The Dirichlet Problem for the Total Variation Flow
Fuensanta Andreu-Vaillo, José M. Mazón, Vicent Caselles · Birkhäuser Basel eBooks · 2004
Suppose that Ω is an open bounded domain with a Lipschitz boundary. The purpose of this chapter is to study the Dirichlet problem $$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = div\left( {\frac{{Du}} {{\left| {Du} \right|}}} \right)in Q = \left( {0,\infty } \right) \times \Omega , \hfill \\ u\left( {t,x} \right) = \phi \left( x \right)on S = \left( {0,\infty } \right) \times \partial \Omega , \hfill \\ u\left( {0,x} \right) = u_0 \left( x \right)in x \in \Omega \hfill \\ \end{gathered} \right. $$ (5.1) where u0 ∈ L1(Ω) and ϕ ∈ L1 (∂Ω). This evolution equation is related to the gradient descent method used to solve the problem $$ \begin{gathered} Minimize \int {_\Omega \left\| {Du} \right\| + } \int {_\Omega f udx + \int {_{\partial \Omega } \left| {u - \phi } \right|d\mathcal{H}^{N - 1} } } \hfill \\ u \in BV\left( \Omega \right) \hfill \\ \end{gathered} $$ (2) where f ∈ L1(Ω), ϕ ∈ L∞ (∂Ω) (existence for this variational problem was proved in [118], Theorem 1.4).