| Abstract: |
| In this talk, we will briefly introduce the free- or moving-boundary problems and their applications in many areas of mathematics and science. Then, we will focus on our recent work, where we showed a bifurcation phenomenon in a two-phase, singularly perturbed, free-boundary problem of phase transition. We show that the uniqueness of the solution breaks down for boundary values below a threshold and three solutions appear, the harmonic or trivial solution, a minimizer of the functional, and a mountain-pass solution. Then, we discuss the stability of the solutions by considering the convergence of a related evolution problem. |
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