BOUNDARY BEHAVIOR OF CAPILLARY SURFACES POSSIBLY WITH EXTREMAL BOUNDARY ANGLES

For solutions to the capillarity problem possibly with the
boundary contact angle θ being 0 and/or π in a relatively open portion of the boundary which is C2, we will show that if the solution is locally bounded up to this portion of
boundary, the trace of the solution on this portion is piecewise
Lipschitz continuous and the solution is Holder continuous up to
the boundary, provided the prescribed mean curvature is bounded
from above and from below. In the case where θ is not required to be bounded away from π/2, 0, and π, and the mean curvature H(x,t0) belongs to Lp(Ω) for some t0∈ℝ and p>n, under the assumption that in a neighborhood of a relatively open portion of the 
boundary the solution is of rotational symmetry, the trace of the solution on
this portion of the boundary is shown to be Holder continuous with exponent 
1/n if n≥3 and with exponent 1/3 if n=2.


Introduction
Given a domain Ω ⊂ R n , we are interested in regularity near the boundary ∂Ω for solutions u ∈ C 2 (Ω) of the mean curvature equation div Tu = H(x,u) on Ω, (1.1) subject to the "contact angle" boundary condition for a piecewise continuous function θ which takes its values in [0, π], where and ν is the outward-pointing unit normal of ∂Ω.Thus, geometrically, we are looking for a function u on Ω whose graph has the prescribed mean curvature H and which meets the cylinder over the boundary in the prescribed angle θ.The function H is assumed to be locally Lipschitz continuous on Ω × R satisfying the structural conditions ∂H ∂t (x,t) ≥ 0, for x ∈ Ω, t ∈ R. (1.4)One of our main interests is in the case when θ takes the value 0 and/or π in a relatively open portion of ∂Ω, at which the solution graph is required to meet the boundary cylinder vertically upward or downward; thus, when approaching this portion, the Euclidean norm of the gradient becomes unbounded, which makes this case mathematically more subtle than the case when θ is bounded away from 0 and π.We will show in this paper that if this portion of ∂Ω is C 2 and if the prescribed mean curvature is bounded from above and from below, then if the solution is bounded locally up to this portion of the boundary, the trace of the solution on this portion is piecewise Lipschitz continuous and the solution is Hölder continuous up to the boundary.
For the regularity result, no a priori regularity is imposed at the boundary and write the equation together with the contact angle boundary condition in the weak form as follows: let ν be any C 0 (Ω) function which coincides on ∂Ω with the outward-pointing unit normal of ∂Ω and let θ : ∂Ω → [0,π] be a given function which is measurable with respect to (n − 1)-dimensional Hausdorff measure Ᏼ n−1 on ∂Ω ∩ B ρ .Suppose that u ∈ C 2 (Ω) ∩ W 1,1 (Ω). ( We say u satisfies the contact angle problem div Tu = H(x,u) in Ω, Tu • ν = cos θ on ∂Ω (1.6) in the weak sense provided that (1.7) We will establish the following theorem.

Main Theorem
Fei-Tsen Liang 3927 which is completely determined by H * , n, and (sup Ω u − inf Ω u).Furthermore, u is Hölder continuous up to ∂Ω with exponent determined by the same set of quantities.
We notice that the last statement follows from [14].Suppose that θ, ∂Ω are of the rotational symmetry with respect to the same symmetry axis.Furthermore, suppose that the same rotational symmetry of the function H(x,z) is imposed on the first variable for every value of t.
We will establish in Section 4 the following, which is valid in particular in the case where θ is not required to be bounded away from π/2, 0, and π.
Main Theorem 1.2.Suppose that ∂Ω is a relatively open subset of ∂Ω which is C 2 .Let β = cos θ.Suppose that, at x 0 ∈ ∂Ω, we have (βx 0 ) = 0, and is Lipschitz continuous in ∂Ω with Lipschitz constant L. Suppose that in a neighborhood U of ∂Ω in Ω, the function u| U is of rotational symmetry, and suppose that the prescribed mean curvature H(x,t) is locally Lipschitz continuous on Ω × R, satisfies (1.4), and H x,t 0 ∈ L p (Ω), for some p, p > n, t 0 ≤ u x 0 . (1.10) ) and u is bounded locally up to the boundary in Ω ∩ B ρ .Then, and u satisfies the boundary condition in a classical way (i.e., (1.2) holds with θ ≡ 0 on ∂Ω ∩ B ρ/2 ).Furthermore, considered as a function on the graph of u| Ω∩Bρ/2 , T * u is Lipschitz continuous.Furthermore, We notice that this result does not yield the regularity of the trace of the solution obtained in this present work.[15] the case where Ω is C 4 , θ in (1.2) is C 1,α for some 0 < α < 1, and H(x,t) is strictly monotone in t:

Simon and Spruck treat in
In case 0 < θ < π, the existence of a solution u ∈ C 2 (Ω) of (1.1) and (1.2) is established in [15].In case θ is allowed to take the values 0 and/or π, setting ) is shown to exist in [15] which satisfies (1.1) in Ω and satisfies (1.2) on S 2 ; furthermore, u is Hölder continuous at each point of S + 1 ∪ S − 1 , has a restriction to ∂Ω which is Lipschitz continuous at each point of S + 1 ∪ S − 1 , and (1.2) is satisfied on S ± 1 in the sense that lim assuming that Tu is extended to some boundary strip Ω ε with width ε so that it is constant along the normals to ∂Ω.This regularity result is obtained by first establishing estimates of the tangential derivatives under the condition that | cos θ| ≤ γ < 1 for some positive constant γ; in case θ is constant in a neighborhood of the point under consideration, this estimate of tangential derivatives is independent of γ.This proves the Lipschitz continuity of the trace of u on ∂Ω, which yields Hölder continuity of u together with [14,Theorem 2].Estimates for the tangential derivatives are obtained by performing a transformation of coordinates near the boundary analogously to that in [14], together with a subsequent differentiation of (1.1), (1.2), and inserting (2.5) into the resultant identities.The disadvantage of these proofs is that H is supposed to satisfy the strict inequality (2.5) rather than the less restrictive condition (1.4).

2.3.
In contrast, the following estimates for the boundary oscillation of u is established in [10,Main Theorem III].
Proposition 2.2.Let u ∈ C 2 (Ω) ∩ W 1,1 (Ω) be a bounded solution to (1.1) and (1.2) in the weak sense of (1.4).Suppose that for positive constants β, β ≤ 1 and a ball B R (x 0 ) intersecting the interior of Ω, the function cos θ is continuous on ∂Ω ∩ B R (x 0 ) and such that either of the inequalities holds for all x ∈ ∂Ω ∩ B R (x 0 ), and such that (2.9) Suppose ∂Ω is piecewise Lipschitz continuous with possible outward and/or inward cusps.
Then the trace of We notice that (2.9) hold in particular if This result is established by modifying the approach taken in [3,4,5], which is based on the minimizing property satisfied by u and the following result due to Stampacchia [16,Lemma 4.1].
Lemma 2.3 (Stampacchia).Suppose that φ(t) is a nonnegative nondecreasing function defined on R such that for some constant C, k 0 , and γ, there holds

10)
Then, ) where Let φ(t) be the area of the level set of a function ηu, where η is a suitably chosen cutoff function in a suitably chosen domain.The identity (2.12) is applied to prove [10, Proposition 2], once the inequality (2.11) is shown to hold for some γ > 1.To make this feasible, the condition (2.8) is crucial.At neighborhoods of points where θ = 0,π, or π/2, the condition (2.8), however, is not satisfied.Main Theorem 1.1 treats the cases that θ ≡ 0 or π in a relatively open subset of ∂Ω.Main Theorem 1.2 treats the case that θ = π/2 at a point of ∂Ω near which β is Lipschitz continuous.
The proof of Main Theorem 1.2 will be based on another modification of the technique used in [3,4,5].The inequalities (2.10) will be shown to hold for some γ i < 1, i = 0,1,..., iteratively with φ being the level set of some function η i u, where η i are suitably chosen cut-off functions in some suitably chosen domains depending on i.Using (2.11), the exponent γ i will strictly increase as the number of times i of iteration increases.After a finite number of times of iteration, the exponent becomes greater than 1 and the identity (2.12) can be applied.
We will obtain the inequality (4.32) below.In order to estimate the third and last terms on the right-hand side of (4.32), we have to control the size of the level set of the function η i u.For this purpose, we impose the restrictions that Ω, β, and hence u are of the same rotational symmetry.The rotational symmetry of Ω enables us to choose η i to be of the same rotational symmetry, and hence control the size of the level set of the function η i u.In order to estimate the fifth and sixth terms on the right-hand side of (4.32), we impose the restriction that β(x 0 ) = 0 and β is Lipschitz continuous in a neighborhood of x 0 or ∂Ω.

2.4.
As for the regularity near the boundary, the following result is established in [11] in which the region A r ( x), for small positive number r, is chosen as follows.Namely, setting we let the boundary ∂(Ω ∩ A r ( x)) be made up of three parts: such that where we let ν Ω∩Ar ( x) be the unit outward normal to and ∂Ω ∩ A r ( x) and ∂ * A r ( x) include balls of radius (r/2) n−1 centered, respectively, at x and a point at distance r from x.We established in [11] the following proposition.

2.5.
The main part of [11] is devoted to estimating the L 1 -norm of u which begins with writing (1.1) and (1.2) in weak form (1.7) in which the assumed boundedness of |u| allows us to take the test function to be (u(x ).The resultant inequalities suggest to us to restrict u to a small region Ω ∩ A of the type indicated in the beginning of [11,Section 3], which is analogous to that of A r (x 0 ) above.In [11,Section 4], a modified version of Sobolev inequality is applied to obtain estimates of In order to apply this modified version of Sobolev inequality, we have to apply the condition (2.8) to treat the boundary integrals At neighborhoods of points where θ = 0,π, or π/2, the condition (2.8), however, is not satisfied.The proof of Proposition 2.4 is based on a modification of the reasoning by Giusti [7, pages 312-313], which leads to estimates for the oscillation of u in terms of the L 1norm of u, under the conditions that either the subgraph of u or the complement of the subgraph of u includes a large portion of a sufficiently small cylinder-type region around u(x 0 ), x 0 ∈ ∂Ω.Such conditions are proved in [7,Theorem 3.2] to be fulfilled by capillary surfaces with boundary contact angle bounded away from 0 or π.In order to prove Main Theorem 1.1, we will rephrase this result in [7] as Theorem 3.1 below.

3.1.
Estimates of the oscillation of u in terms of the L 1 -norm of u.

Equation (1.1) is the Euler equation of the functional
and corresponding to the capillarity problem with boundary contact angle θ is the problem of minimizing the functional among all v ∈ BV(Ω).Miranda [13] introduced the notion of generalized solutions for the minimal surface equation.Giusti [6,7] used the analogous notion of generalized solutions for the mean curvature equation.The idea of generalized solutions originates from the observation that a function u : Ω → R is a solution of (1.1) if and only if its subgraph minimizes the functional locally in Ω × R, in the sense that for every V coinciding with U outside some compact set K ⊂ Ω × R, we have here and in the following, φ V is the characteristic function of the set V : Moreover, a function u ∈ BV(Ω) minimizes Ᏺ in Ω if and only if its subgraph minimizes the functional Minimization is here to be understood in the following sense: for T > 0, set and for U ⊂ Q, We say that U minimizes

Fei-Tsen Liang 3933
A generalized solution can take the values ±∞ on sets of positive measure.However, a locally bounded generalized solution u(x) is a classical solution of (1.1) in Ω.A bounded generalized solution of the functional Ᏺ is a solution of (1.1) and (1.2) in the weak sense.Thus, by the uniqueness (up to an additive constant) of the weak solution of (1.1) and (1.2) in the sense (1.7) (cf.[2, Theorems 7.6 and 7.9]), each weak solution of (1.1) and (1.2) in the sense (1.7) is a generalized solution of the functional Ᏺ * .
Let u be a weak solution to (1.1) and (1.2) in the sense (1.4).For points z = ( x, t) ∈ Ω × R, we now consider the sets and let Based on the minimizing property of the subgraph of the function u, the following result can be established, whose proof is essentially identical with that of corresponding results in [7].
Theorem 3.1.Let u be a weak solution to (1.1) and (1.2) in the sense (1.7).Let U be the subgraph of u.Suppose that there exists a constant γ, 0 ≤ γ < 1, such that there holds either Suppose further that for some constant µ with µ γ < 1 and C Ω depending only on Ω, an inequality holds for all v ∈ BV(Ω).If then there exist positive constants T * and α * determined by n, inf Ω×R H, γ, ∂Ω∩ C ± ε ( z) , R 0 , and the geometry of Ω such that and hence The value of α * can be specified as with k (n+1) being the constant for the isoperimetric inequality in R n+1 .
We also notice that an inequality of the form (3.15) appears first by Emmer in [1] with µ = √ 1 + L 2 for any Lipschitz domain with Lipschitz constant L. By Finn [2, pages 141-143], this result is extended to include domains in which one or more corners with inward opening angle appear.

3.1.2.
The result in Theorem 3.1 is connected with the following estimates of u(x) − u(x 0 ) in terms of the measure of an associated level set.

The Lipschitz continuity of the trace on the boundary: Proof of MainTheorem 1.1.
Assume that there exists a nonnegative constant H * such that Analogously to [11,Section 4], we restrict our consideration to a small region Ω ∩ A δ0 (x 0 ) of the type indicated in the beginning of Section 2.4, for some positive number δ 0 .Let and let Let us set (3.29)

3.2.1.
We notice that the following result holds when θ ≡ 0 or θ ≡ π on a portion of ∂Ω.

Proof of Main Theorem 1.2
Let u be a bounded solution to (1.1) subject to the "contact angle" boundary condition (1.2).Choose a point x 0 ∈ ∂Ω.Let C x0 be the level set of u through x 0 .Since |β| is bounded away from 1, the level set C x0 divides Ω into two components Ω 1 and Ω 2 in which the respective inequalities u(x) ≥ u(x 0 ) and u(x) ≤ u(x 0 ) are valid at points near to C x0 .
We choose, for sufficiently small t 0 , the set Ꮽ δ t0 (x 0 ) for 0 ≤ δ < 1 as follows.Namely, setting d(x) = dist(x,C x0 ) for x ∈ Ω and letting we let the boundary ∂Ꮽ δ t0 (x 0 ) be made up of five parts, namely, such that and -dimensional balls of the respective radii (t 0 ) 1−δ /2, t 0 /2, and here c n is a constant depending only on n.
Let η 0 be a smooth function on Ω such that Let k be a number greater than max(u(x 0 ),t 0 ,0).We set Then, u k belongs to BV(Ω) and from the minimizing property of u, we know that setting the restriction u| Ꮽ0(k,η0) is a capillary surface such that where for v ∈ C 1 (Ω), we set where ν Ꮽ0(k,η0) is the unit outward normal to Ꮽ 0 (k,η 0 ) and we set ( f ) − = min( f ,0) for all real-valued functions f .Assume for a moment that u is smooth, we obtain analogously to [9, equation (1.8)] that Fei-Tsen Liang 3939 where we set being the unit outward normal to Ꮽ δ 3ε/4 (x 0 ) and ν 0 is the unit outward normal to Ꮽ 0 (k,η 0 ).

4.3.
In view of the choice of Ꮽ δ εm (x 0 ) and η m , we have Moreover, it is easy to see that n 0 can be chosen to be n − 1 and q m , 0 ≤ m ≤ n 0 − 1, can be chosen to be arbitrarily close to the number n/(n − (m + 1)) and q n0 can be chosen arbitrarily large; that is, for each positive number ε, we can choose q m , 0 With such a choice of n 0 and q m , 0 ≤ m ≤ n 0 , we have where C n = (3/2)n(n − 1) + 2. Inserting these into (4.32),we obtain where Fei-Tsen Liang 3947 In view of the rotational symmetry of Ω and u, we choose η m to be of the same rotational symmetry so that the level sets of η m u are of the same rotational symmetry.Hence, ∂Ꮽ m (k,η m ) consists of ∂ * Ꮽ δ εm−1 , and another region which is parallel to ∂ * Ꮽ δ εm−1 , together with some other portion of ∂Ꮽ εi (x 0 ) ⊂ ∂ * Ꮽ δ εm−1 .Hence,   Choose δ = 1/2.We obtain where ) 2 allow us to set γ = −1/2 in (3.19) and (3.22) with T = t +− A and A + T (x 0 ) = (Ω ∩ A δ0 (x 0 )) + .Inserting (3.35) into (3.22),we establish Main Theorem 1.1 in case θ ≡ π near x 0 .
is the graph of a Lipschitz continuous function with Lipschitz constant L such that β The Lipschitz constant of the trace of u on ∂Ω ∩ B R (x 0 ) depends only on H, n, together with the constants β, β, and ∂Ω∩BR(x0) , where ∂Ω∩BR(x0) is an upper bound for the absolute value of the principal curvatures of ∂Ω √ 1 + L 2 < 1.