A PDE Approach to the Existence and Regularity of Surfaces of Minimum Mean Curvature Variation

We develop an analytic theory of existence and regularity of surfaces (given by graphs) arising from the geometric minimization problem

$$\begin{aligned} \min _\mathcal {M}\frac{1}{2}\int _\mathcal {M}|\nabla _{\mathcal {M}}H|^2\,{\text {d}}A, \end{aligned}$$

where \(\mathcal {M}\) ranges over all n-dimensional manifolds in \(\mathbb {R}^{n+1}\) with a prescribed boundary, \(\nabla _{\mathcal {M}}H\) is the tangential gradient along \(\mathcal {M}\) of the mean curvature H of \(\mathcal {M}\) and dA is the differential of surface area. The minimizers, called surfaces of minimum mean curvature variation, are central in applications of computer-aided design, computer-aided manufacturing and mechanics. Our main results show the existence of both smooth surfaces and of variational solutions to the minimization problem together with geometric regularity results in the case of graphs. These are the first analytic results available for this problem.