Singularity analysis of a semilinear Bernoulli-type free boundary problem near the stagnation point

Abstract

This study presents a thorough singularity analysis of Bernoulli-type free boundary problem for semilinear elliptic equation, with a particular emphasis on the asymptotic behavior near stagnation points where the gradient of solution vanishes. The findings, derived from variational and weak solutions, rely on the monotonicity formula to construct the blow-up limit, thereby identifying that the possible singular profiles near stagnation points are constrained to corner, cusp, or flat singularity. Additionally, the application of frequency formula eliminates the possibility of flat singularity. Through a further symbol limitation at the right hand side of equation, we show that cusp singularity is impossible. The only admissible singular profile is a corner, whose angle depends on the decay rate of the solution near the stagnation point.