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

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.

### 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