Parabolic Equations Minimizing Linear Growth Functionals: L1-Theory
Fuensanta Andreu-Vaillo, José M. Mazón, Vicent Caselles · Birkhäuser Basel eBooks · 2004
Let Ω be an open bounded set in ℝ N with boundary ∂Ω of class C1. We are interested in the Dirichlet problem 1 $$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = diva\left( {x,Du} \right)in Q = \left( {0,\infty } \right) \times \Omega , \hfill \\ u\left( {t,x} \right) = \phi \left( x \right)on S = \left( {0,\infty } \right) \times \partial \Omega , \hfill \\ u\left( {0,x} \right) = u_0 \left( x \right)in x \in \Omega , \hfill \\ \end{gathered} \right. $$ where ϕ ∈ L1(∂Ω), u0 ∈ L1(Ω) and a(xξ) = ∇ξf(xξ), f being a function with linear growth in ‖ξ‖ as ‖ξ‖ → ∞. In the previous chapter we proved existence and uniqueness of solutions of problem (7.1) for initial data in L2 (Ω). Our aim here is to solve this problem for initial and boundary data in L1 (Ω) using the technique introduced in Chapter 5 to solve the Dirichlet problem for the total variation flow. To do that we use some techniques introduced by Bénilan et al. in [41] to get an existence and uniqueness L1-theory of solutions of nonlinear elliptic equations in divergence form when the associated variational energy has growth in |∇u| of order p with p > 1, and also the doubling variables technique introduced by Kruzhkov to prove uniqueness of scalar conservation laws. Let us give a brief description of these ideas.