Degenerate parabolic PDEs with discontinuities and generalized evolutions of surfaces. (English) Zbl 0841.35057

Ishii, Hitoshi

Adv. Differ. Equ. 1, No. 1, 51-72 (1996).

It is considered the degenerate parabolic equation \[ u_t+ F(Du(x, t), D^2 u(x, t))= 0\quad\text{in}\quad Q_T\equiv \Omega\times (0, T),\quad \Omega\subset \mathbb{R}^n\tag{1} \] where \(F(p, X)\) having discontinuities in a continuum of directions of \(p\)’s. Existence and uniqueness of a viscosity solution of the Cauchy problem to (1) is proved. For viscosity solutions of (1) are established comparison theorem both for bounded and for unbounded domains \(\Omega\). These results give a chance to define the generalized evolution of sets by the map \(E_t\) such that \(E_t(\Gamma_0, D^+_0, D^-_0)= (\Gamma_t, D^+_t, D^-_t)\), where \(\Gamma_t= \{x\in \mathbb{R}^2: u(x, t)= 0\}\), \(D^\pm_t= \{x\in \mathbb{R}^n: \pm u(x, t)> 0\}\).
The work is illustrated by two examples with discontinuities which arise in the level set approach to evolutions of surfaces.

Reviewer: V.N.Grebenev (Novosibirsk)

MSC:

35K65	Degenerate parabolic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35R05	PDEs with low regular coefficients and/or low regular data

Keywords:

generalized evolution of sets; level set approach