When dealing with local perimeter minimizers, volume-constrained perimeter minimizers, and minimizers in prescribed mean curvature problems, the C1,γ- regularity theory from the previous chapters provides only preliminary information on the actual degree of regularity of reduced boundaries. In Section 27.2 we prove some higher regularity theorems, which are based on the fruitful connection between Euler–Lagrange equations for variational integrals and elliptic equations in divergence form presented in Section 27.1.

Elliptic equations for derivatives of Lipschitz minimizers

A convex function f ∈ C2(ℝn) is called locally uniformly convex if for every R > 0 there exists λ(R) > 0 such that

This is the case of the area integrand with

and (M(ξ)e) · e ≥ (1 + R2)−3/2∣e∣2 for every ∣ξ∣ ≤ R and e ∈ ℝn. As turns out, the regularity of local C1,γ minimizers of an integral functional defined by a locally uniformly convex integrand f, can be investigated through the classical Schauder theory for second order elliptic equations. The starting point here is the fact, proved in Theorem 23.4, that u is a solution to the weak Euler–Lagrange equation associated with f,

Recall that, if u is twice differentiable, then (27.3) takes the form

In turn, if both f and u are smooth, then we can differentiate in the xi direction the non-linear PDE (27.4), commute div and and find

This apparently complicated PDE has in fact a nice structure.