The Cauchy problem for linear growth functionals

F. Andreu (), V. Caselles () and J. M. Mazón ()
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F. Andreu: Universitat de Valencia, Departamento de Análisis Matemático
V. Caselles: Universitat Pompeu-Fabra, Departament de Tecnologia
J. M. Mazón: Universitat de Valencia, Deptartamento de Análisis Matemático

A chapter in Nonlinear Evolution Equations and Related Topics, 2003, pp 39-65 from Springer

Abstract: Abstract In this paper we are interested in the Cauchy problem (1.1) $$ \left\{ \begin{gathered} \frac{{\partial u}}{{\partial t}} = div a (x, Du) in Q = (0,\infty ) x {\mathbb{R}^{{N }}} \hfill \\ u (0,x) = {u_{0}}(x) in x \in {\mathbb{R}^{N}}, \hfill \\ \end{gathered} \right. $$ 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.

Keywords: Cauchy Problem; Radon Measure; Entropy Solution; Linear Growth Condition; Parabolic Quasilinear Problem (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-7924-8_4

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DOI: 10.1007/978-3-0348-7924-8_4

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