A note on higher regularity boundary Harnack inequality

We show that the quotient of two positive harmonic functions vanishing on the boundary of a $C^{k,\alpha}$ domain is of class $C^{k,\alpha}$ up to the boundary.


Introduction
In this note we obtain a higher order boundary Harnack inequality for harmonic functions, and more generally, for solutions to linear elliptic equations.
Theorem 1.1. Let u > 0 and v be two harmonic functions in Ω ∩ B 1 that vanish continuously on ∂Ω ∩ B 1 . Assume u is normalized so that u (e n /2) = 1, then with C depending on n, k, α.
We remark that if v > 0, then the right hand side of (1.1) can be replaced by v u (e n /2) as in the classic boundary Harnack inequality. For a more general statement for solutions to linear elliptic equations, we refer the reader to Section 3.
The classical Schauder estimates imply that u, v are of class C k,α up to the boundary. Using that on ∂Ω we have u = v = 0 and u ν > 0, one can easily conclude that v/u is of class C k−1,α up to the boundary.
Theorem 1.1 states that the quotient of two harmonic functions is in fact one derivative better than the quotient of two arbitrary C k,α functions that vanish on the boundary. To the best of our knowledge the result of Theorem 1.1 is not known in the literature for k ≥ 1. The case when k = 0 is well known as boundary Harnack inequality: the quotient of two positive harmonic functions as above must be C α up to the boundary if ∂Ω is Lipschitz, or the graph of a Hölder function, see [HW, CFMS, JK, HW, F].
A direct application of Theorem 1.1 gives smoothness of C 1,α free boundaries in the classical obstacle problem without making use of a hodograph transformation, see [KNS, C]. Corollary 1.2. Let ∂Ω ∈ C 1,α and let u solve Assume that u is increasing in the e n direction. Then ∂Ω ∈ C ∞ .
The corollary follows by repeateadly applying Theorem 1.1 to the quotient u i /u n . Our motivation for the results of this paper comes from the question of higher regularity in thin free boundary problems, which we recently began investigating in [DS].
The idea of the proof of Theorem 1.1 is the following. Let v be a harmonic function vanishing on ∂Ω. The pointwise C k+1,α estimate at 0 ∈ ∂Ω is achieved by approximating v with polynomials of the type x n P with deg P = k. It turns out that we may use the same approximation if we replace x n by a given positive harmonic function u ∈ C k,α that vanishes on ∂Ω. Moreover, the regularity of ∂Ω does not play a role since the approximating functions u P already vanish on ∂Ω.
In order to fix ideas we treat the case k = 1 separately in Section 2, and we deal with the general case in Section 3.
In this section, we provide the proof of our main Theorem 1.1 in the case k = 1. We also extend the result to more general elliptic operators.
Let Ω ⊂ R n with ∂Ω ∈ C 1,α . Precisely, Let u be a positive harmonic function in Ω∩B 1 , vanishing continuously on ∂Ω∩B 1 . Normalize u so that u(e n /2) = 1. Throughout this section, we refer to positive constants depending only on n, α as universal.
Theorem 2.1. Let v be a harmonic function in Ω ∩ B 1 vanishing continuously on First we remark that from the classical Schauder estimates and Hopf lemma, u satisfies Thus, after a dilation and multiplication by a constant we may assume that where the constant δ will be specified later. We claim that Theorem 2.1 will follow, if we show that there exists a linear function To obtain (2.4), we prove the next lemma.
Lemma 2.2. Assume that, for some r ≤ 1 and P as in Then, there exists a linear function for some ρ > 0 universal, and Define also,ũ We have, Moreover, from (2.2) we have ∇u − e n L ∞ (Ω∩Br ) ≤ δr α .
Thus,ṽ solves Hence, as δ → 0 (using also (2.2))ṽ must converge (up to a subsequence) uniformly to a solution v 0 of for some ρ universal and By compactness, if δ is chosen sufficiently small, then from which the desired conclusion follows by choosinḡ Remark 2.3. Notice that, from boundary Harnack inequality,ṽ satisfies (see (2.5) and recall that u( 1 2 e n ) = 1) sinceũ is bounded below in such region. This, together with the identity v u = P + r 1+αṽ u Proof of Theorem 2.1. After multiplying v by a small constant, the assumptions of the lemma are satisfied with P = 0 and r = r 0 small. Thus, if we choose r 0 small universal, we can apply the lemma indefinitely and obtain a limiting linear function P 0 such that |v − uP 0 | ≤ Cr 2+α , r ≤ r 0 . In fact, from Remark 2.3 we obtain v u − P 0 ≤ C|x| 1+α which together with (2.6) gives the desired conclusion.
It is easy to see that our proof holds in greater generality. For example, if v solves ∆v = f ∈ C α in Ω ∩ B 1 and vanishes continuously on ∂Ω ∩ B 1 , then we get To obtain this estimate it suffices to take in Lemma 2.2 linear functions P (x) = a 0 + n i=1 a i x i satisfying 2a n u n (0) = f (0). In fact, the following more general Theorem holds.

Theorem 2.4. Let
Lu with C depending on α, λ, Λ and n.
Remark 2.5. We emphasize that the conditions on the matrix A and the right hand side f are those that guarantee interior C 2,α Schauder estimates. However the conditions on the domain Ω and the lower order coefficients b, c are those that guarantee interior C 1,α Schauder estimates.
Remark 2.6. The theorem holds also for divergence type operators The proof of Theorem 2.4 follows the same argument of Theorem 2.1. For convenience of the reader, we give a sketch of the proof.
Sketch of the proof of Theorem 2.4. After a dilation we may assume that (2.2) holds and also (2.7) with δ to be chosen later. Again, it suffices to show the analogue of Lemma 2.2 in this context, with the x n coefficient of P andP satisfying 2a n = 2ā n = f (0).
Defineṽ as before. Then On the other hand, L(uP ) = (Lu)P + 2(∇u) T A∇P + u b · ∇P thus, using (2.2)-(2.7) and the fact that 2a n = f (0) From this we conclude that |Lṽ| ≤ Cδ inΩ ∩ B 1 and we can argue by compactness exactly as before.
From now on, a positive constant depending on n, k, α, λ, Λ is called universal.
Remark 3.2. If we are interested only in C k,α estimates for v u on ∂Ω ∩ B 1/2 , then the regularity assumption on c can be weakened to c C k−3,α ≤ Λ.
If u and v solve (3.1)-(3.2) respectively, the rescalings satisfy the same problems with Ω, A, b, c and f replaced bỹ c(x) = r 2 0 c(r 0 x),f (x) = r 0 f (r 0 x). Thus, as in the case k = 1, we may assume that ∇u(0) = e n , A(0) = I and that the following norms are sufficiently small: with δ to be specified later. The proof of Theorem 3.1 is essentially the same as in the case k = 1. However, we now need to work with polynomials of degree k rather than linear functions.
We introduce some notation. A polynomial P of degree k is denoted by with the a m non-zero only if m ≥ 0 and |m| ≤ k. We use here the summation convention over repeated indices and the notation Also, in what follows,ī denotes the multi-index with 1 on the ith position and zeros elsewhere and P = max |a m |.
Given u a solution to (3.1), we will approximate a solution v to (3.2) with polynomials P such that L(uP ) and f are tangent at 0 of order k − 1.
Below we show that the coefficients of such polynomials must satisfy a certain linear system. Indeed, L(uP ) = (Lu)P + 2(∇u) T A∇P + u tr(AD 2 P ) + u b · ∇P.
Since Lu = 0, we find Using the first order in the expansions below (l.o.t = lower order terms), we write each g i , g ij as a sum of a polynomial of degree k − 1 and a reminder of order O(|x| k−1+α ). We find In the case P = x m we obtain Also in view of (3.3) Thus, if P = a m x m , with P ≤ 1 then with w as above and the coefficients of R satisfying (3.6) d l = (l n + 1)(l n + 2)a l+n + i =n Definition 3.3. We say that P is an approximating polynomial for v/u at 0 if the coefficients d l of R(x) coincide with the coefficients of the Taylor polynomial of order k − 1 for f at 0.
We think of (3.6) as an equation for a l+n in terms of d l and a linear combination of a m 's with either |m| < |l| + 1 or when |m| = l + 1 with m n < l n + 1. Thus the a m 's are uniquely determined from the system (3.6) once d l and a m with m n = 0 are given.
The proof of Theorem 3.1 now follows as in the case k = 1, once we establish the next lemma.
Lemma 3.4. Assume that for some r ≤ 1 and an approximating polynomial P for v/u at 0, with P ≤ 1, we have v − uP L ∞ (Ω∩Br ) ≤ r k+1+α .
However P + r k+α Q( x r ) is not approximating for v/u at 0, and we need to modify Q into a slightly different polynomialQ.
Thus, by subtracting the last two equations, the coefficients of Q −Q solve the system (3.8) with left hand side bounded by Cδ, and we can findQ such that Q −Q L ∞ (B1) ≤ Cδ.