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

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 boundary in two dimensions for fully nonlinear elliptic operators. Along the way, several n-dimensional results of independent interest are obtained such as BMOestimates, C 1,1 regularity up to the fixed boundary, and a description of the behavior of blow-up solutions.

Boundary Regularity and Nontransversal Intersection for the Fully Nonlinear Obstacle Problem

“…u 0 . It follows that @ x 1 u 0 .´/ D M´n and proceeding as in [17], one deduces that u 0 .x/ D ax 1 x n C z bx 2 n for a ¤ 0 and z b 2 R (up to a translation along the x 1 -axis), contradicting that u 0.…”

“…In [17] it was shown that W 2;p -solutions are C 1;1 . Furthermore, given u 2 P C r .0; M; /, the free boundary is denoted by…”

“…Note that N < 1 by C 1;1 regularity for the class P C 1 .0; M; / and the boundary condition (see [17]). By compactness, e k !…”

“…The class of uniformly bounded solutions is denoted by P C 1 .0; M; /; where kuk L 1 .B C 1 / Ä M . In what follows, the tangential touch problem is considered for D .fu ¤ 0g [ fru ¤ 0g/ \ fx n > 0g R n C : It has been conjectured that the free boundary intersects the fixed boundary nontransversally, and in two dimensions this was proved in [17] (partial results have also been obtained in [22]). The case of the Laplacian was treated in [8,24].…”

Regularity of solutions in semilinear elliptic theory

“…We consider the L 2 projection of D 2 u on the space of Hessians generated by second order homogeneous harmonic polynomials on balls with radius r > 0 and show that the projections stay uniformly bounded as r → 0 + . Although this approach has proven effective in dealing with a variety of free boundary problems [2,6,8,9], Theorem 1.1 illustrates that it is also useful in extending and refining the classical elliptic theory.…”