Prescribed mean curvature equation on torus (Q2031290)

To extend the prescribed mean curvature problem on the \(1\)-dimensional torus \(\mathbb T\) \[ \left(\frac{u'}{\sqrt{1+|u'|^2}}\right)'=H(x,u,u') \] to the \(n\)-dimensional torus \(\mathbb T^n\), this nice paper utilizes the Hölder inequality, a Poincare-type inequality, the Lax-Milgram theorem and a fixed-point theorem to prove that for \[(n+1)/2<p<n+1\quad\text{and}\quad q=np/(n+1-p)\] there is a constant \(\epsilon_1>0\) such that if \[ 0<\epsilon<\epsilon_1\quad\text{and}\quad g=(g',g^{n+1})\in W^{1,p}\big(\mathbb T^n\times(-1,1);\mathbb R^{n+1}\big) \] obey \[ \begin{cases} \|g\|_{W^{1,p}(\mathbb T^n\times(-1,1))}<\sqrt{\epsilon}^3,\\ \partial_{n+1}g^{n+1}(x,x^{n+1})>\sqrt{\epsilon}(\sqrt{\epsilon}+|\partial_{n+1}g'(x,x^{n+1})|,\\ \int_{\mathbb T^n}g^{n+1}(x,0) =0, \end{cases} \] then there is \(u\in W^{2,q}(\mathbb T^n)\) such that \[ \begin{cases} -\text{div}\Bigg(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\Bigg)=\nu(\nabla u)\cdot g(x,u(x))\ \ \text{on}\ \ {\mathbb T}^n,\\ \big\|u-\int_{{\mathbb T}^n}u\big\|_{W^{2,q}(\mathbb T^n)}\le\sqrt{\epsilon}. \end{cases} \]