Prescribed mean curvature equation on torus

1 Introduction

In this paper, we consider the following prescribed mean curvature problem on the torus 𝕋n:=ℝn/ℤn:

where ν is the unit normal vector of u, that is,

The vector field g⁢(x,xn+1):𝕋n×ℝ→ℝn+1 is given, and we seek a solution u satisfying (1.1). The left-hand side of (1.1) represents the mean curvature of the graph of u, and the right-hand side is the normal component of the vector field g on the graph.

In the case of Dirichlet conditions of a bounded domain Ω⊂ℝn, prescribed mean curvature problems have been studied by numerous researchers. Bergner [3] solved the Dirichlet problem in the case where the right-hand side of (1.1) is H=H⁢(x,u,ν⁢(∇⁡u)) under the assumptions of boundedness (|H|<∞), monotonicity (∂n+1⁡H≥0), and convexity of Ω. Under the same conditions for the function H, Marquardt [9] imposed a condition on ∂⁡Ω depending on H that guarantees the existence of solutions even for a domain Ω that is not necessarily convex. In [13], we proved the existence of a solution only under the condition that the Sobolev norm of H is sufficiently small. In the case of a compact Riemannian manifold, Aubin [2] solved the linear elliptic problem -∂i⁡[ai⁢j⁢(x)⁢∂j⁡u]=H⁢(x) if the integrated value of H is zero. The assumption of the integrated value plays an important role in the existence of solutions to elliptic equations on a compact Riemannian manifold. Denny [4] solved the quasilinear elliptic problem -div⁡(a⁢(u⁢(x))⁢∇⁡u)=H⁢(x) on the torus 𝕋n with n=2,3. Prescribed mean curvature problems on the one-dimensional torus

have been investigated for a wide variety of conditions H (we refer to, for example, [5, 7, 8, 10, 11, 14]).

As we noted in [13], the motivation for the present study comes from a singular perturbation problem, and we proved the following in [12]. Suppose a constant ε>0 and functions ϕε∈W1,2 and gε∈W1,p, with p>n+12, satisfy

where W is a double-well potential such as W⁢(ϕ)=(1-ϕ2)2. Then the interface {ϕε=0} converges locally in the Hausdorff distance to a surface having a mean curvature given by ν⋅g as ε→0. Here, ν is the unit normal vector of the surface, and g is the weak W1,p limit of gε. If the surface is represented locally as a graph of a function u on 𝕋n, we can observe that u satisfies (1.1). In this paper, we prove the existence of solutions to (1.1) assuming that the Sobolev norm of g is sufficiently small, gn+1 for the (n+1)-st component is monotonous, and the integrated value of gn+1 is zero. The following theorem is the main result.

Assumptions (1.2) and (1.3) guarantee the existence and uniqueness of solutions to the linearized problem of (1.1) where a given function depends on ∇⁡u. Equation (1.4) is necessary for the existence of solutions to elliptic equations on the torus. To the best of our knowledge, prescribed mean curvature problems on the torus in the general dimension have been insufficiently studied. However, we have proved the existence of the solution under natural assumptions.

The following is the method of proof. First, we find the conditions of H for the linearized problem of (1.1), i.e.

to have a unique solution. If we add a suitable constant term for any v, the function ν⁢(∇⁡v)⋅g⁢(x,v⁢(x)) satisfies the conditions. By estimating the norm of this solution with g, the mapping T⁢(v)=u has a fixed point using a fixed-point theorem, and Theorem 1.1 follows.

2 Proof of Theorem 1.1

A theorem that holds in the Euclidean space also holds on a torus, as we consider a function on a torus to be a periodic function in the Euclidean space.

Let X⁢(𝕋n) be a function space on 𝕋n. We define a subspace Xave⁢(𝕋n)⊂X⁢(𝕋n) as

We define a mollifier as follows:

where the constant C>0 is selected such that ∫ℝn+1η=1. We define

For any f∈L2⁢(𝕋n×(-1,1)) and xn+1∈(-1+λ,1-λ),

where Bn+1⁢(x,λ) is an open ball with center x and radius λ in 𝕋n×ℝ. Moreover, for any

we define gλ:=(gλ1,…,gλn,gλn+1)=(gλ′,gλn+1).

Let us define an operator T:𝒜⁢(s)→Wave1,2⁢(𝕋n)×[-14,14] by T⁢(v)=(T1⁢(v),T2⁢(v)):=(u,cv) that satisfies (2.6), where