Parabolic Equations Minimizing Linear Growth Functionals: L1-Theory

Abstract

Let Ω be an open bounded set in ℝN with boundary ∂Ω of class C1. We are interested in the Dirichlet problem $$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = diva\left( {x,Du} \right)in Q = \left( {0,\infty } \right) \times \Omega , \hfill \\ u\left( {t,x} \right) = \phi \left( x \right)on S = \left( {0,\infty } \right) \times \partial \Omega , \hfill \\ u\left( {0,x} \right) = u_0 \left( x \right)in x \in \Omega , \hfill \\ \end{gathered} \right. $$ (1) 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.KeywordsCauchy ProblemDirichlet ProblemRadon MeasureEntropy SolutionNonlinear Elliptic EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.