PRINCIPAL CURVATURE ESTIMATES FOR THE LEVEL SETS OF HARMONIC FUNCTIONS AND MINIMAL GRAPHS IN R

We give a sharp lower bound for the principal curvature of the level sets of harmonic functions and minimal graphs defined on convex rings in R3 with homogeneous Dirichlet boundary conditions.


1.
Introduction. The convexity of the level sets of the solutions of elliptic partial differential equations has been studied for a long time. For instance, Ahlfors [1] contains the well-known result that level curves of Green function on simply connected convex domain in the plane are the convex Jordan curves. In 1931, Gergen [10] proved the star-shapeness of the level sets of Green function on 3-dimensional starshaped domain. For the minimal annulus whose boundary consists of two closed convex curves in parallel planes P 1 and P 2 , in 1956 Shiffman [22] proved that the intersection of the surface with any plane P , between P 1 and P 2 , is a convex Jordan curve. For elliptic partial differential equations on domains in R n , the convexity of level set was first considered by Gabriel [9] in 1957. He proved that the level sets of the Green function on a 3-dimensional convex domain are strictly convex. Later, in 1977, Lewis [15] extended Gabriel's result to p-harmonic functions in higher dimensions and obtained the following theorem. Theorem 1.1 (Gabriel [9], Lewis [15]). Let u satisfy    div(|∇u| p−2 ∇u) = 0 in Ω = Ω 0 \Ω 1 , u = 0 on ∂Ω 0 , u = 1 on ∂Ω 1 , where 1 < p < +∞, Ω 0 and Ω 1 are bounded convex domains in R n , n ≥ 2,Ω 1 ⊂ Ω 0 . Then all the level sets of u are strictly convex.
For the minimal graphs, Korevaar (see Remark 13 in [14]) proved the following result.
In 1982, Caffarelli-Spruck [6] generalized the Lewis [15] results to a class of semilinear elliptic partial differential equations. A good survey of this subject is given by Kawohl [13]. For more recent related extensions, please see the papers by Bianchini-Longinetti-Salani [3] and Bian-Guan-Ma-Xu [2]. Now we turn to the curvature estimates of the level sets of the solutions of elliptic partial differential equations. For 2-dimensional harmonic functions and minimal surfaces with convex level curves, Ortel-Schneider [20], Longinetti [17] and [18] proved that the curvature of the level curves attains its minimum on the boundary (see also Talenti [23] for related results). Jost-Ma-Ou [12] proved that the Gaussian curvature of the convex level sets of 3-dimensional harmonic function attains its minimum on the boundary. For the other related results and their application to free boundary problem, please see the papers by Rosay-Rudin [21], Dolbeault-Monneau [8].
In this paper, using the strong maximum principle, we obtain a sharp principal curvature estimates for the level set of lower dimensional p-harmonic functions and minimal graphs defined on convex ring. Our theorems are the principal curvature counterpart of the Gaussian curvature estimates in [12]. Now we state our theorems.
Assume |∇u| > 0 in Ω. If n+1 2 ≤ p ≤ 3 and the level sets of u are strictly convex with respect to normal ∇u, then the smallest principal curvature of the level sets of u cannot attain its minimum in Ω, unless it is constant.
Using Theorem 1.1 and Theorem 1.3, we have the following corollary.
where Ω 0 and Ω 1 are smooth bounded convex domains in R 3 ,Ω 1 ⊂ Ω 0 . If 3 ≤ n ≤ 5 and n+1 2 ≤ p ≤ 3, then the principal curvature of the level sets of u attains its minimum on ∂Ω. Now we turn on the 3-dimensional minimal graphs.
Let Ω be a bounded domain in R 3 and u be a minimal graph over Ω, i.e., u satisfy the minimal surface equation div( ∇u Assume |∇u| > 0 in Ω. If the level sets of u are strictly convex with respect to normal ∇u, then the smallest principal curvature of the level sets of u cannot attain its minimum in Ω, unless it is constant.
Similarly, we have the following corollary.
where Ω 0 and Ω 1 are smooth bounded convex domains in R 3 ,Ω 1 ⊂ Ω 0 . Then the principal curvature of the level sets of u attains its minimum on ∂Ω.
Let (a ij ) be the symmetry curvature matrix on the strictly convex level sets defined in (2.4), and let (a ij ) be its inverse matrix. We consider the auxiliary function We shall derive the following differential inequality n α,β=1 is the associated elliptic operator in (1.1) or (1.2). In (1.7), we have suppressed the terms containing the gradient of ∇ϕ with locally bounded coefficients, then we apply the strong maximum principle to obtain the results. Now let us mention that three dimensional harmonic function always has very special properties. The famous theorem of Lewy [16] states that if u is a harmonic function on a domain in R 3 and the map x −→ ∇u(x) is a homeomorphism, then x −→ ∇u(x) is a diffeomorphism. In 1991, Gleason-Wolff [11] extended this results to higher dimensions, but needed some extra conditions, and gave a counterexample in the higher dimensional case without these additional conditions.
In section 2, we first give brief definition on the convexity of the level sets, then obtain the curvature matrix a ij of the level sets of a function, which appeared in [2]. The main technique in the proof of theorems consists in rearranging the second and third derivatives terms using the equation and the first derivatives condition for ϕ. In the 3-dimensional case, we get "good" sign for the second and third derivatives terms, which allows us to reach our conclusions.
2. The curvature formulas of level sets. In this section, we shall give the brief definition on the convexity of the level sets, then introduce the curvature matrix (a ij ) of the level sets of a function, which appeared in [2]. Firstly, we recall some fundamental notations in classical surface theory. Assume a surface Σ ⊂ R n is given by the graph of a function v in a domain in R n−1 : The principal curvature κ = (κ 1 , · · · , κ n−1 ) of the graph of v, being the eigenvalues of the second fundamental form relative to the first fundamental form. We have the following well-known formula.
where the summation convention over repeated indices is employed.
Now we give the definition of the convex level sets of the function u.
Let Ω be a domain in R n and u ∈ C 2 (Ω), its level sets can be usually defined in the following sense.
and v(x ) satisfies the following equation Then the first fundamental form of the level set is g ij = δ ij + uiuj u 2 n , and W = (1 + |∇v| 2 ) 1 2 = |∇u| |un| . The upward normal direction of the level set is then the second fundamental form of the level set of function u is Definition 2.4. For the function u ∈ C 2 (Ω) we assume |∇u| > 0 in Ω. Without loss of generality we can let u n (x o ) = 0 for x o ∈ Ω. We define locally the level set Σ u(xo) = {x ∈ Ω|u(x) = u(x o )} is convex with respect to the upward normal direction ν if the second fundamental form b ij is nonnegative definite.
Remark 2.5. If we let ∇u be the upward normal of the level set Σ u(xo) at x o , then u n (x o ) > 0 by (2.2). And from the definition 2.4, if the level set Σ u(xo) is convex with respect to the normal direction ∇u, then the matrix (h ij (x o )) is nonpositive definite.
Now we obtain the representation of the curvature matrix (a ij ) of the level sets of the function u with the derivative of the function u, From now on we denote and then the symmetric curvature matrix of the level sets of u could be represented as (2.7) 3. Principal curvature estimates of level set of p-harmonic function. In this section, we prove the Theorem 1.3. We study the following equation and we shall prove this theorem using strong minimum principle. In the following proof, the Greek indices (α, β, γ, δ, ...) run from 1 to n, the Roman indices (i, j, k, l, ...) run from 1 to n − 1. Denote Proof of Theorem 1.3: Since the level sets of u are strictly convex with respect to normal ∇u, then the curvature matrix (a ij ) of the level sets is positive definitive in Ω. Let (a ij ) be the inverse matrix of (a ij ). We consider the auxiliary function We shall derive the following differential inequality n α,β=1 where we modify the terms of the gradient of ϕ with locally bounded coefficients. Then by the standard strong maximum principle, we get the result immediately.
In order to prove (3.4) at an arbitrary point x o ∈ Ω, as in Caffarelli-Friedman [4], we choose the normal coordinate at x o . We have mentioned in Remark 2.5, since the level sets of u are strictly convex with respect to normal ∇u, by rotating the coordinate system suitably by T xo , we may assume that u i (x o ) = 0 , 1 ≤ i ≤ n − 1 and u n (x o ) = |∇u| > 0. And we can further assume ξ = e 1 , the matrix (u ij (x o )) (1 ≤ i, j ≤ n − 1) is diagonal and u ii (x o ) < 0. Consequently we can choose T xo to vary smoothly with x o . If we can establish (3.4) at x o under the above assumption, Taking the first derivative of ϕ, we get it follows that Taking derivative of equation (3.5) once more, we have In order to get (3.4), we only need to prove u 3 n a 2 11 n α,β=1 where we modify the terms of the gradient of ϕ with locally bounded coefficients. We shall prove (3.8) in two steps.

(3.22)
Now we work on the term T 4 , we will first calculate u n u 11 n α,β=1 F αβ u nαβ in   F αβ u αβ1 = 2u n u 1n n α,β=1 − 4u 4 n u n1 a 11,1 − 4(p − 2)u 4 n u 11 a 11,n .    Let us use the equation (3.1) to substitute the term u nn1 in (3.33). We take derivative of (3.1) with respect to x 1 to get F αα u nα a 11,α . (3.36) Step 2: The end of the theorem. Now we calculate the following term in (3.8) F αβ a kk a 1k,α a 1k,β .
Recall that the level sets are strictly convex with respect to the normal direction ∇u, we have u ii < 0 for 1 ≤ i ≤ n − 1. Hence, for p ≥ n+1 2 , we have We also need p − 1 ≥ 0 and 3 − p ≥ 0, i.e. 1 ≤ p ≤ 3. Hence, for n+1 2 ≤ p ≤ 3 we obtain n α,β=1 We complete the proof of the Theorem 1.3.

4.
Principal curvature estimates of level set of minimal graphs. In this section, we study the following equation div( ∇u and prove the Theorem 1.5. We prove this theorem as in the last section. In the following proof, the Greek indices (α, β, γ, δ, ...) run from 1 to 3, the Roman indices (i, j, k, l, ...) run from 1 to 2. Denote F αβ (∇u) = (1 + |∇u| 2 )δ αβ − u α u β . Proof of Theorem 1.5: Since the level sets of u are strictly convex with respect to the normal direction ∇u, the curvature matrix (a ij ) of the level sets is positive definite in Ω. Let (a ij ) be the inverse matrix of (a ij ). As in the last section, Let ϕ = a 11 . We will derive the following differential inequality F αβ a 11,αβ . (4.5) In the following, we shall prove F αβ a 11,αβ ≥ 0 mod ∇ϕ in Ω, (4.6) Let us prove (4.6) in two steps.
Step 2: The end of the theorem. Similar to (3.37) in the last section, we have   F αβ ϕ αβ ≥ 0 mod ∇ϕ.
We complete the proof of Theorem 1.5.
Remark 4.1. Using another auxiliary function, recently Chang-Ma-Yang [7] and Ma-Ou-Zhang [19] got the lower bound estimates of the principal curvature and the Gaussian curvature for the convex level sets of higher-dimensional harmonic functions in convex rings.