PhD Dissertation Defense - Nurul Raihen
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Nurul Raihen, PhD Candidate
A bifurcation Phenomenon of Regularized Free Boundary Problems of Two-Phase Elliptic-Parabolic Type
In this thesis, we prove that a bifurcation phenomenon exists in a two-phase singularly perturbed free boundary problem of phase transition. This extends the current knowledge for the one-phase problem. More precisely, the uniqueness of a solution of the two-phase problem breaks down as the boundary data decreases through a threshold value. The minimizer of the functional under consideration separates from the harmonic solution which is treated as a trivial solution in the absence of a free boundary. Moreover, we prove a third solution, a critical point of the functional being minimized, exists in this case by using the Mountain Pass Lemma. We prove convergence of the evolution with initial data, near a stable critical point to the stable solution, while the evolution deviates away from a saddle point solution of the free boundary problem as time goes by.