The Dirichlet problem for the total variation flow (Q5931948)

The following Cauchy-Dirichlet problem (connected with minimization of total variation) is studied NEWLINE\[NEWLINE {\partial u\over\partial t}= \text{div}\left({Du\over |Du|}\right) \quad \text{in } (0,+\infty)\times\Omega; NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(0,x)=u_0(x)\in L^1(\Omega); \quad u(t,x)=\varphi(x), \;x\in\partial\Omega. NEWLINE\]NEWLINE The authors introduce the notion of entropy solution to this problem. Using the nonlinear semigroup theory they prove existence of the entropy solution. Uniqueness and the comparison principle are established by Kruzhkov's method of doubling variables. In the case of nonnegative initial and boundary data the entropy solution is proved to be a strong solution.