## Abstract

We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the p(x)-Laplacian. Under the assumption of Lipschitz continuity of the order of the power growth p(x) > 1, we use the growth rate of the solution near the free boundary to obtain its porosity, which implies that the free boundary is of Lebesgue measure zero for p(x)-Laplacian type heterogeneous obstacle problems. Under additional assumptions on the operator heterogeneities and on data we show, in two different cases, that up to a negligible singular set of null perimeter the free boundary is the union of at most a countable family of C^{1} hypersurfaces: (i) by extending directly the finiteness of the (n - 1)-dimensional Hausdorff measure of the free boundary to the case of heterogeneous p-Laplacian type operators with constant p, 1 < p < ∞; (ii) by proving the characteristic function of the coincidence set is of bounded variation in the case of non degenerate or non singular operators with variable power growth p(x) > 1.

Original language | English |
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Pages (from-to) | 359-394 |

Number of pages | 36 |

Journal | Interfaces and Free Boundaries |

Volume | 16 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

## Keywords

- Heterogeneous p-Laplacian
- Obstacle problem
- Quasi-linear elliptic operators
- Regularity of the free boundary

## ASJC Scopus subject areas

- Applied Mathematics