\(C^{\infty}\)-regularity of the interface of the evolution \(p\)-Laplacian equation
The authors study the regularity of the interface of the evolution \(p\)-Laplacian equation \[ u_t = \text{div}(| Du| ^{p-2}Du),\qquad (x,t)\in \mathbb R^n\times [0,\infty), \] in the range of exponents \(p>2\) and \(n\geq 2\). The equation becomes degenerate when \(Du=0\), therefore solutions are not always smooth. The \(C^{\infty}\)-regularity of the interface in dimension \(n=1\) is the well-known result. In order to prove regularity properties the authors introduce into consideration a convenient ''pressure'' function \(f\) setting \[ u = (p-2)/(p-1)f^{(p-1)/(p-2)}, \] near the free boundary \(u=0\). The main goal of the article under review is to show that, under certain assumptions on the initial data, the function \(f\), defined as above in terms of \(u\), is smooth up to the interface, for time \(0<t\leq T\), for some \(T>0\). As a consequence, the free boundary is smooth. The proof is based on the idea of introducing an appropriate change of variables which transforms the free boundary problem to an initial value problem with fixed boundary.
scientific article; zbMATH DE number 1309087