Regularity of a free boundary in parabolic potential theory

Abstract

We study the regularity of the free boundary in a Stefan-type problem \[ Δ u − ∂ t u = χ Ω in D ⊂ R n × R , u = | ∇ u | = 0 on D ∖ Ω \Delta u - \partial _t u = \chi _\Omega \quad \text {in $D\subset \mathbb {R}^n\times \mathbb {R}$}, \qquad u = |\nabla u| = 0 \quad \text {on $D\setminus \Omega $} \] with no sign assumptions on u u and the time derivative ∂ t u \partial _t u .

C1, 1 Regularity in semilinear elliptic problems

“…Finally, we mention that the parabolic case (see [4]), which can be solved for problems such as that in Example 1, seems to introduce certain difficulties. We leave as an open question whether one can improve the above proof to adapt the main theorem to the parabolic case.…”

On the regularity of the free boundary in the parabolic obstacle problem. Application to American options

“…x,t;loc (R 2 ). We recall Lemma 2.6 in [7] (see also Lemma 5.1 in [8]): Lemma 2.1 (Non-degeneracy lemma) Under Assumption (1.6), consider a solution u of (1.4) in Q R (P 0 ). Let R ∈ (0, R), P 1 ∈ {u > 0} be such that Q − r (P 1 ) ⊂ Q R (P 0 ) for some r > 0 small enough.…”

“…Recently in [8], L. Caffarelli, A. Petrosyan and H. Shahgholian considered the case of the parabolic potential problem (i.e. with constant coefficients in any dimension and without any sign assumptions on the solution).…”

A parabolic two-phase obstacle-like equation

“…where the inequality is strict unless both 2 have polynomial growth with respect to the space variables. Then…”

“…From [13, Theorem 4.1] we infer now that each blow-up limit z at (t 1 , x 1 ) is a nonnegative backward self-similar solution. Concerning those, it has been shown in [2,Lemma 6.3] and [2, Theorem 8.1] that either z is a half-plane solution of the form…”