Prescribed mean curvature equation on torus

Prescribed mean curvature problems on the torus has been considered in one dimension. In this paper, we prove the existence of a graph on the $n$-dimensional torus $\mathbb {T}^n$, the mean curvature vector of which equals the normal component of a given vector field satisfying suitable conditions for a Sobolev norm, the integrated value, and monotonicity.


Introduction
In this paper, we consider the following prescribed mean curvature problem on torus T n := R n /Z n : −div ∇u where ν is the unit normal vector of u, that is, ν(z) = 1 √ 1+|z| 2 (−z, 1). The vector field g(x, x n+1 ) : T n × R → R n+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 ⊂ R n , 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 [11] 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 [a ij (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 T n with n = 2, 3. Prescribed mean curvature problems on the one-dimensional torus = H(x, u, u ′ ) have been investigated for a wide variety of conditions H (refer to [5,7,8,9,10,14], for example).
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 φ ε ∈ W 1,2 and g ε ∈ W 1,p , with p > n+1 where W is a double-well potential such as W (φ) = (1 − φ 2 ) 2 . Then, the interface {φ ε = 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 W 1,p limit of g ε . If the surface is represented locally as a graph of a function u on T n , 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, g n+1 for the n + 1th component is monotonous, and the integrated value of g n+1 is zero.
The following theorem is the main result.
Theorem 1.1. Fix n+1 2 < p < n + 1 and q = np n+1−p . Then, there exists a constant ε 1 = ε 1 (n, p) > 0 with the following property. If ε < ε 1 , and g = (g 1 , . . . , g n , g n+1 ) = (g ′ , g n+1 ) ∈ W 1,p (T n × (−1, 1); R n+1 ) satisfies (1.4)-(1.6), then there exists a function u ∈ W 2,q (T n ) such that −div ∇u Moreover, the following inequality holds: The assumptions (1.4) and (1.5) guarantee the existence and uniqueness of solutions to the linearized problem of (1.1) where a given function depends on ∇u. (1.6) is necessary for the existence of solutions to elliptic equations on the torus. To 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 method of proof. We first find the conditions of H for the linearized problem of (1.1) −div ∇u √ 1+|∇v| 2 = H 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.

Proof of Theorem 1.1
A theorem that holds in Euclidean space also holds on a torus, as we consider a function on a torus to be a periodic function in Euclidean space.
Let X(T n ) be a function space on T n . We define a subspace X ave (T n ) ⊂ X(T n ) as Proof. We define a function B : By the Hölder inequality, we obtain Using the Poincaré inequality, we have By (2.1), (2.2), and the Lax-Milgram theorem, for any H ∈ L 2 ave (T n ), there exists a unique function u ∈ W 1,2 Thus, Theorem 2.1 follows.
We define a mollifier as follows.
(2.18) Using Lemma 2.3, we obtain Using the Sobolev inequality, we find that there exists a constant c 7 = c 7 (n, p) > 0 such that Next, we estimate the term T 1 (v) L q (T n ) . If q ≤ 2, then, by (2.2) and Lemma 2.3, we obtain If q > 2, by (2.21) and the Riesz-Thorin theorem, we obtain , we obtain For the general case of v ∈ W 2,q (T n ), suppose that {v m } m∈N ∈ C ∞ (T n ) converges to v in the sense of C 1 (T n ). By (2.24), there exists a subsequence {v m k } k∈N ⊂ {v m } m∈N such that T 1 (v m k ) converges to a function w ∞ ∈ W 2,q (T n ) in the sense of C 1 (T n ) and T 2 (v m k ) converges to a constant d ∞ ∈ − 1 4 , 1 4 . For any φ ∈ W 1,2 (T n ), we obtain (2.26) By (2.25) and (2.26), we obtain T n ∇w ∞ · ∇φ . By (2.24) and (2.27), Theorem 2.5 follows.
Next, we write the fixed-point theorem, which is needed later ([1, Theorem 1]). An operator T : X → A is considered weakly sequentially continuous if, for every sequence {x m } m∈N ⊂ X and x ∞ ∈ X such that x m weakly converges to x ∞ , T (x m ) weakly converges to T (x ∞ ).
Theorem 2.6. Let X be a metrizable, locally convex topological vector space and Ω be a weakly compact convex subset of X. Then, any weakly sequentially continuous map T : Ω → Ω has a fixed point.