The Cauchy problem for linear growth functionals

Abstract

In this paper we are interested in the Cauchy problem

$$ \left\{ \begin{gathered} \frac{{\partial u}}{{\partial t}} = div a (x, Du) in Q = (0,\infty ) x {\mathbb{R}^{{N }}} \hfill \\ u (0,x) = {u_{0}}(x) in x \in {\mathbb{R}^{N}}, \hfill \\ \end{gathered} \right. $$

((1.1))

where ${u_{0}} \in L_{{loc}}^{1}({\mathbb{R}^{N}})$ and $a(x,\xi ) = {\nabla _{\xi }}f(x,\xi ),f:{\mathbb{R}^{N}}x {\mathbb{R}^{N}} \to \mathbb{R}$being a function with linear growth as ‖ξ‖ satisfying some additional assumptions we shall precise below. An example of function f(x, ξ) covered by our results is the nonparametric area integrand $f(x,\xi ) = \sqrt {{1 + {{\left\| \xi \right\|}^{2}}}}$; in this case the right-hand side of the equation in (1.1) is the well-known mean-curvature operator. The case of the total variation, i.e., when f(ξ)= ‖ξ‖ is not covered by our results. This case has been recently studied by G. Bellettini, V. Caselles and M. Novaga in [8]. The case of a bounded domain for general equations of the form (1.1) has been studied in [3] and [4] (see also [18], [11] and [15]). Our aim here is to introduce a concept of solution of (1.1), for which existence and uniqueness for initial data in L ¹ _loc (ℝ^N) is proved.