Some new monotonicity theorems with applications to free boundary problems.(English)Zbl 1142.35382

This paper is devoted to some new monotonicity theorems with applications to free boundary problems. More precisely, the authors reformulate the theorem so that it applies to inhomogeneous equations in which the right-hand side of the equation need not vanish at the free boundary. Here particular interest is the case in which $$\Delta u$$ takes two different constant values on the set where $$u> 0$$ and on the set where $$u< 0$$. The new versions of the monotonicity theorem presented in the paper lead to existence of certain classical solutions to the Prandtl-Batchelor equation in two dimensions and partial regularity results in higher dimensions.

MSC:

 35J25 Boundary value problems for second-order elliptic equations 35B65 Smoothness and regularity of solutions to PDEs 35R35 Free boundary problems for PDEs
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