class C , where U is an open neighbourhood of Ω (cid:48) . If Ω (cid:48) ε , < | ε | < ε , are theapproximating domains given by the theorem for Ω (cid:48) for ε > ε = ϕ (Ω (cid:48) ε ) are a family of bounded domains of class C ∞ such that (cid:84) − ε <ε< Ω ε = ¯Ω , (cid:83) <ε<ε Ω ε = Ω. Remark 5.3.
If Ω is Lipschitz, then the homeomorphism between Ω ε and Ω defined in the proofof Theorem 5.1 is a bi-Lipschitz map (with Lipschitz constants bounded independently of ε for0 < | ε | < ε ). In order to check this it suffices to show that the functions t and β (for theinterior approximation, say) are Lipschitz. This can be seen in the case of t , for example, byapplying ∂∂x i to F ( t ( x ) , x ) = 0 with F as in (5.4), obtaining thus that ∂t∂x i = − ∂F∂x i / ∂F∂t . Given x ∈ ∂ Ω with a corresponding good direction n , there exist δ = δ ( x ) and c ( δ ) , c ( δ ) such that0 < c ( δ ) < ∇ ρ ( x ) · n, |∇ ρ ( x ) | < c ( δ ) for all x ∈ B ( x , δ ) ∩ Ω (see [41, p. 63, relation (A.3), LemmaA.1 and the line after (A.7)]). We can then apply compactness and Remark 3.1 to show that forsome δ > c , c depending only on δ we have 0 < c < ∇ ρ ( x ) · G ( x ) , |∇ ρ ( x ) | ≤ c for all x ∈ Ω with dist ( x, ∂ Ω) < δ . Hence ∂F ( t ( x ) ,x ) ∂t is bounded away from zero, and ∂F ( t ( x ) ,x ) ∂x i isbounded, for all x ∈ Ω with dist ( x, ∂ Ω) < δ . Thus |∇ f ( x ) | is bounded for such x and hence for all x ∈ Ω. Hence f ∈ W , ∞ (Ω , R m ). Since Ω is Lipschitz this implies that f is Lipschitz. This follows,for example, by noting that by Stein [60, Chapter VI, Theorem 5] f may be extended to a function˜ f ∈ W , ∞ ( R m , R m ) so that ˜ f is Lipschitz (a more general result can be found in [25, Theorem 4.1]).Related results concerning the approximation of Lipschitz domains were obtained in [50, 61, 62] (forthe last two papers see the comments in the introduction). The recent paper of Amrouche, Ciarlet& Mardare [1], which refers to an earlier version of our paper, contains a statement (their Theorem2.2) of a special case of Theorem 5.1 for Ω Lipschitz. Remark 5.4.
Theorem 5.1 implies in particular that if Ω ⊂ R m is a bounded domain of class C ,then ¯Ω and Ω c have differential structures making them C ∞ m -manifolds with boundary, and ∂ Ω hasa differential structure making it a C ∞ ( m − − manifold, since ¯Ω (resp. Ω c , ∂ Ω) is homeomorphicto ¯Ω ε (resp. Ω cε , ∂ Ω) for 0 < | ε | < ε , which is such a C ∞ manifold. The field of good directionscan be thought of as a transverse field to ∂ Ω, and more generally the existence of a transverse fieldto a topological manifold embedded in R m is known to imply the existence of a smooth differentialstructure following the work of Cairns [6] and Whitehead [64] (see also Pugh [53]).A referee has drawn our attention to the possibility of using the techniques of the theory of smoothingof manifolds to prove the existence of diffeomorphic interior and exterior approximations by C ∞ ⊂ R m whose boundaries are locally flat in thesense of Brown [5], that is ¯Ω and Ω c are topological manifolds with boundary (one advantage ofthis class of domains is that it is invariant under homeomorphisms). Indeed this seems likely tobe the case for m ≥ m = 2 , f ( ε, · ) of ¯Ω onto ¯Ω ε for 0 < | ε | < ε that is adiffeomorphism of Ω onto Ω ε . In general, if U ⊂ R m , V ⊂ R m are homeomorphic open sets, then if m = 1 , , U and V are diffeomorphic, while if m ≥ U is the whole of R m (or equivalently U is an open ball) then U and V are also diffeomorphic, since in these cases U hasa unique differential structure up to diffeomorphism. These results are due to [47], [49], [59] andare surveyed in [45]. This is not true if m = 4 because of the existence (following from the work ofFreedman [16] and Donaldson [12]) of ‘small exotic R s’, which implies that there is a bounded opensubset of R which is homeomorphic but not diffeomorphic to an open ball in R ; for discussionsof this work see [55], and [31, Chapter XIV]. Whether two arbitrary homeomorphic open subsets of R m , m ≥
5, are diffeomorphic seems not to be known in general.
Remark 5.5.
For an arbitrary open set Ω ⊂ R m with ¯Ω (cid:54) = R m one can easily find smooth setsapproximating it and contained in Ω (respectively R m \ ¯Ω), provided that one does not require theapproximating sets to preserve the topology or diffeomorphism class of Ω. For instance, as shown in[60, Chapter 6] one can always find a regularised distance ˜ ρ and then for almost all ε small enoughthe sets Ω ε defined as in (5.1) (but with ρ replaced by ˜ ρ ) will be of class C ∞ (due to Sard’s theorem,see for instance [42, p. 35]).We now discuss counterexamples to the conclusions of Theorem 5.1 when Ω is not of class C . It iseasy to construct such examples for the exterior approximation. Example 5.1.
Let Ω ⊂ R be the simply-connected domain defined byΩ = B (0 , \ B ((0 , / , / . Then Ω is not of class C because the boundary cannot be represented as a graph in the neigh-bourhood of the boundary point (0 , ( j ) ⊂ R , j = 1 , , . . . , which are each homeomorphic to Ω and such that ¯Ω = (cid:84) ∞ j =1 Ω ( j ) .Indeed, for any such sequence we have that S = ∂B (0 , ⊂ Ω ( j ) for all j , but (0 , / (cid:54)∈ Ω ( j ) forsufficiently large j , so that Ω ( j ) is not simply-connected for sufficiently large j .In order to give counterexamples for the interior approximation we will consider bounded opensubsets Ω ⊂ R m which are topological manifolds with boundary (strictly speaking, such that ¯Ω is atopological manifold with boundary), i.e. each point x ∈ ∂ Ω has a neighbourhood in ¯Ω that can bemapped homeomorphically onto a relatively open subset of H m = { ( x , . . . , x m ) ∈ R m : x m ≥ } .Any bounded domain of class C is a topological manifold with boundary (see, for example, [2,Appendix A]), but not conversely; for example, the interior of a Jordan curve in R is a topologicalmanifold with boundary by the Schoenflies theorem.18 roposition 5.1. Let Ω ⊂ R m be a bounded open set that is a topological manifold with boundary.Let Ω (cid:48) ⊂ R m be a bounded open set that is homeomorphic to Ω . Then ∂ Ω has a finite number N ofboundary components, and ∂ Ω (cid:48) has a finite number N (cid:48) of boundary components, where N (cid:48) ≤ N .Proof. We first claim that ∂ Ω has a finite number N of boundary components Γ i , ≤ i ≤ N .If there were infinitely many then there would be a point ξ ∈ ∂ Ω and a sequence ξ i → ξ with ξ i ∈ Γ i , i = 1 , , . . . , ∞ . Since Ω is a topological manifold with boundary, there exist δ > ψ : B ( ξ, δ ) ∩ ¯Ω → V , where V is a relatively open subset of H m . Then for ¯ δ > ψ − ( B ( ψ ( ξ ) , ¯ δ ) ∩ ∂H m ) is a connected open neighbourhood of ξ in ∂ Ω to which ξ i belongs for sufficiently large i , a contradiction.By the collaring theorem of Brown [5] (for an alternative proof see [8]), for each i = 1 , . . . , N thereis a homeomorphism ψ i mapping Γ i × [0 ,
1) onto a relatively open neighbourhood U i of Γ i in ¯Ω suchthat ψ i ( x,
0) = x for all x ∈ Γ i . Consider the open subset U i,ε = ψ i (Γ i × (0 , ε )) of Ω. Since theproduct of connected sets is connected, and ψ i is continuous, U i,ε is connected.By assumption there is a homeomorphism ϕ : Ω → Ω (cid:48) . Therefore ϕ ( U i,ε ) is connected. Hence ϕ ( U i,ε )is a closed connected subset of Ω (cid:48) . Let ε j →
0. We may assume that the sets ϕ ( U i,ε j ) converge toa subset V i of Ω (cid:48) in the Hausdorff metric, and V i is connected. We claim that V i ⊂ ∂ Ω (cid:48) . Indeedif z ∈ Ω (cid:48) ∩ V i there would exist a sequence x k ∈ Ω with x k → ¯ x ∈ Γ i and ϕ ( x k ) → z . But then x k → ϕ − ( z ) ∈ Ω, a contradiction. Hence for each i we have a corresponding connected subset V i of ∂ Ω (cid:48) . We claim that ∂ Ω (cid:48) ⊂ (cid:83) Ni =1 V i . Indeed suppose that z ∈ ∂ Ω (cid:48) . Then there exists a sequence z k ∈ Ω (cid:48) with z k → z . Then ϕ − ( z k ) ∈ Ω for each k , and we may assume that ϕ − ( z k ) → x ∈ ¯Ω.If x ∈ Ω then z k → ϕ ( x ) = z ∈ Ω (cid:48) , a contradiction. Hence ϕ − ( z k ) → x ∈ ∂ Ω, and so x ∈ Γ i forsome i . So there exists a subsequence z k j with ϕ − ( z k j ) ∈ U i,ε j , and hence z ∈ V i . Thus ∂ Ω (cid:48) has N (cid:48) components for some N (cid:48) ≤ N . Remark 5.6.
In general N (cid:48) < N . For example if Ω consists of the union of two balls with disjointclosures while Ω (cid:48) consists of two disjoint open balls whose boundaries touch, then we have N = 2and N (cid:48) = 1. It is perhaps true that N (cid:48) = N if Ω is connected, but we do not need this. Example 5.2.
Let Ω = B (0 , \ ∞ (cid:91) j =1 B ((1 − /j ) e , / j ) ⊂ R m , where e = (1 , , . . . , ∂ Ω has infinitely many boundarycomponents, and thus Ω cannot be homeomorphic to a bounded domain of class C ∞ (or of class C ). Ω and the properties of the map of gooddirections In Section 2 we constructed special smooth fields of good directions, that we called canonical. Inthis section we study the properties of arbitrary continuous fields of good directions that are notnecessarily canonical.We start with an illustrative case that provides significant insight into more general situations. Weconsider a standard solid torus in R given by Ω T = T ([0 , π ] × [0 , π ] × [0 , T : [0 , π ] × [0 , π ] × [0 , → R , T ( θ, ϕ, r ) = (cos θ (2 + r cos ϕ ) , sin θ (2 + r cos ϕ ) , r sin ϕ ) , ∂ Ω T = T = T b ([0 , π ] ), where T b : [0 , π ] × [0 , π ] → R , T b ( θ, ϕ ) = (cos θ (2 + cos ϕ ) , sin θ (2 + cos ϕ ) , sin ϕ ) . (6.1)A continuous field C : T → S is a field of pseudonormals with respect to Ω T if and only if C ( P ) · ν T ( P ) > P ∈ T , (6.2)where ν T ( P ) denotes the interior normal to Ω T at P ∈ T . The geometrical condition (6.2) imposesa constraint on the image of the field C : Proposition 6.1.
For any continuous field C : T → S satisfying the geometrical condition (6.2) there exists a band of size δ around the equator on the unit sphere, namely E δ def = { n ∈ S : | n · e | < δ } , (6.3) such that E δ ⊂ C ( T ) , where e = (0 , , .Conversely, for any given γ > there exists a continuous field C γ : T → S satisfying (6.2) suchthat C γ ( T ) ⊂ E γ .Proof. In order to prove the first claim we use degree theory for the map C , on a domain ω ⊂ T with respect to the point n ∈ S , denoted d ( C, ω, n ). Since the domain ω we use is diffeomorphicto a bounded open subset of R we can use the theory of degree for subsets of Euclidean space asdescribed, for example, in [13]. It suffices to show that d ( C, ω, n ) = 1 for all n ∈ E δ with suitable δ > ω ⊂ T (because this implies that E δ ⊂ C ( ω )). In order to show this we use the homotopyinvariance of the degree [13, Theorem 2.3] for the homotopy H : [0 , × ω → S defined by H ( λ, P ) = λC ( P ) + (1 − λ ) ν T ( P ) | λC ( P ) + (1 − λ ) ν T ( P ) | that connects the smooth negative Gauss map ν T with the field C . We choose ω def = T b ([0 , π ] × { (0 , π/ ∪ (3 π/ , π ] } )to be the ‘exterior part’ of the torus. We first note that d ( ν T , ω, n ) = 1 for all n ∈ S \ {± e } . Thisis because the Jacobian of ν T equals the Gaussian curvature of the torus which is positive in ω , andbecause such n do not belong to ν T ( ∂ω ) and have exactly one inverse image in ω under ν T . Nextobserve that the condition C ( P ) · ν T ( P ) > C ensures that there exists a δ > C ( P ) (cid:54)∈ E δ for P ∈ ∂ω . Thus for n ∈ E δ the condition n (cid:54)∈ { H ( λ, P ) , λ ∈ [0 , , P ∈ ∂ω } is satisfied and we can apply the homotopy invariance of the degree to conclude that d ( C, ω, n ) = 1for n ∈ E δ , and hence E δ ⊂ C ( T ).To prove the second claim of the proposition we follow a suggestion of an anonymous referee, whichprovides a simpler example than in our original approach. We denote by ˜ n ( θ, ϕ ) ∈ S the interiornormal at T b ( θ, ϕ ) (where T b was defined in (6.1)). We let t : [0 , π ] → S be given by t ( θ ) :=( − sin θ, cos θ, t ( θ ) · ˜ n ( θ, ϕ ) = 0 for all ( θ, ϕ ) ∈ [0 , π ] . Defining p ε : [0 , π ] → S for ε ∈ (0 ,
1) by p ε ( θ, ϕ ) := (1 − ε ) t ( θ ) + ε ˜ n ( θ, ϕ ) √ − ε + 2 ε
20e have | p ε ( θ, ϕ ) | = 1 for all ( θ, ϕ ) ∈ [0 , π ] and p ε · ˜ n = ε √ − ε + ε , so that p ε is a pseudonormalfield. On the other hand | p ε · e | = (cid:12)(cid:12)(cid:12) ε sin ϕ √ − ε +2 ε (cid:12)(cid:12)(cid:12) ≤ ε √ − ε +2 ε . Hence for any γ < ε suchthat p ε ( T ) ⊂ E γ .We now explore what happens for more general bounded domains. Theorem 6.1.
Let Ω ⊂ R m be a bounded domain of class C . (i) If m ≥ , and the Euler characteristic of Ω is non-zero then any continuous pseudonormal field n : ∂ Ω → S m − is surjective.If m = 3 and the Euler characteristic of Ω is zero then a continuous pseudonormal field is notnecessarily surjective. (ii) If m = 2 any continuous pseudonormal field n : ∂ Ω → S is surjective.Proof. (i) We first prove that if the Euler characteristic of Ω is non-zero then n : ∂ Ω → S m − issurjective. We assume for contradiction that n is not surjective. Thus there exists a δ > n : ∂ Ω → S m − such that (cid:107) ¯ n − n (cid:107) C ( ∂ Ω) < δ is also not surjective.We claim that there exists a continuous field of good directions ˜ n defined on a neighbourhood V of ∂ Ω so that (cid:107) ˜ n − n (cid:107) C ( ∂ Ω) < δ/ n : ∂ Ω → S m − is continuous and ∂ Ω is bounded,there exists δ (cid:48) > | n ( x ) − n ( y ) | < δ x, y ∈ ∂ Ω with | x − y | < δ (cid:48) . (6.4)As Ω is of class C there exist some ¯ δ > P , . . . , P l ∈ ∂ Ω such that ∂ Ω ⊂ ∪ lj =1 B ¯ δ ( P j ),and such that for any j = 1 , , . . . , l and R ∈ B δ ( P j ) we have that n ( P j ) is a good direction withrespect to Ω at R . We assume, without loss of generality, that 8¯ δ < δ (cid:48) .Recalling the definition (5.1) of the sets Ω ε and from (3.2) that ≤ ρ ( x ) d ( x ) ≤ x ∈ R m , wehave that Ω − ¯ δ/ \ Ω ¯ δ/ ⊂ ∪ lj =1 B δ ( P j ). Consider a partition of unity α j , j = 1 , . . . , l , such that α j ∈ C ∞ ( B δ ( P j )) and Σ lj =1 α j ( x ) = 1 , for all x ∈ ∪ lj =1 B δ ( P j ). Letˆ n ( P ) def = l (cid:88) j =1 α j ( P ) n ( P j ) . Then, by Lemma 2.2, ˆ n ( P ) (cid:54) = 0 and ˆ n ( P ) | ˆ n ( P ) | is a good direction at P for any P ∈ Ω − ¯ δ/ \ Ω ¯ δ/ . Takinginto account that α j ( P ) = 0 for | P − P j | > δ , that 4¯ δ < δ (cid:48) , and (6.4) we obtain | n ( P ) − ˆ n ( P ) | ≤ l (cid:88) j =1 α j ( P ) | n ( P ) − n ( P j ) | ≤ δ P ∈ ∂ Ω . (6.5)On the other hand, for P ∈ ∂ Ω we have (cid:12)(cid:12)(cid:12)(cid:12) ˆ n ( P ) − ˆ n ( P ) | ˆ n ( P ) | (cid:12)(cid:12)(cid:12)(cid:12) = || ˆ n ( P ) | − | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l (cid:88) j =1 α j ( P ) n ( P j ) − l (cid:88) j =1 α j ( P ) n ( R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈J P α j ( P )( n ( P j ) − n ( R )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) j ∈J P α j ( P ) | n ( P j ) − n ( R ) | < δ , (6.6)21here J P def = { j ∈ { , , . . . , l } ; | P − P j | ≤ δ } and R = P i for some arbitrary i ∈ J P . For the lastinequality we used that | R − P j | ≤ δ ≤ δ (cid:48) , for all j ∈ J P and (6.4) together with (cid:80) j ∈J P α j ( P ) = 1.We take ˜ n ( P ) def = ˆ n ( P ) | ˆ n ( P ) | on V def = Ω − ¯ δ/ \ Ω ¯ δ/ and (6.5), (6.6) prove our claim about the existence of˜ n .We denote by n ε ( P ) the interior normal to ∂ Ω ε at P , so that − n ε : ∂ Ω ε → S m − is the Gauss map;note that n ε is parallel to ∇ ρ ( P ) and has the same degree as − n ε up to (possible) change of sign.As noted above ˜ n | ∂ Ω is not surjective. Hence, as ˜ n is continuous on V , there exists ε > − ε \ Ω ε ⊂ V and ˜ n | Ω − ε \ Ω ε is also not surjective. Moreover, by Theorem 5.1 the sets Ω ε and Ωare homeomorphic and thus they have the same Euler characteristic [23]. On the other hand for thesmooth domain Ω ε the Euler characteristic equals the degree of the Gauss map ([4, p. 384]) andhence the Gauss map has non-zero degree. For any P ∈ ∂ Ω ε we have that both ˜ n ( P ) and n ε ( P )are good directions at P , so that by Lemma 2.1 we have that ˜ n ( P ) · n ε ( P ) (cid:54) = −
1. The homotopy h : [0 , × ∂ Ω ε → S m − connecting ˜ n | ∂ Ω ε and n ε given by h ( t, P ) = t ˜ n ( P ) + (1 − t ) n ε ( P ) | t ˜ n ( P ) + (1 − t ) n ε ( P ) | is thus well defined. Hence n ε has the same non-zero degree as ˜ n | ∂ Ω ε and thus ˜ n | ∂ Ω ε is surjective (see[27, pp.123,125]), a contradiction. Hence n is surjective.If m = 3 and the Euler characteristic of Ω is zero the second part of Proposition 6.1 provides therequired counterexample.(ii) In the same way as for part (i) it suffices to show that ˜ n | ∂ Ω ε is surjective for nonzero | ε | sufficientlysmall. We observe that for ε (cid:54) = 0 the set Ω ε is a smooth 2D manifold with boundary. By theclassification theorem for 1D connected, compact smooth manifolds (see for instance [44]) we havethat each connected component Γ of ∂ Ω ε is diffeomorphic to S . Thus Γ can be parametrized asa smooth closed curve γ : S → R with constant speed | ˙ γ ( t ) | = s >
0. Let N ( t ) = n ε ( γ ( t )) for t ∈ S . Then ∆( t ) = ˙ γ ( t ) N ( t ) − ˙ γ ( t ) N ( t ) equals ± s for each t ∈ S , and since ∆( t ) is continuousthe sign of ∆( t ) is independent of t ∈ S . The Umlaufsatz theorem of Hopf (see for instance [7, p.275]) guarantees that ˙ γ ( t ) / | ˙ γ ( t ) | has degree ± S into S . Thefact that ∆( t ) has constant sign implies that N : S → S is homotopic to ˙ γ ( t ) / | ˙ γ ( t ) | , and so it hasdegree ± N and ˜ n | Γ are also homotopic (by the same argument as before), we havethat ˜ n | Γ has non-zero degree and hence is surjective. Remark 6.1.
The proof of Part (i) of Theorem 6.1 shows that if m is odd, then any continuouspseudonormal field is surjective, provided that a connected component of the boundary has non-zeroEuler characteristic. This is due to the fact that for m odd the degree of the Gauss map of a closed,smooth ( m − R m is half of its Euler characteristic (see for example[19, p. 196]).We continue by investigating properties of the multivalued map of all pseudonormals. For Ω ⊂ R m a bounded domain of class C and P ⊂ ∂ Ω we let G ( P ) def = { n ∈ S m − : n is a pseudonormal at P } , and for any E ⊂ ∂ Ω let G ( E ) def = ∪ P ∈ E G ( P ). We denote by P ( S m − ) the set of all subsets of S m − and begin by noting that the map G : ∂ Ω → P ( S m − ) is lower semicontinuous. Indeed, by definitionthis means that given any P ∈ ∂ Ω and n ∈ G ( P ), for any neighbourhood V of n in S m − there is a22eighbourhood U of P in ∂ Ω such that G ( Q ) ∩ V (cid:54) = ∅ for all Q ∈ U ; this is obvious since n ∈ G ( Q )for all Q in a neighbourhood of P .We use the following topological fact. Lemma 6.1. If Ω ⊂ R m is a connected open set and U is a connected component of R m \ Ω then ∂ U is connected.Proof. This is a consequence of [36, 49.VI,Theorem 2 and 57.I.9(i),57.III.1] (also noted in [34, Lem-mas 4(i), 5]).In addition we have the following structural result.
Lemma 6.2.
Let Ω ⊂ R m be a bounded domain of class C . Then R m \ ¯Ω has a single unboundedconnected component D and finitely many bounded connected components U i , i = 1 , . . . , k , each ofwhich is a bounded domain of class C with R m \ ¯ U i connected. Furthermore ∂ Ω can be written asthe disjoint union ∂ Ω = ∂ D ∪ ∂ U ∪ · · · ∪ ∂ U k (6.7) and the connected components of ∂ Ω are the sets ∂ D , ∂ U i , i = 1 , . . . , k .Proof. Since Ω is bounded, R m \ ¯Ω has a single unbounded connected component D . If there wereinfinitely many bounded connected components U i then we would have x i → x ∈ ∂ Ω for somesequence with x i ∈ U i , which is easily seen to contradict that Ω is of class C . Since Ω is of class C we have that ∂ Ω = ∂ ( R m \ ¯Ω), from which (6.7) follows. The fact that all the sets in the union aredisjoint follows easily from Ω being of class C . Also R m \ U i = ∪ j (cid:54) = i U j ∪ D ∪ Ω, which is connectedbecause all the sets in the union are connected and, for example, each point of ∂ D (resp. ∂ U j , j (cid:54) = i )has a neighbourhood consisting of points in ¯ D ∪
Ω (resp. ¯ U j ∪ Ω). Finally, by Lemma 6.1 each of thedisjoint compact sets in (6.7) is connected, and so they are the connected components of ∂ Ω.We can now provide some properties of the image of G : Proposition 6.2.
Let Ω ⊂ R m , m ≥ be a bounded domain of class C . Let C be a connectedcomponent of ∂ Ω . Then (i) Span G ( C ) = R m , (ii) G ( C ) is connected.Proof. (i) We assume for contradiction that this is not true. Then Span G ( C ) is contained in an( m − R m and thus G ( C ) ⊂ S m − ∩ { z ∈ R m : z · N ≥ } for some N ∈ S m − . By Lemma 6.2 either C = ∂ D or C = ∂ U i for some i . If C = ∂ D then sliding a hyperplanewith normal N from x · N = + ∞ until it touches C for the first time at some P , we find a gooddirection at P belonging to { z ∈ R m : z · N < } , a contradiction. Similarly, if C = ∂ U i for some i ,then sliding such a hyperplane from x · N → −∞ until it touches C for the first time, and recallingthat by Lemma 6.2 U i is of class C , gives a similar contradiction.(ii) We assume for contradiction that G ( C ) is not connected. Then G ( C ) can be decomposed as G ( C ) = A ∪ B where A, B are nonempty sets and A ∩ ¯ B = ¯ A ∩ B = ∅ . Let P ∈ C . We claim that23ither G ( P ) ⊂ A or G ( P ) ⊂ B . Indeed if n ∈ A, n ∈ B where n , n ∈ G ( P ) then since G ( P ) isconvex n ( λ ) def = λn + (1 − λ ) n | λn + (1 − λ ) n | ∈ G ( P ) for all λ ∈ [0 , . But the sets { λ ∈ [0 ,
1] : n ( λ ) ∈ A } and { λ ∈ [0 ,
1] : n ( λ ) ∈ B } are relatively open and their unionis [0 , , C A = { P ∈ C : G ( P ) ⊂ A } and C B = { P ∈ C : G ( P ) ⊂ B } , whose disjointunion is C . Since C is connected, one of the sets C A ∩ ¯ C B , C B ∩ ¯ C A is nonempty. Suppose, for example,that P ∈ C A ∩ ¯ C B . Then there exists a sequence P j → P with G ( P j ) ⊂ B . Let n ∈ G ( P ). Then n ∈ A but also n ∈ G ( P j ) for sufficiently large j , and hence n ∈ B , a contradiction. C domains In this section we show that if Ω ⊂ R m is a bounded domain of class C then Ω has a Lipschitzboundary portion.
Lemma 7.1.
Let Ω ⊂ R m , m ≥ , be a bounded domain of class C . If the set of good directions ( pseudonormals ) at a point P ∈ ∂ Ω contains m linearly independent directions then ∂ Ω is Lipschitzin a neighbourhood of P , that is for some δ > and a suitable orthonormal coordinate system Y def = ( y (cid:48) , y m ) with origin at P Ω ∩ B ( P, δ ) = { y ∈ R m : y m > f ( y (cid:48) ) , | y | < δ } (7.1) where f : R m − → R is Lipschitz.Proof. Let { n , . . . , n m } be a set of linearly independent good directions at P and let ˜ n be an interiorpoint of the geodesically convex hull co g { n , . . . , n m } of { n , . . . , n m } , which by Lemma 2.2 is alsoan interior point of the set of good directions at P . For example we can take˜ n = (cid:80) mi =1 n i | (cid:80) mi =1 n i | . Choosing an orthonormal coordinate system Y = ( y (cid:48) , y m ) with origin at P and ˜ n = e m we have byLemma 2.2 that the representation (7.1) holds for some δ > f : R m − → R continuous. Weclaim that f is Lipschitz in a neighbourhood of 0. Then extending f outside this neighbourhood toa Lipschitz map on R m − , and choosing δ smaller if necessary, gives the result.If the claim were false there would exist sequences S j → T j → R m − such that S j (cid:54) = T j and f ( S j ) − f ( T j ) | S j − T j | → ∞ as j → ∞ . But for large enough j the unit vector N j def = ( S j , f ( S j )) − ( T j , f ( T j )) (cid:112) | S j − T j | + | f ( S j ) − f ( T j ) | if m ≥
24s not a good direction, since the line t → ( T j , f ( T j )) + tN j meets ∂ Ω twice in B ( P, δ ) at ( T j , f ( T j ))and ( S j , f ( S j )). But lim j →∞ N j = e m , contradicting that ˜ n = e m is an interior point of the set ofgood directions at P .We continue by showing that if part of the boundary of a 2D domain is the graph of a nowheredifferentiable function then there is exactly one good direction for each point on that part of theboundary. Lemma 7.2.
Let f : ( a, b ) → R be a continuous nowhere differentiable function. Let G def = { ( x, f ( x )) ∈ R : x ∈ ( a, b ) } be a subset of the boundary ∂ Ω of a bounded domain Ω ⊂ R of class C .Then at any point P ∈ G there exists a unique good direction namely ± (0 ,
1) ( the sign being independent of P ∈ G ) .Proof. We can assume without loss of generality that (0 ,
1) is a good direction for all P ∈ G , i.e.that Ω lies locally above G . Let us assume for contradiction that there exists a point P ∈ G withanother good direction n ∈ S with n (cid:54) = (0 , P in G with two good directions (0 ,
1) and n . Restricting the interval ( a, b ) if necessary we may thusassume that for any P ∈ G there are two good directions (0 ,
1) and n .Then for any m = ( m , m ) ∈ S in the interior of the geodesically convex hull of (0 ,
1) and n we haveby Lemma 7.1 that G is the graph, along m , of a Lipschitz function, say g . Then g is differentiablealmost everywhere, so that for almost all points in G there exists a tangent to G . Let G ∗ def = { P ∈ G : the tangent to G at P exists and is parallel to (0 , } . We claim now that if the function g is differentiable at the point P ∈ G\G ∗ then so is f . More preciselylet us denote P = (¯ x ∗ , ¯ y ∗ ) in the system of coordinates with axis O ¯ y = m and O ¯ x = m ⊥ = ( m , − m )respectively P = ( x ∗ , y ∗ ) in the system of coordinates with axis Oy = (0 ,
1) and Ox = (1 , g is differentiable means that g (¯ x ) − g (¯ x ∗ )¯ x − ¯ x ∗ → g (cid:48) (¯ x ∗ ) , as ¯ x → ¯ x ∗ . (7.2)Let θ be the angle between the ¯ x axis and the x axis (see Figure 3). Letting R ( θ ) def = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) we have that (cid:18) ¯ x ¯ y (cid:19) = R ( θ ) (cid:18) xy (cid:19) , which implies that¯ x − ¯ x ∗ = cos θ ( x − x ∗ ) + sin θ [ f ( x ) − f ( x ∗ )] g (¯ x ) − g (¯ x ∗ ) = − sin θ ( x − x ∗ ) + cos θ [ f ( x ) − f ( x ∗ )] (7.3)Hence f ( x ) − f ( x ∗ ) x − x ∗ = sin θ + cos θ (cid:16) g (¯ x ) − g (¯ x ∗ )¯ x − ¯ x ∗ (cid:17) cos θ − sin θ (cid:16) g (¯ x ) − g (¯ x ∗ )¯ x − ¯ x ∗ (cid:17) . (7.4)Note now that (7.3) and the continuity of f imply that ¯ x → ¯ x ∗ when x → x ∗ , which together with(7.2) and (7.4) implies f ( x ) − f ( x ∗ ) x − x ∗ → sin θ + cos θ g (cid:48) (¯ x ∗ )cos θ − sin θ g (cid:48) (¯ x ∗ ) as x → x ∗
25 x¯ x ¯ y P = ( x ∗ , y ∗ ) = (¯ x ∗ , ¯ y ∗ ) θ OFigure 3: Changing the coordinate system(note that cos θ − sin θg (cid:48) (¯ x ∗ ) (cid:54) = 0 by our assumption that P = (¯ x ∗ , g (¯ x ∗ )) (cid:54)∈ G ∗ ). Hence f isdifferentiable at P as claimed.There are now two cases: Case I:
The measure of G ∗ is strictly smaller than the measure of G . Then at all points P =( x, f ( x )) ∈ G \ G ∗ we have that f is differentiable at x . Thus we obtain a contradiction with theassumption that f is nowhere differentiable. Case II:
The measure of G ∗ is the same as that G , hence g (cid:48) is almost everywhere cot θ . Then G iscontained in a straight line in the direction (0 , , Theorem 7.1.
Let Ω be a bounded domain of class C in R m , m ≥ . If Ω has non-zero Eulercharacteristic then there exists a point P ∈ ∂ Ω in the neighbourhood of which ∂ Ω is Lipschitz.Proof. We consider the canonical field of good directions obtained in the proof of Proposition 2.1,namely G ( P ) def = Σ ki =1 α i ( P ) n P i | Σ ki =1 α i ( P ) n P i | (7.5)with n P i a good direction at P i with respect to the ball B ( P i , δ i ) and thus at each point R ∈ B ( P i , δ P i ) , i = 1 , . . . , k, and α i , i = 1 , . . . , k, a partition of unity subordinate to the covering B ( P i , δ P i ) , i = 1 , . . . , k . Then G | ∂ Ω is a continuous pseudonormal field and hence Theorem 6.1,part (i), provides that G : ∂ Ω → S m − is surjective. We claim now that there exist m linearlyindependent good directions ¯ n , . . . , ¯ n m ∈ { n P i , i = 1 , . . . , k } such that ∂ Ω and the m correspondingballs, on which they are good directions, have a non-empty intersection. Lemma 7.1 then shows that ∂ Ω is Lipschitz in a neighbourhood of any point P in their intersection.In order to prove the claim we assume, for contradiction, that at each point Q ∈ ∂ Ω the subset of { n P i ∈ S m − , i = 1 , . . . , k } that are good directions at Q is contained in a hyperplane. For each i = 1 , . . . , k , let E i = { P ∈ ∂ Ω : α i ( P ) > } . Then each ∂E i is a closed nowhere dense subset of ∂ Ω,and so by the Baire Category theorem ∪ ki =1 ∂E i is a closed nowhere dense subset of ∂ Ω. Consider26ny nonempty subset of the form A i ,...,i r = ∩ rj =1 E i j , where 1 ≤ r ≤ k and 1 ≤ i < · · · < i r ≤ k . For P ∈ A i ,...,i r we have by our assumption that the vectors n P ij , ≤ j ≤ r, lie in a hyperplane, andthus from (7.5) we have that G ( A i ,...,i r ) is contained in this hyperplane. Hence G ( ∂ Ω \ ∪ ki =1 ∂E i ) iscontained in a closed nowhere dense subset A of S m − (the union of the intersection with S m − of afinite number of hyperplanes). Since G is continuous and ∪ ki =1 ∂E i is nowhere dense, it follows that G ( ∂ Ω) ⊂ A , contradicting the surjectivity of G .We continue by studying the set of all pseudonormals in the general case of C domains with notopological restrictions imposed on ∂ Ω. Lemma 7.3.
Let Ω be a bounded domain of class C in R m , m > . For each connected component C of ∂ Ω there exists a point P ∈ C at which the set of good directions at P is not a singleton.Proof. Assume for contradiction that for any P ∈ C there exists only one good direction n ( P ) ∈ S m − .Note that if N is a good direction at some point x ∈ ∂ Ω then N is also a good direction at points z ∈ ∂ Ω sufficiently close to x . Pick P ∈ C and define E = { P ∈ C : n ( P ) = n ( P ) } . The precedingproperty implies that E is both open and closed in C . Since C is connected it follows that E = C , incontradiction to Proposition 6.2(i).The last lemma, combined with the characterization of Lipschitz regularity of the boundary inLemma 7.1 immediately imply: Theorem 7.2.
Let Ω be a bounded domain of class C in R . For each connected component C of ∂ Ω there exists a point P ∈ C in the neighbourhood of which ∂ Ω is Lipschitz. Remark 7.1.
Note that Theorem 7.2 is not in general true for unbounded domains in R of class C . Indeed, by Lemma 7.2, if f : R → R is nowhere differentiable then the domain Ω = { ( x , x ) : x > f ( x ) } has no Lipschitz boundary portions.We also have a similar result in 3D, but the proof is considerably more intricate. Theorem 7.3.
Let Ω be a bounded domain of class C in R . For each connected component of ∂ Ω there exists a point P ∈ ∂ Ω in the neighbourhood of which ∂ Ω is Lipschitz.Proof. We consider again the canonical field of good directions obtained in the proof of Proposi-tion 2.1, namely G ( P ) def = (cid:80) ki =1 α i ( P ) n P i | (cid:80) ki =1 α i ( P ) n P i | (7.6)with n P i a good direction at P i with respect to the ball B ( P i , δ i ) and thus at each point R ∈ B ( P i , δ P i ) , i = 1 , . . . , k, and α i , i = 1 , . . . , k, a partition of unity subordinate to the covering B ( P i , δ P i ) , i = 1 , . . . , k .We continue by analyzing the image of G when restricted to an arbitrary connected component S = S ε of ∂ Ω ε for the approximating Ω ε as in Theorem 5.1, with ε > G ( S ) has non-empty interior and that this allows one to infer that there exist threelinearly independent directions at some point Q ε ∈ ∂ Ω ε for all ε >
0. Then, thanks to the specialstructure of the canonical field G , it will be shown that the same can be claimed at some point Q ∈ ∂ Ω. 27e start by noting that if the connected component S of ∂ Ω ε has non-zero Euler characteristic thenTheorem 6.1 and Remark 6.1 ensure that G ( S ) = S hence G ( S ) has non-empty interior.The case when S has zero Euler characteristic is more delicate as in this situation the field G | S isnot necessarily surjective, as was shown in Proposition 6.1. The proof continues in two steps. Inthe first we show that we can assume, without loss of generality, that S = T is the standard torus T = S × S embedded in R , for which we know by Proposition 6.1 that any smooth field of gooddirections G : S → S has an image G ( S ) with non-empty interior. Then, in the second step, weshow that G ( S ) having non-empty interior implies (irrespective of the Euler characteristic of S beingzero or not) that there exist three linearly independent directions at some point Q ε ∈ ∂ Ω ε for all ε >
0, and that the same holds also at some point Q ∈ ∂ Ω. Step 1 (reduction to the standard torus and consequences):
We note that S is a smooth2-dimensional compact, connected and orientable manifold without boundary, that has zero Eulercharacteristic. Let T = { (cos θ (2 + cos ϕ ) , sin θ (2 + cos ϕ ) , sin ϕ ) : θ, ϕ ∈ [0 , π ] } (7.7)denote a standard torus embedded in R . Then T is a 2-dimensional compact, connected andorientable manifold without boundary of zero Euler characteristic and by the theorem of classificationof 2-dimensional compact manifolds [27, Chapter 9] there exists a diffeomorphism D : S → T .We use the diffeomorphism D to transport the field of good directions, in a bijective manner, from S onto T , while preserving the angle between the good direction and the normal. Lemma 7.4.
Let ν S : S → S , ν T : T → S denote the interior normals to S , respectively T .There exist smooth functions e, ˆ e : S → S , f, ˆ f : T → S such that e, ˆ e and ν S , respectively f, ˆ f and ν T are pairwise orthogonal at each point.Proof. The proof is essentially an easy consequence of the fact that S and T are parallelizable mani-folds embedded in R . We consider the torus T as in (7.7). Then ν T ( θ, ϕ ) = − (cos θ cos ϕ, sin θ cos ϕ, sin ϕ )for any θ, ϕ ∈ [0 , π ] × [0 , π ]. We take f ( θ, ϕ ) def = ( − sin θ, cos θ, , ˆ f ( θ, ϕ ) def = ( − cos θ sin ϕ, − sin θ sin ϕ, cos ϕ ) . We consider the derivative
T D : T S → T of the diffeomorphism D : S → T , acting between thetangent bundles T S and T T . Then for any point P ∈ S we have that the linear map T P D : T P S → T D ( P ) T is an invertible linear function, and as such T P D − (cid:0) f ( D ( P )) (cid:1) and T P D − (cid:0) ˆ f ( D ( P )) (cid:1) definea basis in T P S which varies smoothly in P . However this basis need not be an orthogonal basis andin order to obtain an orthogonal, smoothly varying, basis, we take: e ( P ) def = T P D − (cid:0) f ( D ( P )) (cid:1) | T P D − (cid:0) f ( D ( P )) (cid:1) | , ˆ e ( P ) = e ( P ) × ν S ( P ) . Continuing the proof of Theorem 7.3, let G : S → S be a smooth field of good directions on S suchthat G ( P ) · ν S ( P ) > P ∈ S . We define˜ G ( D ( P )) def = (cid:0) G ( P ) · ν S ( P ) (cid:1) ν T ( D ( P )) + (cid:0) G ( P ) · e ( P ) (cid:1) f ( D ( P )) + (cid:0) G ( P ) · ˆ e ( P ) (cid:1) ˆ f ( D ( P ))28e have then that ˜ G ( Q ) · ν T ( Q ) = G ( D − ( Q )) · ν S ( D − ( Q )) > Q ∈ T and thus ˜ G is asmooth field of good directions on T , the transported version to T of the field G .We claim now that G ( S ) has nonempty interior if and only if the transported version ˜ G ( T ) hasnonempty interior.To this end let us define the continuous function H : S × S → T × S by H ( P, n ) def = (cid:16) D, (cid:0) n · ν S (cid:1) ν T ( D ) + (cid:0) n · e (cid:1) f ( D ) + (cid:0) n · ˆ e (cid:1) ˆ f ( D ) (cid:17) ( P )One can check that (cid:101) H : T × S → S × S defined by (cid:101) H ( R, m ) def = (cid:16) D − , (cid:0) m · ν T (cid:1) ν T ( D − ) + (cid:0) m · f (cid:1) e ( D − ) + (cid:0) m · ˆ f )ˆ e ( D − ) (cid:17) ( R )is the continuous inverse of H and thus H is a homeomorphism. Moreover we have: H ( P, G ( P )) = ( D ( P ) , ˜ G ( D ( P ))) . (7.8)Let us assume now that G ( S ) has nonempty interior E . Then, since G is continuous, G − ( E ) isnonempty and open. Since H is a homeomorphism, it takes nonempty open sets into nonempty opensets, so that, by (7.8), H ( G − ( E ) , E )=( D ( G − ( E ) , ˜ G ( D ( G − ( E )))). Thus ˜ G ( D ( G − )( E )) ⊂ ˜ G ( T )is a nonempty open set and therefore it has nonzero measure. One can argue in a similar way, using˜ H to show that if ˜ G ( T ) has nonempty interior then so does G ( S ), thus proving our claim.Proposition 6.1 now shows that ˜ G ( T ) has nonempty interior and therefore so does G ( S ). Step (from the smooth approximating domains back to the rough one): Let Ω ε bea sequence of smooth domains approximating Ω, as given in Theorem 5.1. Let G : V → S be acanonical field of good directions, where V is an open set containing ∂ Ω (hence there exists an ε > ∂ Ω ε ⊂ V for 0 < ε < ε ). The previous step and the remark before it show that for eachconnected component S ε of ∂ Ω ε we have that the interior of G ( S ε ) is nonempty.Let us recall now the definition (7.6) of the canonical field G . Arguing as in the proof of Propo-sition 7.1 (and using the fact that G ( S ε ) has nonempty interior, instead of surjectivity of G ) wehave that there exists a point Q ε ∈ S ε so that there exist three linearly independent good direc-tions n P i ε ) , n P i ε ) , n P i ε ) such that the three corresponding balls B ( P i k ( ε ) , δ i k ( ε ) ) , k = 1 , , , havea nonempty intersection containing Q ε .Since the number of balls in the cover is finite there exist three of them, say B ( ¯ P i , ¯ δ i ) , i = 1 , , Q ε j , ε j → Q ∈ ∂ Ω, a limit point of the Q ε j , with ¯ Q ∈ ∩ i =1 B ( ¯ P i , δ i ) ⊂ ∩ i =1 B ( ¯ P i , δ i ) and such that there arethree linearly independent good directions at ¯ Q . Remark 7.2.
Despite some efforts we have been unable to decide whether Theorem 7.1 remainstrue for m ≥ Remark 7.3.
Theorems 7.1-7.3 remain valid if the hypothesis that Ω is of class C is replaced bythe weaker hypothesis that Ω is the image under a C diffeomorphism ϕ : U → R m of a boundeddomain Ω (cid:48) ⊂ R m of class C , where U is an open neighbourhood of Ω (cid:48) . Indeed, by [28, Theorem4.1] (in which the term strongly Lipschitz is used instead of our Lipschitz) if ∂ Ω (cid:48) is Lipschitz in aneighbourhood of P then ∂ Ω is Lipschitz in a neighbourhood of ϕ ( P ).29 emark 7.4. The partial regularity results of this section crucially use a rigidity specific to thedefinition of C domains. Indeed, just by enlarging a bit the class of domains studied, to topologicalmanifolds with boundaries, one can immediately provide counterexamples to partial regularity. Thiscan be seen for instance by considering the domain in polar coordinates in R : Ω def = { ( r, θ ); 0 ≤ r < f ( θ ) } with f : S → R a nowhere differentiable function, which is a topological manifold withboundary but is not a domain of class C . Acknowledgements
The research of both authors was partly supported by EPSRC grants EP/E010288/1 and by theEPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1).The research of JMB was also supported by the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 291053 andby a Royal Society Wolfson Research Merit Award. The research of AZ was partially supportedby the Basque Government through the BERC 2014-2017 program; and by the Spanish Ministry ofEconomy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323.We are especially grateful to Marc Lackenby for the suggestion of an example forming the basis ofthe second part of Proposition 6.1 as well as for other advice, to Vladimir ˇSver´ak, whose perceptivequestions after a seminar of JMB at the University of Minnesota on an earlier version of the paper ledto radical improvements, and to Rob Kirby for long discussions concerning smoothing of manifoldsthat are incorporated in Remark 5.4. We are also grateful to Moe Hirsch for his interest and variousreferences, and to Nigel Hitchin for useful comments.Finally we are grateful to anonymous referees for pointing out to us the connection with the the-ory of smoothing of manifolds, for a simplification of our original example in the second part ofProposition 6.1, and for mentioning various relevant references.
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