Prescribed mean curvature equation on torus

Yuki Tsukamoto

doi:10.1515/anly-2020-0054

Abstract

Abstract Prescribed mean curvature problems on the torus have been considered in one dimension. In this paper, we prove the existence of a graph on the n-dimensional torus 𝕋 n {\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.

Highlights

In this paper, we consider the following prescribed mean curvature problem on the torus Tn := Rn/Zn:− div( ∇u ) = ν(∇u) ⋅ g(x, u(x)) on Tn, √1 + |∇u|2 (1.1)where ν is the unit normal vector of u, that is, ν(z) = (−z, 1). √1 + |z|2The vector field g(x, xn+1) : Tn × R → Rn+1 is given, and we seek a solution u satisfying (1.1)
Prescribed mean curvature problems on the torus have been considered in one dimension
We prove the existence of a graph on the n-dimensional torus Tn, 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

Summary

Introduction

We consider the following prescribed mean curvature problem on the torus Tn := Rn/Zn:. In [13], we proved the existence of a solution only under the condition that the Sobolev norm of H is sufficiently small. 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. 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. By (2.1), (2.2), and the Lax–Milgram theorem, for any H ∈ L2ave(Tn), there exists a unique function u ∈ Wa1v,e2(Tn).

Let us define an operator

If ε

For any φ