- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:47
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
31
2015
1
Regularity and geometric estimates for minima of discontinuous functionals
Eduardo
Teixeira
Universidade Federal do Ceará, FORTALEZA - CEARA, BRAZIL
Raimundo
Leitão
Universidade Federal do Ceará, FORTALEZA - CEARA, BRAZIL
Discontinuous functionals, free boundary problems, degenerate elliptic equations
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.
Partial differential equations
69
108
10.4171/RMI/827
http://www.ems-ph.org/doi/10.4171/RMI/827