A semilinear PDE with free boundary. (March 2017) - Fotouhi, Morteza; Shahgholian, Henrik
Abstract: We study the semilinear problem Δ u = λ + ( x ) ( u + ) q − 1 − λ − ( x ) ( u − ) q − 1 in B 1, from a regularity point of view for solutions and the free boundary ∂ ± u > 0 . Here B 1 is the unit ball, 1 < q < 2 and λ ± are Lipschitz. Our main results concern local regularity analysis of solutions and their free boundaries. One of the main difficulties encountered in studying this equation is classification of global solutions. In dimension two we are able to present a fairly good analysis of global homogeneous solutions, and hence a better understanding of the behavior of the free boundary. In higher dimensions the problem becomes quite complicated, but we are still able to state partial results; e.g. we prove that if a solution is close to one-dimensional solution in a small ball, then in an even smaller ball the free boundary can be represented locally as two C 1 -regular graphs Γ + = ∂ u > 0 and Γ − = ∂ u < 0 , tangential to each other. It is noteworthy that the above problem (in contrast to the case q = 1 ) introduces interesting and quite challenging features, that are not encountered in the case q = 1 . E.g. one obtains homogeneous global solutions that are not one-dimensional. This complicates the analysis of the free boundary substantially.