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.


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