6533b7cefe1ef96bd1257915

RESEARCH PRODUCT

The Cauchy problem for linear growth functionals

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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 loc 1 (ℝ N ) is proved.