Search results for "Cauchy's convergence test"

The Cauchy problem for linear growth functionals

2003

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 …

Regularity of solutions of cauchy problems with smooth cauchy data

1988

Convergence in discrete Cauchy problems and applications to circle patterns

2005

A lattice-discretization of analytic Cauchy problems in two dimensions is presented. It is proven that the discrete solutions converge to a smooth solution of the original problem as the mesh size ε \varepsilon tends to zero. The convergence is in C ∞ C^\infty and the approximation error for arbitrary derivatives is quadratic in ε \varepsilon . In application, C ∞ C^\infty -approximation of conformal maps by Schramm’s orthogonal circle patterns and lattices of cross-ratio minus one is shown.