JournalsrmiVol. 31, No. 1pp. 69–108

Regularity and geometric estimates for minima of discontinuous functionals

  • Eduardo V. Teixeira

    Universidade Federal do Ceará, Fortaleza, Brazil
  • Raimundo Leitão

    Universidade Federal do Ceará, Fortaleza, Brazil
Regularity and geometric estimates for minima of discontinuous functionals cover
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Abstract

In this paper we study nonnegative minimizers of general degenerate elliptic functionals, F(X,u,u)dXmin\int F(X,u,\nabla u)\,dX \to \min, for variational kernels FF that are discontinuous in uu with discontinuity of order χ{u>0}\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}\{ u > 0 \} has locally finite perimeter and that the reduced free boundary, red{u>0}\partial_\mathrm{red} \{u > 0 \}, has Hn1\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 C1,γC^{1,\gamma} function.

Cite this article

Eduardo V. Teixeira, Raimundo Leitão, Regularity and geometric estimates for minima of discontinuous functionals. Rev. Mat. Iberoam. 31 (2015), no. 1, pp. 69–108

DOI 10.4171/RMI/827