Regularity and geometric estimates for minima of discontinuous functionals

Abstract

In this paper we study nonnegative minimizers of general degenerate elliptic functionals, $\int F(X,u,\nabla u)dX\to \min$, for variational kernels $F$ that are discontinuous in $u$ with discontinuity of order $\sim {\chi }_{\{u>0\}}$. The Euler–Lagrange equation is therefore governed by a nonhomogeneous, degenerate elliptic equation with free boundary between the positive and the zero phases of the minimizer. We show optimal gradient estimate as well as nondegeneracy of minima. We also address weak and strong regularity properties of the free boundary. We show the set $\{u>0\}$ has locally finite perimeter and that the reduced free boundary, ${\partial }_{\mathrm{red}}\{u>0\}$, has ${\mathcal{H}}^{n-1}$-total measure. For more specific problems that arise in jet flows, we show the reduced free boundary is locally the graph of a $C^{1,\gamma }$ function.