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We develop an analytic theory of existence and regularity of surfaces arisingfrom the geometric minimization problem$$\min_{\mathcal{M}}\frac{1}{2}\int_{\mathcal{M}}|\nabla_{\mathcal{M}}H|^2\,dA$$where $\mathcal{M}$ ranges over all $n$-dimensional manifolds in$\mathbb{R}^{n+1}$ with prescribed boundary, $\nabla_{\mathcal{M}}H$ is thetangential 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 inapplications of computer-aided design, computer-aided manufacturing andmechanics. Our main results show the existence of both smooth surfaces and ofvariational solutions to the minimization problem together with geometricregularity results. These are the first analytic results available on theliterature for this problem.

A PDE approach to the existence and regularity of surfaces of minimum mean curvature variation