Bernstein Functions and Radial Limits of Prescribed Mean Curvature Surfaces
BBernstein Functions and Radial Limits ofPrescribed Mean Curvature Surfaces
Mozhgan “Nora” EntekhabiDepartment of MathematicsFlorida A & M UniversityTallahassee, FL 32307&Kirk E. LancasterWichita, Kansas 67226
Abstract
The radial limits at a point y of the boundary of the domain Ω ⊂ IR of a bounded variational solution f of Dirichlet or contact angle boundaryvalue problems for a prescribed mean curvature equation are studied withan emphasis on the effects of assumptions about the curvatures of theboundary ∂ Ω on each side of the point y . For example, at a nonconvexcorner y , we previously proved that all nontangential radial limits of f at y exist; here we provide sufficient conditions for the tangential radiallimits to exist, even when the Dirichlet data φ ∈ L ∞ ( ∂ Ω) has no one-sidedlimits at y or the contact angle γ ∈ L ∞ ( ∂ Ω : [0 , π ]) is not bounded awayfrom 0 or π. We also provide a complement to a 1976 Theorem by LeonSimon on least area surfaces.
Let Ω be a locally Lipschitz domain in IR and define N f = ∇ · T f = div (
T f ) , where f ∈ C (Ω) and T f = ∇ f √ |∇ f | . Consider the Dirichlet problem
N f = H ( · , f ( · )) in Ω (1) f = φ on ∂ Ω (2)and the contact angle problem
N f = H ( · , f ( · )) in Ω (3) T f · ν = cos γ on ∂ Ω , (4)1 a r X i v : . [ m a t h . A P ] A ug here φ : ∂ Ω → IR , γ : ∂ Ω → [0 , π ] , and H : Ω × IR → IR are prescribedfunctions, H ( x , t ) is nondecreasing in t for each x ∈ Ω (cf. [6]) and ν is theexterior unit normal to ∂ Ω . For a smooth domain, some type of boundary curvature condition (whichdepends on H ) must be satisfied in order to guarantee that a classical solutionof (1)-(2) exists for each φ ∈ C ( ∂ Ω) ; when H ≡ , this curvature condition isthat ∂ Ω must have nonnegative curvature (with respect to the interior normaldirection of Ω) at each point (e.g. [17]). However, Leon Simon ([30]) has shownthat if Γ ⊂ ∂ Ω is smooth (i.e. C ), H ≡ , φ ∈ C , ( ∂ Ω) , the curvature Λof ∂ Ω is negative on Γ and Γ is a compact subset of Γ , then the variationalsolution z = f ( x ) , x ∈ Ω , extends to Ω ∪ Γ as a H¨older continuous function withLipschitz continuous trace, even though f may not equal φ on Γ; Simon’s resultholds for least area hypersurfaces in IR n , n ≥ ∂ Ωhas a negative upper bound on Γ ⊂ ∂ Ω (see also [1, 27]).One can look at this in a different way. In the case H ≡ , the requirementthat Λ( p ) < p ∈ ∂ Ω implies that
N f = 0 has a (continuous)Bernstein function ψ at p for Ω (see Definition (1) and Definition (2)). In[8], Bernstein functions for the minimal surface equation in IR are constructedfor C ,α domains Ω ⊂ IR whose curvature Λ (with respect to − ν ) vanishesat a finite number of points and satisfies Λ ≤ ∂ Ω . Usingthese Bernstein functions, we will prove the following generalization of [30] when n = 2 . Corollary 1.
Let Ω be a domain in IR , Γ is a C ,λ open subset of ∂ Ω andthe curvature Λ (with respect to − ν ) of Γ is nonpositive and vanishes at only afinite number of points of Γ , for some λ ∈ (0 , . Suppose φ ∈ L ∞ ( ∂ Ω) , y ∈ Γ , either f is symmetric with respect to a line through y or φ is continuous at y , and f ∈ BV (Ω) minimizes J ( u ) = (cid:90) Ω (cid:112) | Du | d x + (cid:90) ∂ Ω | u − φ | ds (5) for u ∈ BV (Ω) . Then f ∈ C (Ω ∪ { y } ) . If φ ∈ C (Γ) , then f ∈ C (Ω ∪ Γ) . Example 1.
Let
Ω = { ( x, y ) ∈ IR : 1 < ( x + 1) + y < cosh (1) } and φ ( x, y ) = sin (cid:16) πx + y (cid:17) for ( x, y ) (cid:54) = (0 , (see Figure 1 for a rough illustration ofthe graph of φ ). Set O = (0 , . Let f ∈ C (Ω) minimize (5) over BV (Ω) (i.e. f is the variational solution of (1)-(2) with H ≡ ). Then Corollary 1 (with y = O ) implies f ∈ C (cid:0) Ω (cid:1) , even though φ has no limit at O . Variational solutions of (3)-(4) will exist in some sense (e.g. § φ ∈ L ∞ (Ω) but need not be continuous at each point of the boundary. Manyauthors (e.g. [7, 9, 13, 22, 28, 30, 31]) have investigated the boundary behaviorat corners of variational solutions of (1)-(2) and a number of authors (e.g. [4, 10,12, 11, 15, 18, 23, 24, 25, 29]) have done so for variational solutions of (3)-(4).2 x yz Figure 1: Ω and part of the graph of φ We shall investigate the existence and behavior of the radial limits of non-parametric prescribed mean curvature surfaces at corners of the domain, includ-ing “smooth corners” (e.g. Corollary 1). In particular, we shall use Bernsteinfunctions to investigate the behavior of variational solutions of (1)-(2) or (3)-(4)at points of ∂ Ω . Let Q be the operator on C (Ω) given by Qf ( x ) def = N f ( x ) − H ( x , f ( x )) , x ∈ Ω , (6)where H : Ω × IR → IR is prescribed and H ( x , t ) is weakly increasing in t foreach x ∈ Ω . Let ν be the exterior unit normal to ∂ Ω , defined almost everywhereon ∂ Ω . We assume that for almost every y ∈ ∂ Ω , there is a continuous extensionˆ ν of ν to a neighborhood of y . For each point y ∈ ∂ Ω , polar coordinates relative to y are denoted by r y and θ y . We shall assume that for each y ∈ ∂ Ω , there exists a δ > ∂ Ω ∩ B δ ( y ) \ { y } consists of two (open) arcs ∂ y Ω and ∂ y Ω , whose tangent raysapproach the rays L y : θ y = α ( y ) and L y : θ y = β ( y ) respectively, as the point y is approached, with α ( y ) < β ( y ) < α ( y ) + 2 π, in the sense that the tangentcone to Ω at y is { α ( y ) ≤ θ y ≤ β ( y ) , ≤ r y < ∞} . (In particular, { α ( y ) <θ y < β ( y ) , < r y < (cid:15) ( θ y ) } is a subset of Ω for some (cid:15) ∈ C (( α ( y ) , β ( y ))) ,(cid:15) ( · ) > , and { β ( y ) < θ y < α ( y ) + 2 π, < r y < (cid:15) ( θ y ) } ∩ Ω = ∅ for some (cid:15) ∈ C (( β ( y ) , α ( y ) + 2 π )) , (cid:15) ( · ) > . ) When β ( y ) − α ( y ) < π, ∂ Ω is said tohave a convex corner at y and when β ( y ) − α ( y ) > π, ∂ Ω is said to havea nonconvex corner at y . The radial limit of f at y = ( y , y ) ∈ ∂ Ω in thedirection ω ( θ ) = (cos θ, sin θ ) , θ ∈ ( α ( y ) , β ( y )) , is Rf ( θ, y ) def = lim r ↓ f ( y + r cos( θ ) , y + r sin( θ )) . (7) Rf ( α ( y ) , y ) will be defined as the limit at y of the trace of f restricted to ∂ y Ωand Rf ( β ( y ) , y ) as the limit at y of the trace of f restricted to ∂ y Ω . Noticethat if f is a generalized (e.g. variational or Perron) solution of (1)-(2), f neednot equal φ on portions of ∂ Ω and the tangential radial limits Rf ( α ( y ) , y ) and Rf ( β ( y ) , y ) may, for example, differ from φ ( y ) when φ is continuous at y . efinition 1. Given a domain Ω as above, a upper Bernstein pair ( U + , ψ + ) for a curve Γ ⊂ ∂ Ω and a function H is a C domain U + and a function ψ + ∈ C ( U + ) ∩ C (cid:16) U + (cid:17) such that Γ ⊂ ∂U + , ν is the exterior unit normal to ∂U + at each point of Γ (i.e. U + and Ω lie on the same side of Γ; see Figure2), Qψ + ≤ in U + , and T ψ + · ν = 1 almost everywhere on an open subset of ∂U + containing Γ in the same sense as in [3]; that is, for almost every y ∈ Γ , lim U + (cid:51) x → y ∇ ψ + ( x ) · ˆ ν ( x ) (cid:112) |∇ ψ + ( x ) | = 1 . (8) Definition 2.
Given a domain Ω as above, a lower Bernstein pair ( U − , ψ − ) for a curve Γ ⊂ ∂ Ω and a function H is a C domain U − and a function ψ − ∈ C ( U − ) ∩ C (cid:16) U − (cid:17) such that Γ ⊂ ∂U − , ν is the exterior unit normal to ∂U − at each point of Γ (i.e. U − and Ω lie on the same side of Γ ), Qψ − ≥ in U − , and T ψ − · ν = − almost everywhere on an open subset of ∂U − containing Γ in the same sense as in [3]. In the following theorem, we consider a domain with a nonconvex corner y andprove that the radial limits of f at y exist and behave as in [7, 20, 21, 25].In [20], Ω was required to be locally convex at points of ∂ y Ω and ∂ y Ω and, in[7, 21], the curvatures of ∂ y Ω and ∂ y Ω were required to have an appropriatepositive lower bound when these curves were smooth. In [9], no such curvaturerequirement was imposed but only nontangential radial limits were shown toexist. This theorem strengthens Theorem 1 of [9] when the curvatures of ∂ y Ωand ∂ y Ω imply Bernstein functions exist (see § Theorem 1.
Let f ∈ C (Ω) ∩ L ∞ (Ω) satisfy Qf = 0 in Ω and let H ∗ ∈ L ∞ (IR ) satisfy H ∗ ( x ) = H ( x , f ( x )) for x ∈ Ω . Suppose that y ∈ ∂ Ω , β ( y ) − α ( y ) >π, and there exist δ > and upper and lower Bernstein pairs (cid:0) U ± , ψ ± (cid:1) and (cid:0) U ± , ψ ± (cid:1) for (Γ , H ∗ ) and (Γ , H ∗ ) respectively, where Γ = B δ ( y ) ∩ ∂ y Ω and Γ = B δ ( y ) ∩ ∂ y Ω . Then the limits lim Γ (cid:51) x → y f ( x ) = z and lim Γ (cid:51) x → y f ( x ) = z (9) exist, the radial limit Rf ( θ, y ) exists for each θ ∈ [ α ( y ) , β ( y )] , Rf ( α ( y ) , y ) = z ,Rf ( β ( y ) , y ) = z , and Rf ( · , y ) is a continuous function on [ α ( y ) , β ( y )] whichbehaves in one of the following ways:(i) Rf ( · , y ) = z is a constant function and f is continuous at y . (ii) There exist α and α so that α ( y ) ≤ α < α ≤ β ( y ) , Rf = z on [ α ( y ) , α ] , Rf = z on [ α , β ( y )] and Rf is strictly increasing (if z < z ) orstrictly decreasing (if z > z ) on [ α , α ] . (iii) There exist α , α L , α R , α so that α ( y ) ≤ α < α L < α R < α ≤ β ( y ) ,α R = α L + π , and Rf is constant on [ α ( y ) , α ] , [ α L , α R ] , and [ α , β ( y )] andeither strictly increasing on [ α , α L ] and strictly decreasing on [ α R , α ] or strictlydecreasing on [ α , α L ] and strictly increasing on [ α R , α ] . U ± (middle) U ± (right)In the second theorem, we consider a domain with a smooth corner y (i.e. β ( y ) − α ( y ) = π ) and show that the radial limits of f at y exist and behave asexpected. Corollary 1 follows from this theorem and an additional argument. Theorem 2.
Let f ∈ C (Ω) ∩ L ∞ (Ω) satisfy Qf = 0 in Ω and let H ∗ ∈ L ∞ (IR ) satisfy H ∗ ( x ) = H ( x , f ( x )) for x ∈ Ω . Suppose that y ∈ ∂ Ω , β ( y ) − α ( y ) = π, and there exist δ > and upper and lower Bernstein pairs ( U ± , ψ ± ) for (Γ , H ∗ ) , where Γ = B δ ( y ) ∩ ∂ Ω . Then the limits lim Γ (cid:51) x → y f ( x ) = z and lim Γ (cid:51) x → y f ( x ) = z exist, Rf ( θ, y ) exists for each θ ∈ [ α ( y ) , β ( y )] , Rf ( · , y ) ∈ C ([ α ( y ) , β ( y )]) , Rf ( α ( y ) , y ) = z , Rf ( β ( y ) , y ) = z , and Rf ( · , y ) behaves as in (i) or (ii) ofTheorem 1. In the third theorem, we consider a domain with a convex corner y and provethat the radial limits of f at y exist and behave as expected. This theoremstrengthens Theorem 2 of [9]. Theorem 3.
Let f ∈ C (Ω) ∩ L ∞ (Ω) satisfy Qf = 0 in Ω and let H ∗ ∈ L ∞ (IR ) satisfy H ∗ ( x ) = H ( x , f ( x )) for x ∈ Ω . Suppose that y ∈ ∂ Ω and thereexist δ > and upper and lower Bernstein pairs (cid:0) U ± , ψ ± (cid:1) for (Γ , H ∗ ) , where Γ = B δ ( y ) ∩ ∂ y Ω . Suppose further that z = lim Γ (cid:51) x → y f ( x ) exists, where Γ = B δ ( y ) ∩ ∂ y Ω . Then lim Γ (cid:51) x → y f ( x ) = z exists, Rf ( θ, y ) exists for each θ ∈ [ α ( y ) , β ( y )] , Rf ( · , y ) ∈ C ([ α ( y ) , β ( y )]) , Rf ( α ( y ) , y ) = z , Rf ( β ( y ) , y ) = z , and Rf ( · , y ) behaves as in (i), (ii) or (iii)of Theorem 1. In the fourth theorem, we generalize Theorem 2 of [10].
Theorem 4.
Let f ∈ C (Ω) satisfy Qf = 0 in Ω and let H ∗ ∈ L ∞ (IR ) satisfy H ∗ ( x ) = H ( x , f ( x )) for x ∈ Ω . Suppose that y ∈ ∂ Ω , β ( y ) − α ( y ) < π, and thereexist δ > and upper and lower Bernstein pairs (cid:0) U ± , ψ ± (cid:1) for (Γ , H ∗ ) , where Γ = B δ ( y ) ∩ ∂ y Ω . Suppose further that f ∈ C (cid:0) Ω ∪ ∂ y Ω ∪ ∂ y Ω (cid:1) , T f ( x ) · ν ( x ) =cos( γ ( x )) for x ∈ ∂ y Ω , and lim ∂ y Ω (cid:51) x →O γ ( x ) = γ . uppose also that there exist λ , λ ∈ [0 , π ] with < λ − λ < β ( y ) − α ( y )) such that λ ≤ γ ( x ) ≤ λ for x ∈ ∂ y Ω and π − α − λ < γ < π + 2 α − λ . Then the conclusions of Theorem 3 hold.
In the fifth theorem, we generalize Theorem 1 of [25] at the cost of extra bound-ary assumptions; Theorem 1 of [5] also generalizes the Lancaster-Siegel theorembut only obtains nontangential radial limits while here the existence of all radiallimits is established while not requiring the contact angle to be bounded awayfrom zero or π. Theorem 5.
Let f ∈ C (Ω) ∩ L ∞ (Ω) satisfy (1) and (2) almost everywhere on ∂ Ω . Let H ∗ ∈ L ∞ (IR ) satisfy H ∗ ( x ) = H ( x , f ( x )) for x ∈ Ω . Let y ∈ ∂ Ω andsuppose there exist δ > and upper and lower Bernstein pairs (cid:0) U ± , ψ ± (cid:1) and (cid:0) U ± , ψ ± (cid:1) for (Γ , H ∗ ) and (Γ , H ∗ ) respectively, where Γ = B δ ( y ) ∩ ∂ y Ω and Γ = B δ ( y ) ∩ ∂ y Ω . If β ( y ) − α ( y ) ≤ π, assume there exist constants γ ± , γ ± , ≤ γ ± ≤ γ ± ≤ π, satisfying π − ( β ( y ) − α ( y )) < γ + + γ − ≤ γ + + γ − < π + β ( y ) − α ( y ) such that γ ± ≤ γ ± ( s ) ≤ γ ± for all s ∈ (0 , s ) , for some s > . Then theconclusions of Theorem 1 hold.
Example 2.
Let
Ω = { ( r cos θ, r sin θ ) : 0 < r < , − α < θ < α } with α > π . (see Figure 3(a)). Let φ ( x, y ) = sin (cid:16) πx + y (cid:17) for ( x, y ) (cid:54) = (0 , (see Figure 3(b)for a rough illustration of the graph of φ. ). Let f satisfy (1) in Ω with H ≡ and f = φ on ∂ Ω \ {O} . Then [9] shows that Rf ( θ ) exists when | θ | < α. Since Ω is locally convex at each point of ∂ Ω \ {O} , we see that f ∈ C (Ω \ {O} ) and f = φ on ∂ Ω \ {O} . Since φ has no limit at O , Rf ( ± α ) do not exist; however lim θ ↓− α Rf ( θ ) and lim θ ↑ α Rf ( θ ) both exist (e.g. from the behavior of Rf ( θ ) established in [9, 21, 25]) and, by symmetry, are equal.Suppose we replace Ω with a slightly larger (and still symmetric) domain Ω , Ω ⊂ Ω ⊂ B ( O ) , such that ∂ Ω ∩ B ( O ) has negative curvature (with respect tothe exterior normal to Ω ) and ∂ Ω and ∂ Ω are tangent at O (see Figure 3 (c)for an illustration of Ω ). Let f ∈ C (Ω) minimize (5) over BV (Ω ) , so that f is the variational solution of (1)-(2) in Ω with H ≡ . Then Theorem 1 implies Rf ( θ ) exists when | θ | ≤ α and symmetry implies Rf ( − α ) = Rf ( α ) . Onewonders, for example, about the relationship between Rf ( α ) and lim θ ↑ α Rf ( θ ) . Remark 1.
The proofs of these Theorems are similar to those in [9] (and [5]).One difference is that the results in [5, 9] were only concerned with nontangentialradial limits at one point, O , and so restricting the solution (“ f ”) to a subdomainwhich is tangent to the domain Ω at O and therefore assuming f ∈ C (Ω \ {O} )6 a) (b) (c) Figure 3: (a) Ω (b) The graph of φ over ∂ Ω (c) Ω caused no difficulties. Since we wish to show that tangential radial limits alsoexist and describe the behavior of f on ∂ Ω , we cannot make such simplifyingassumptions and so we have to modify the proofs in [5, 9]. Proof of Theorem 1:
We may assume Ω is a bounded domain. Set S = { ( x , f ( x )) : x ∈ Ω } . From the calculation on page 170 of [25], we see that thearea of S is finite; let M denote this area. For δ ∈ (0 , , set p ( δ ) = (cid:115) πM ln (cid:0) δ (cid:1) . Let E = { ( u, v ) : u + v < } . As in [7, 25], there is a parametric descriptionof the surface S ,Y ( u, v ) = ( a ( u, v ) , b ( u, v ) , c ( u, v )) ∈ C ( E : IR ) , (10)which has the following properties:( a ) Y is a diffeomorphism of E onto S .( a ) Set G ( u, v ) = ( a ( u, v ) , b ( u, v )) , ( u, v ) ∈ E. Then G ∈ C ( E : IR ) . ( a ) Set σ ( y ) = G − ( ∂ Ω \ { y } ) ; then σ ( y ) is a connected arc of ∂E and Y maps σ ( y ) onto ∂ Ω \ { y } . We may assume the endpoints of σ ( y ) are o ( y ) and o ( y ) . (Note that o ( y ) and o ( y ) are not assumed to be distinct.)( a ) Y is conformal on E : Y u · Y v = 0 , Y u · Y u = Y v · Y v on E .( a ) (cid:52) Y := Y uu + Y vv = H ∗ ( Y ) Y u × Y v on E .Notice that for each C ∈ IR , Q ( ψ + j + C ) = Q ( ψ + j ) ≤ ∩ U + j and Q ( ψ − j + C ) = Q ( ψ − j ) ≥ ∩ U − j , j = 1 , , and so N ( ψ + j + C )( x ) ≤ H ( x , f ( x )) = N f ( x ) for x ∈ Ω ∩ U + j , j = 1 , N ( ψ − j + C )( x ) ≥ H ( x , f ( x )) = N f ( x ) for x ∈ Ω ∩ U − j , j = 1 , . (12)Let q denote a modulus of continuity for ψ ± and ψ ± . Let ζ ( y ) = ∂E \ σ ( y ); then G ( ζ ( y )) = { y } and o ( y ) and o ( y ) are theendpoints of ζ ( y ) . There exists a δ > w ∈ E and dist ( w , ζ ( y )) ≤ δ , then G ( w ) ∈ (cid:0) U +1 ∪ U +2 (cid:1) ∩ (cid:0) U − ∪ U − (cid:1) . Now
T ψ ± j · ν = ± ± j of ∂U ± j which contains Γ j ; thereexists a δ > (cid:0) ∂U ± j \ Υ ± j (cid:1) ∩ { x ∈ IR : | x − y | ≤ p ( δ ) } = ∅ . Set δ ∗ = min { δ , δ } and V ∗ = { w ∈ E : dist( w , ζ ( y )) < δ ∗ } . Notice if w ∈ V ∗ , then G ( w ) ∈ U +1 ∪ U +2 and G ( w ) ∈ U − ∪ U − . Claim: Y is uniformly continuous on V ∗ and so extends to a continuous func-tion on V ∗ . Pf:
Let (cid:15) > . Choose δ ∈ (cid:16) , ( δ ∗ ) (cid:17) such that p ( δ )+2 q ( p ( δ )) < (cid:15). Let w , w ∈ V ∗ with | w − w | < δ ; then G ( w ) , G ( w ) ∈ (cid:0) U +1 ∪ U +2 (cid:1) ∩ (cid:0) U − ∪ U − (cid:1) . Set C r ( w ) = { u ∈ E : | u − w | = r } and B r ( w ) = { u ∈ E : | u − w | < r } . Fromthe Courant-Lebesgue Lemma (e.g. Lemma 3 . ρ = ρ ( δ ) ∈ (cid:16) δ, √ δ (cid:17) such that the arclength l ρ ( w ) of Y ( C ρ ( w )) is less than p ( δ ) . Notice that w ∈ B ρ ( δ ) ( w ) . Let k ( δ )( w ) = inf u ∈ C ρ ( δ ) ( w ) c ( u ) = inf x ∈ G ( C ρ ( δ ) ( w )) f ( x )and m ( δ )( w ) = sup u ∈ C ρ ( δ ) ( w ) c ( u ) = sup x ∈ G ( C ρ ( δ ) ( w )) f ( x );then m ( δ )( w ) − k ( δ )( w ) ≤ l ρ < p ( δ ) . Fix x ∈ C (cid:48) ρ ( δ ) ( w ) . For j = 1 , , set C + j = inf x ∈ U + j ∩ C (cid:48) ρ ( δ ) ( w ) ψ + j ( x ) and C − j = sup x ∈ U − j ∩ C (cid:48) ρ ( δ ) ( w ) ψ − j ( x ) . Then ψ + j − C + j ≥ U + j ∩ C (cid:48) ρ ( δ ) ( w ) and ψ − j − C − j ≤ U − j ∩ C (cid:48) ρ ( δ ) ( w ) . Therefore, for j, l ∈ { , } and x ∈ U + j ∩ U − l ∩ C (cid:48) ρ ( δ ) ( w ) , we have k ( δ )( w ) + (cid:0) ψ − l ( x ) − C − l (cid:1) ≤ f ( x ) ≤ m ( δ )( w ) + (cid:0) ψ + j ( x ) − C + j (cid:1) . For j = 1 , , set b + j ( x ) = m ( δ )( w ) + (cid:0) ψ + j ( x ) − C + j (cid:1) for x ∈ U + j ∩ G (cid:0) B ρ ( δ ) ( w ) (cid:1) and b − j ( x ) = k ( δ )( w ) + (cid:0) ψ − j ( x ) − C − j (cid:1) for x ∈ U − j ∩ G (cid:0) B ρ ( δ ) ( w ) (cid:1) . Now ρ ( δ ) < √ δ < δ ∗ ≤ δ ; notice that if w ∈ B ρ ( δ ) ( w ) , then | w − w | < δ and | G ( w ) − y | < p ( δ ) and thus if x ∈ G (cid:0) B ρ ( δ ) ( w ) (cid:1) ∩ ∂U ± j , then x ∈ Υ ± j . b − l ≤ f on U − l ∩ C (cid:48) ρ ( δ ) ( w ) and f ≤ b + j on U + j ∩ C (cid:48) ρ ( δ ) ( w ) for j, l = 1 , , and the general comparison principle (Theorem5.1, [12]), we have (see Figure 4) b − l ≤ f on U − l ∩ G (cid:0) B ρ ( δ ) ( w ) (cid:1) for l = 1 , f ≤ b + j on U + j ∩ G (cid:0) B ρ ( δ ) ( w ) (cid:1) for j = 1 , . (14)Since the diameter of G (cid:0) B ρ ( δ ) ( w ) (cid:1) ≤ p ( δ ) , we have (cid:12)(cid:12) ψ ± j ( x ) − C ± j (cid:12)(cid:12) ≤ q ( p ( δ ))Figure 4: General comparison principle applied on U ± (left) and U ± (right)for x ∈ U ± j ∩ G (cid:0) B ρ ( δ ) ( w ) (cid:1) . Thus, whenever x , x ∈ G (cid:0) B ρ ( δ ) ( w ) (cid:1) , at leastone of the cases (a) x , x ∈ U +1 ∩ U − , (b) x , x ∈ U +2 ∩ U − , (c) x ∈ U +1 and x ∈ U − or (d) x ∈ U +2 and x ∈ U − holds. Since c ( w ) = f ( G ( w )) ,G ( w ) ∈ U + i ∩ U − j for some i = 1 , j = 1 , , and G ( w ) ∈ U + l ∩ U − n forsome l = 1 , n = 1 , , we have b − j ( G ( w )) − b + l ( G ( w )) ≤ c ( w ) − c ( w ) ≤ b + i ( G ( w )) − b − n ( G ( w ))or − (cid:2) m ( δ )( w ) − k ( δ )( w ) + (cid:0) ψ + l ( G ( w )) − C + l (cid:1) − (cid:0) ψ − j ( G ( w )) + C − j (cid:1)(cid:3) ≤ c ( w ) − c ( w ) ≤ (cid:2) m ( δ )( w ) − k ( δ )( w ) + (cid:0) ψ + i ( G ( w )) − C + i (cid:1) − (cid:0) ψ − n ( G ( w )) + C − n (cid:1)(cid:3) . Since | ψ ± j ( G ( w )) − C ± j | ≤ q ( p ( δ )) for w ∈ B ρ ( δ ) ( w ) ∩ U ± j , we have | c ( w ) − c ( w ) | ≤ p ( δ ) + 2 q ( p ( δ )) < (cid:15). Thus c is uniformly continuous on V ∗ and, since G ∈ C ( E : IR ) , we see that Y is uniformly continuous on V ∗ . Therefore Y extends to a continuous function,still denote Y, on V ∗ . Notice that lim Γ (cid:51) x → y f ( x ) = lim ∂E (cid:51) w → o ( y ) c ( w ) = c ( o ( y ))9nd lim Γ (cid:51) x → y f ( x ) = lim ∂E (cid:51) w → o ( y ) c ( w ) = c ( o ( y ))and so, with z = c ( o ( y )) and z = c ( o ( y )) , we see that (9) holds.Now we need to consider two cases:( A ) o ( y ) = o ( y ) . ( B ) o ( y ) (cid:54) = o ( y ) . These correspond to Cases 5 and 3 respectively in Step 1 of the proof of Theo-rem 1 of [25].
Case (A):
Suppose o ( y ) = o ( y ); set o = o ( y ) = o ( y ) . Then f extends toa function in C (Ω ∪ { y } ) and case (i) of Theorem 1 holds. Pf:
Notice that G is a bijection of E ∪ { o } and Ω ∪ { y } . Thus we may define f = c ◦ G − , so f ( G ( w )) = c ( w ) for w ∈ E ∪ { o } ; this extends f to a functiondefined on Ω ∪ { y } . Let { δ i } be a decreasing sequence of positive numbersconverging to zero and consider the sequence of open sets { G ( B ρ ( i ) ( o )) } in Ω , where ρ ( i ) = ρ ( δ i ( o )) . Now y / ∈ G ( C ρ ( i ) ( o )) and so there exist σ i > P ( i ) = { x ∈ Ω : | x − y | < σ i } ⊂ G ( B ρ ( i ) ( o ))for each i ∈ IN . Thus if x ∈ P ( i ) , we have | f ( x ) − f ( y ) | < p ( δ i ) + 2 q ( p ( δ i )) . Thecontinuity of f at y follows from this. Case (B):
Suppose o ( y ) (cid:54) = o ( y ) . Then one of case (ii) or (iii) of Theorem 1holds.
Pf:
As at the end of Step 1 of the proof of Theorem 1 of [25], we define X : B → IR by X = Y ◦ g and K : B → IR by K = G ◦ g, where B = { ( u, v ) ∈ IR : u + v < , v > } and g : B → E is either a conformal or an indirectlyconformal (or anticonformal) map from B onto E such that g (1 ,
0) = o ( y ) ,g ( − ,
0) = o ( y ) and g ( u, ∈ o ( y ) o ( y ) for each u ∈ [ − , , where ab denotes the (appropriate) choice of arc in ∂E with a and b as endpoints.Notice that K ( u,
0) = y for u ∈ [ − , . Set x = a ◦ g, y = b ◦ g and z = c ◦ g, so that X ( u, v ) = ( x ( u, v ) , y ( u, v ) , z ( u, v )) for ( u, v ) ∈ B. Now, from Step 2 ofthe proof of Theorem 1 of [25], X ∈ C (cid:0) B : IR (cid:1) ∩ C ,ι (cid:0) B ∪ { ( u,
0) : − < u < } : IR (cid:1) for some ι ∈ (0 ,
1) and X ( u,
0) = ( y , z ( u, − , . Define Θ( u ) = arg ( x v ( u,
0) + iy v ( u, . Fromequation (12) of [25], we see that α = lim u ↓− Θ( u ) and α = lim u ↑ Θ( u );here α < α . As in Steps 2-5 of the proof of Theorem 1 of [25], we see that Rf ( θ ) exists when θ ∈ ( α , α ) ,G − ( L ( α )) ∩ ∂E = { o ( y ) } (& K − ( L ( α )) ∩ ∂B = { (1 , } ) when α < β ( y )10 − ( L ( α )) ∩ ∂E = { o ( y ) } (& K − ( L ( α )) ∩ ∂B = { ( − , } ) when α > α ( y )where L ( θ ) = { y +( r cos( θ ) , r sin( θ )) ∈ Ω : 0 < r < δ ∗ } , and one of the followingcases holds:(a) Rf is strictly increasing or strictly decreasing on ( α , α ).(b) There exist α L , α R so that α < α L < α R < α , α R = α L + π , and Rf isconstant on [ α L , α R ] and either increasing on ( α , α L ] and decreasing on [ α R , α )or decreasing on ( α , α L ] and increasing on [ α R , α ).We may argue as in Case A to see that f is uniformly continuous onΩ + = { y + ( r cos( θ ) , r sin( θ )) ∈ Ω : 0 < r < δ, α ≤ θ < β ( y ) + (cid:15) } and f is uniformly continuous onΩ − = { y + ( r cos( θ ) , r sin( θ )) ∈ Ω : 0 < r < δ, α ( y ) − (cid:15) < θ ≤ α } for some small (cid:15) > δ > , since G is a bijection of E ∪ { o ( y ) } and Ω ∪ { y } and a bijection of E ∪ { o ( y ) } and Ω ∪ { y } . (Also see [5, 10].) Theorem 1 thenfollows, as in [9], from Steps 2-5 of the proof of Theorem 1 of [25] (replacingStep 3 with [6]). Proof of Theorem 2:
The proof of this theorem is essentially the same asthat of Theorem 1.
Proof of Corollary 1:
From pp.1064-5 in [8], we see that there exist upperand lower Bernstein pairs ( U ± , ψ ± ) for (Γ , H ∗ ) . From Theorem 2, we see thatthe radial limits Rf ( θ, y ) exist for each θ ∈ [ α ( y ) , β ( y )] . (Since β ( y ) − α ( y ) = π, case (iii) of Theorem 1 cannot occur.) Set z = Rf ( α ( y ) , y ) , z = Rf ( β ( y ) , y )and z = φ ( y ) . If z = z , then case (i) of Theorem 1 holds. (If f is symmetricwith respect to a line through y , then z = z and we are done.)Suppose otherwise that z (cid:54) = z ; we may assume that z < z and z < z . Then there exist α , α ∈ [ α ( y ) , β ( y )] with α < α such that Rf ( θ, y ) is constant(= z ) for α ( y ) ≤ θ ≤ α strictly increasing for α ≤ θ ≤ α constant(= z ) for α ≤ θ ≤ β ( y ) . From Theorem 2, we see that Rf ( θ, y ) exists for each y ∈ Γ and θ ∈ [ α ( y ) , β ( y )]and f is continuous on Ω ∪ Γ \ Υ for some countable subset Υ of Γ . Let z ∈ ( z , min { z , z } ) and θ ∈ ( α , α ) satisfy Rf ( θ , y ) = z . Let C ⊂ Ω be the z − level curve of f which has y and a point y ∈ ∂ Ω \ { y } as endpoints. Let y ∈ ∂ y Ω ∩ Γ \ Υ and y ∈ C such that the (open) line segment L joining y and y is entirely contained in Ω . Let M = inf L f, Π be the plane containing( y , z ) and L × { M } , and h be the affine function on IR whose graph is Π . LetΩ be the component of Ω \ ( C ∪ L ) whose closure contains B δ ( y ) ∩ ∂ y Ω forsome δ > . Then there is a curve C ⊂ Ω on which f = h whose endpoints are y and y , for some y ∈ ∂ y Ω between y and y , such that h > f in Ω , whereΩ ⊂ Ω is the open set bounded by C and the portion of ∂ y Ω between y and11 . Notice that h < f in L ∪ C . (In Figure 5, on the left, { ( x , h ( x )) : x ∈ C } isin red, L is in dark blue, C is in yellow, and the light blue region is a portionof ∂ y Ω × IR , and, on the right, Ω is in light green and ∂ y Ω is in magenta.) Nowlet g ∈ C (Ω) be defined by g = f on Ω \ Ω and g = h on Ω and observe that J ( g ) < J ( f ) , which contradicts the fact that f minimizes J. Thus it must bethe case that z = z , case (i) of Theorem 1 holds and f is continuous at y . Figure 5: Side View of Π ∩ (Ω × IR) (left) and Ω (right) Remark 2.
Corollary 1 can be generalized to minimizers of J ( u ) = (cid:90) Ω (cid:112) | Du | d x + (cid:90) Ω (cid:32)(cid:90) u ( x ) c H ( x , t ) dt (cid:33) d x + (cid:90) ∂ Ω | u − φ | ds for u ∈ BV (Ω) and the conclusion remains the same; here c is a reference height(e.g. c = 0 ). In the proof of Corollary 1, the only change is a replacement ofthe plane Π with an appropriate surface (e.g. a portion of a sphere) over asubdomain like Ω such that the test function g satisfies J ( g ) < J ( f ) . Proof of Example 1:
By Corollary 1, f is continuous on Ω ∪ { (0 , } . Clearly f is continuous at ( x, y ) when ( x + 1) + y = cosh (1) . By [30], f is continuousat ( x, y ) when ( x + 1) + y = 1 and ( x, y ) (cid:54) = (0 , . The parametrization(10) of the graph of f (restricted to Ω \ { ( x,
0) : x < } ) satisfies Y ∈ C ( E ) . Notice that ζ ((0 , { o } (since β ((0 , − α ((0 , π and z = z ) for some o ∈ ∂E. Suppose G in ( a ) is not one-to-one. Then there exists a nondegeneratearc ζ ⊂ ∂E such that G ( ζ ) = { y } for some y ∈ ∂ Ω and therefore f is notcontinuous at y , which is a contradiction. Thus f = g ◦ G − and so f ∈ C (cid:0) Ω (cid:1) . (The continuity of G − follows, for example, from Lemma 3 . Proof of Theorem 3:
The proof of Theorem 2 of [9] uses unduloids as Bern-stein functions (i.e. comparison surfaces) on subdomains of Ω (see Figure 7 of[9]). The proof of Theorem 3 is essentially the same, using the Bernstein pairs( U ± , ψ ± ) rather than unduloids, staying on ∂ y Ω rather than on an arc of a circleinside Ω , and arguing as in the proof of Theorem 1. Proof of Theorem 4:
The proof of Theorem 2 of [10] uses portions of tori asBernstein functions (i.e. comparison surfaces) on subdomains of Ω (see Figure7 of [10]). The proof of Theorem 4 is essentially the same, using the Bernsteinpairs ( U ± , ψ ± ) rather than tori, staying on ∂ y Ω rather than on an arc of a circleinside Ω , and arguing as in the proof of Theorem 1.12 roof of Theorem 5: The proof of Theorem 1 of [5] uses Theorem 2 of [10];the proof of Theorem 5 is essentially the same, using Theorem 4 in place ofTheorem 2 of [10] and arguing as in the proof of Theorem 1.
The value of Theorems 1 - 5 is dependent on the existence of Bernstein functions.The results of [8] provide Bernstein pairs for minimal surfaces.
Proposition 1.
Let a < b, λ ∈ (0 , , ψ ∈ C ,λ ([ a, b ]) and Γ = { ( x, ψ ( x )) ∈ IR : x ∈ [ a, b ] } such that ψ (cid:48) ( x ) < for x ∈ [ a, b ] , ψ (cid:48)(cid:48) ( x ) < for x ∈ [ a, b ] \ J, there exist C > and (cid:15) > such that if ¯ x ∈ J and | x − ¯ x | < (cid:15) , then ψ (cid:48)(cid:48) ( x ) ≤ − C | x − ¯ x | λ , and tψ ( x ) + (1 − t ) ψ ( x ) < ψ ( tx + (1 − t ) x ) foreach t ∈ (0 , and x , x ∈ [ a, b ] with x (cid:54) = x , where J is a finite subset of ( a, b ) . Then there exists an open set U ⊂ IR with Γ ⊂ ∂U and a function h ∈ C ( U ) ∩ C (cid:0) U (cid:1) such that ∂U is a closed, C ,λ curve, Γ lies below U in IR (i.e. the exterior unit normal ν = ( ν ( x ) , ν ( x )) to ∂U satisfies ν ( x ) < for a ≤ x ≤ b ), N h = 0 in U and (8) holds for each y ∈ Γ , where ˆ ν is a continuousextension of ν to a neighborhood of Γ . Proof:
We may assume that a, b > . There exists c > b and k ∈ C ,λ ([ − c, c ])with k ( − x ) = k ( x ) for x ∈ [0 , c ] such that k ( x ) = − ψ ( x ) for x ∈ [ a, b ] , k (cid:48)(cid:48) ( x ) > x ∈ [ − c, c ] \ J, where J is a finite set, k (cid:48)(cid:48) (0) > , and the set K = { ( x, k ( x )) ∈ IR : x ∈ [ − c, c ] } is strictly concave (i.e. tk ( x ) + (1 − t ) k ( x ) > k ( tx + (1 − t ) x ) for each t ∈ (0 ,
1) and x , x ∈ [ − c, c ] with x (cid:54) = x ). From [8] (pp.1063-5), we canconstruct a domain Ω( K, l ) such that K ⊂ ∂ Ω( K, l ) and Ω(
K, l ) lies below K (i.e. the outward unit normal to Ω( K, l ) at ( x, k ( x )) is ν ( x ) = ( − k (cid:48) ( x ) , ) √ k (cid:48) ( x )) ; seeFigure 4 of [8]) and a function F + ∈ C (Ω( K, l )) ∩ C (cid:16) Ω( K, l ) (cid:17) such that µ ( x ) def = ( ∇ F + ( x ) , − (cid:112) |∇ F + ( x ) | , x ∈ Ω( K, l ) , extends continuously to a function on Ω( K, l ) ∪ K and µ ( x, k ( x )) · ν ( x ) = 1for x ∈ [ − c, c ] . Now let V be an open subset of Ω with C ,λ boundary suchthat { ( x, − ψ ( x )) : x ∈ [ a, b ] } ⊂ ∂V and ∂V ∩ ( ∂ Ω( K, l ) \ K ) = ∅ and then let U = { ( x, − y ) : ( x, y ) ∈ V } and h ( x, y ) = F + ( x, − y ) for ( x, y ) ∈ U .
Remark 3.
Let Ω ⊂ IR be an open set, Γ ⊂ ∂ Ω be a C ,λ curve and y ∈ Γ bea point at which we wish to have upper and lower Bernstein pairs for H ≡ . Let Σ ⊂ Γ be the intersection of ∂ Ω with a neighborood of y and suppose there is rigid motion ζ : IR → IR such that ζ (Σ) and ζ (Ω) satisfy the hypotheses ofProposition 1. Then (cid:0) ζ − ( U ) , h ◦ ζ (cid:1) will be an upper Bernstein pair for Σ and H ≡ and (cid:0) ζ − ( U ) , − h ◦ ζ (cid:1) will be a lower Bernstein pair for Σ and H ≡ . When H ( x , z ) is independent of z, the existence of (bounded) Bernsteinfunctions is tied to boundary curvature conditions; in Theorem 3.1 of [15] (andTheorem 6.6 of [12]), we see that Proposition 2.
Suppose Ω is a C domain in IR such that | (cid:90) (cid:90) A H ( x ) d x | < (cid:90) | Dχ A | for all A ⊂ Ω , A (cid:54) = ∅ , Ω (15) and (cid:82) (cid:82) Ω H ( x ) d x = (cid:82) | Dχ Ω | ; that is, Ω is an extremal domain. Let y ∈ ∂ Ω andsuppose Λ( y ) < H ( y ) , (16) where Λ( y ) is the (signed) curvature of ∂ Ω at y with respect to the interiornormal direction. Then the (unique up to vertical translations) solution g of N g ( x ) = H ( x ) for x ∈ Ω is bounded and continuous in W = Ω ∩ B (cid:15) ( y ) ,T g extends continuously to a function on W and T g ( x ) = ν ( x ) for each x ∈ B (cid:15) ( y ) ∩ ∂ Ω for some (cid:15) > , where ν is the exterior unit normal to Ω . Using Proposition 2 and a similar procedure to that in the proof of Proposition 1,we can obtain Bernstein pairs near y when ∂ Ω ∩ B (cid:15) ( y ) is a subset of the boundaryof an extremal domain W for some (cid:15) > W are on the sameside of ∂ Ω ∩ B (cid:15) ( y ) and the boundary curvature condition Λ W ( y ) < | H ( y ) | issatisfied. In the same manner, we can obtain Bernstein pairs near y , illustratedin Figure 2 by the sets U ± and U ± , when ∂ y Ω ∩ B (cid:15) ( y ) and ∂ y Ω ∩ B (cid:15) ( y ) aresubsets of the boundaries of extremal domains W and W for some (cid:15) > , Ωand W j are on the same side of ∂ j y Ω ∩ B (cid:15) ( y ) for j = 1 , , Λ W ( y ) < | H ( y ) | andΛ W ( y ) < | H ( y ) | , where Λ W j ( y ) denotes (signed) curvature of ∂W j . Remark 4.
In Proposition 2, the sets A are Caccioppoli sets; that is, Borel setssuch that the distributional (first) derivatives of the characteristic function χ A of A are Radon measures. The notation A (cid:54) = ∅ , Ω means that neither A nor Ω \ A has (two-dimensional) measure zero and the notation (cid:82) | Dχ Ω | means the totalvariation of χ A ∈ BV (Ω) (e.g. § H. We may use § U of a point y ∈ Γ when Γ ⊂ ∂ Ω is a C curvesatisfying Λ( x ) < | H ( x ) | for x ∈ Γ ∩ U and H ∈ C (cid:0) U ∩ Ω (cid:1) is either non-positive or non-negative in U ∩ Ω . Lemma 1.
Suppose Ω is a C ,λ domain in IR for some λ ∈ (0 , . Let y ∈ ∂ Ω and Λ( y ) denote the (signed) curvature of ∂ Ω at y with respect to the interior ormal direction (i.e. − ν ). Suppose Λ( y ) < | H ( y ) | and H ∈ C (cid:0) U ∩ Ω (cid:1) iseither non-positive or non-negative in U ∩ Ω , where U is some neighborhood of y . Then there exist δ > and upper and lower Bernstein pairs ( U ± , ψ ± ) for (Γ , H ) , where Γ = B δ ( y ) ∩ ∂ Ω . Proof:
There exists δ > B δ ( y ) ⊂ U and Λ( x ) < | H ( x ) | for each x ∈ ∂ Ω ∩ B δ ( y ) . There exists a δ ∈ (0 , δ /
2) such thatΛ def = sup { Λ( x ) : x ∈ ∂ Ω ∩ B δ ( y ) } < inf { | H ( x ) | : x ∈ ∂ Ω ∩ B δ ( y ) } def = 2 H . If Λ > , set R = ; otherwise let R be a small positive number. Now let W be a C ,λ domain in IR such that ∂ Ω ∩ B δ ( y ) ⊂ ∂W, Ω and W lie on thesame side of ∂ Ω ∩ B δ ( y ) and W satisfies an interior sphere condition of radius R at each point of ∂ Ω ∩ B δ ( y ) . Continuously extend H outside U to W in sucha manner that H is either non-positive or non-negative in W. From inequality(14.73) of [14], there exists
L > u ( x ) − u ( x ) ≤ L for x ∈ ∂W ∩ B δ ( y ) , where u is any solution of (1) in W and u ( x ) = sup { u ( t ) : t ∈ ∂W \ B R ( x ) } . We may assume 2 δ < R and set u ∗ = sup { u ( t ) : t ∈ ∂W \ B R − δ ( y ) } . Then u ( x ) ≤ u ∗ for each x ∈ ∂W ∩ B δ ( y ) and u ≤ L + u ∗ on ∂ Ω ∩ B δ ( y ) . Now let φ ∈ C ∞ ( ∂W ) such that φ = 0 on ∂W \ B R − δ ( y ) and φ > L on ∂W ∩ B δ ( y )and let h ∈ C ( W ) be the solution of (1)-(2) in W with Dirichlet data φ. (Justas [14] ignores in Theorem 14.11 the question of whether u = φ on ∂ Ω \ B R ( y ) , we may assume that W satisfies curvature conditions (i.e. Λ W ≥ | H | ) on ∂W \ B R − δ ( y ) so that h = φ on ∂ Ω \ B R ( y ) and so h ∗ = 0 . ) It then follows(e.g. [1]) that h ∈ C ( W ) and ∂h∂ν = + ∞ on B δ ( y ) ∩ ∂ Ω . Thus h is an upper Bernstein function. The existence of a lower Bernsteinfunction is similar. Remark 5.
In a similar manner, given y ∈ ∂ Ω we can establish the existenceof upper and lower Bernstein pairs for the intersections of ∂ y Ω and ∂ y Ω with aneighborhood of y when these sets are each subsets of the boundaries of smooth(i.e. C ,λ ) domains W and W which satisfy appropriate boundary curvatureconditions at y . (For capillary surfaces in positive gravity (and prescribed meancurvature surfaces with ∂H∂z ( x , z ) ≥ κ > , one can examine Theorem 2 of [19].) ∂ y Ω and ∂ y Ω In [9], the existence of nontangential radial limits of bounded, nonparametricprescribed mean curvature surfaces at nonconvex corners was proven; in The-orem 1, we showed that all radial limits of such surfaces at nonconvex corners15xist when Bernstein functions exist. On the other hand, [22] and Theorem3 of [25] provide examples in which no radial limit exists at a point y of ∂ Ωat which the boundary of Ω is smooth. In this section, we shall focus on thepoints y ∈ ∂ Ω at which β ( y ) − α ( y ) ≤ π and ask which type of behavior (i.e.(a) no radial limits exist, (b) nontangential radial limits exist or (c) all radiallimits exist) occurs, depending essentially on the curvatures of ∂ y Ω and ∂ y Ω . The following lemma shows that (a), (b) and (c) are the only possible behaviorsof radial limits when H ( x , t ) is weakly increasing in t for each x ∈ Ω , providedthat we include in (b) all of the cases in which Rf ( θ, y ) exists for θ in one ofthe three intervals ( α ( y ) , β ( y )) , [ α ( y ) , β ( y )) , and ( α ( y ) , β ( y )] . Lemma 2.
Let f ∈ C (Ω) ∩ L ∞ (Ω) satisfy Qf = 0 in Ω and let H ∗ ∈ L ∞ (IR ) satisfy H ∗ ( x ) = H ( x , f ( x )) for x ∈ Ω . Let y ∈ ∂ Ω and suppose there exists a θ ∈ [ α ( y ) , β ( y )] such that Rf ( θ , y ) exists. Then Rf ( θ, y ) exists for each θ ∈ ( α ( y ) , β ( y )) , Rf ( · , y ) ∈ C (( α ( y ) , β ( y ))) and Rf ( · , y ) behaves as in Theorem1 of [9].Suppose, in addition, that there exist δ > and upper and lower Bernsteinpairs (cid:0) U ± , ψ ± (cid:1) and (cid:0) U ± , ψ ± (cid:1) for (Γ , H ∗ ) and (Γ , H ∗ ) respectively, where Γ = B δ ( y ) ∩ ∂ y Ω and Γ = B δ ( y ) ∩ ∂ y Ω . Then the conclusions of Theorem 1hold.
Proof:
The first part follows from Theorem 2 of [9]. The second part followsfrom Theorem 3.Now suppose f ∈ C (Ω) satisfies N f ( x ) = H ( x ) for x ∈ Ω and y ∈ ∂ Ωsatisfies β ( y ) − α ( y ) ≤ π. Under what conditions do types of behavior (a), (b)or (c) occur?
Lemma 3.
Suppose Ω , y and H are as above and Λ( x ) ≥ | H ( x ) | for almostall x ∈ B (cid:15) ( y ) ∩ ∂ y Ω ∪ ∂ y Ω , for some (cid:15) > . Then there exists φ ∈ L ∞ ( ∂ Ω) such that the solution f ∈ C (Ω) ∩ L ∞ (Ω) of N f = H in Ω and f = φ almosteverywhere on ∂ Ω has no radial limits at y . Proof:
This follows from Theorem 16.9 of [14] and the “gliding hump” argumentin [22].
Lemma 4.
Suppose Ω , y and H are as above and Λ( x ) < | H ( x ) | for almostall x ∈ B (cid:15) ( y ) ∩ ∂ y Ω ∪ ∂ y Ω , for some (cid:15) > . Then there exist δ > and upperand lower Bernstein pairs (cid:0) U ± j , ψ ± j (cid:1) for (Γ j , H ) , where Γ j = B δ ( y ) ∩ ∂ j y Ω , for j = 1 , , and the conclusions of Theorems 2–5 hold when their other hypothesesare satisfied. Proof:
This follows from Remark 5. 16 heorem 6.
Suppose Ω is a C ,λ domain in IR and f ∈ C (Ω) ∩ L ∞ (Ω) isa variational (i.e. BV) solution of (1)-(2) for some φ ∈ L ∞ (Ω) and λ ∈ (0 , . Let y ∈ ∂ Ω and let Λ( y ) denote the (signed) curvature of ∂ Ω at y with respectto the interior normal direction (i.e. − ν ).(i) Suppose Λ( y ) < | H ( y ) | . Then the conclusions of Theorem 2 hold.(ii) Suppose Λ( y ) > | H ( y ) | . Then the conclusions of Theorem 2 hold if φ restricted to ∂ j y Ω has a limit z j at y for j = 1 , , while for certain φ ∈ L ∞ (Ω) , Rf ( · , y ) does not exist for any θ ∈ [ α ( y ) , β ( y )] . Proof:
The first part follows from Lemma 4. The second part follows fromTheorem 16.9 of [14], [21] (see also [7, 25]) and Lemma 3.
Remark 6.
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