Nontransversal intersection of free and fixed boundaries for fully nonlinear elliptic operators in two dimensions

Indrei, Emanuel; Minne, Andreas

doi:10.2140/apde.2016.9.487

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Nontransversal intersection of free and fixed boundaries for fully nonlinear elliptic operators in two dimensions

Emanuel Indrei and Andreas Minne

Vol. 9 (2016), No. 2, 487–502

DOI: 10.2140/apde.2016.9.487

Abstract

In the study of classical obstacle problems, it is well known that in many configurations, the free boundary intersects the fixed boundary tangentially. The arguments involved in producing results of this type rely on the linear structure of the operator. In this paper, we employ a different approach and prove tangential touch of free and fixed boundaries in two dimensions for fully nonlinear elliptic operators. Along the way, several $n$ -dimensional results of independent interest are obtained, such as BMO-estimates, $C^{1, 1}$ -regularity up to the fixed boundary, and a description of the behavior of blow-up solutions.

Keywords

obstacle problem, tangential touch, fully nonlinear equations, nontransverse intersection, free boundary problem

Mathematical Subject Classification 2010

Primary: 35JXX, 35QXX

Secondary: 49SXX

Milestones

Received: 12 June 2015

Revised: 6 January 2016

Accepted: 9 February 2016

Published: 24 March 2016

Authors

Emanuel Indrei
Center for Nonlinear Analysis Carnegie Mellon University Pittsburgh, PA 15213 United States

Andreas Minne
Department of Mathematics KTH Royal Institute of Technology 100 44 Stockholm Sweden

Vol. 9, No. 2, 2016