A parabolic two-phase obstacle-like equation

Abstract

For the parabolic obstacle-problem-like equation ∆u − ∂tu = λ + χ {u>0} − λ − χ {u<0} , where λ + and λ − are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary ∂{u > 0} ∪ ∂{u < 0} is in a neighborhood of each "branch point" the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper [11] to the parabolic case. There are substantial difficulties in the parabolic case due to the fact that the time derivative of the solution is in general not a continuous function. Our result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counterexample .

On regularity properties of solutions to the hysteresis-type problem

“…Proof. Continuity of ∂ t u across Γ * can be proved by using the same arguments as in (the proof of) Lemma 7.1 [SUW09]. For the readers convenience we sketch the details.…”

Recent Advances in Reaction-Diffusion Equations with Non-ideal Relays

“…Inequalities (15)-(17) allow one to apply methods from the theory of free boundary problems (see, e.g., [18,19]) and estimate |u t (x, t)| and |u xixj (x, t)| for a.e.…”

“…Due to [24, Theorem 2.5], problem (18), (19) admits a unique solution in the class of functions satisfying sup s∈[0,t] |u ε n (s)| ≤ Ae B|n| , n ∈ Z, t ≥ 0, with some A = A(t, ε) ≥ 0 and B = B(t, ε) ∈ R. Thus, we are now in a position to discuss the dynamics of solutions for each fixed grid step ε and analyze the limit ε → 0. First, we observe that ε in (18), (19) can be scaled out. Indeed, setting…”

Free boundaries in problems with hysteresis

“…(1) is, in fact, the two-phase parabolic free boundary problem. The properties of solutions of this two-phase problem as well as the behaviour of the corresponding free boundary were completely studied in [11]. Remark 1.3.…”