Nonlocal geometric expansion of convex planar curves
The expansion is described by the equation \[ V = F(t, \kappa, \gamma) \quad \text{on} \;\Gamma_t , \tag{1} \] where \(\Omega_t\) is a time dependent bounded convex set in two dimensional space; \(\Gamma_t = \partial \Omega_t,\) \(V = V_{\Gamma_t}(y)\) is the outer normal velocity of \(\Gamma_t\) at \(y \in \Gamma_t,\) \(\gamma = \gamma_{\Omega_t}(t)\) is the radius of the largest disk in \(\overline \Omega_t\) that touches \(\Gamma_t\) at \(y,\) and \(\kappa = \kappa_{\Gamma_t}(y)\) is the curvature of \(\Gamma_t\) at \(y.\) Due to the definitions \(\kappa \gamma \leq 1,\) thus the function \(F(t, \kappa, \gamma)\) need only be defined for \(\kappa \gamma \leq 1.\) The functions allowed in the model include \[ F(t, \kappa, \gamma) = {\gamma \over 3(t + a)} {3 - 2 \kappa \gamma \over 2 - \kappa \gamma} , \qquad F(t, \kappa, \gamma) = \gamma \Big(1 - {\kappa \gamma \over 2} \Big) \] which arise in the modelling of collapsing sandpiles and compression molding, respectively. Under suitable hypotheses the authors show that, given an open convex and bounded set \(\Omega\) there exists a Lipschitz continuous family of open bounded convex sets \(\{\Omega_t; t \geq 0\}\) which is a viscosity solution of (1) with initial condition \(\Omega_0 = \Omega.\) The main tool is a version of Perron's method.
scientific article; zbMATH DE number 1797563
scientific article