Cauchy problem

Consider a smooth hypersurface $S\in \mathbb {R} ^{n}$ having a continuous, non-tangential direction field described by unitary vectors ${\hat {\mathbf {d} (\mathbf {x} )}},\mathbf {x} \in S$, i.e.

${\hat {\mathbf {d} (\mathbf {x} )}}\cdot {\hat {\mathbf {n} (\mathbf {x} )}}\neq 0,\forall \mathbf {x} \in S$

where ${\hat {\mathbf {n} (\mathbf {x} )}}$ is the unitary vector perpendicular to $S$.

The general Cauchy problem, consists on finding the solution $u$ of a $\kappa -$order differential equation that also satisfies the conditions:

$u(\mathbf {x} )=f_{0}(\mathbf {x} ),\mathbf {x} \in S$

${\frac {\partial ^{m}u(\mathbf {x} )}{\partial d^{m}}}=f_{m}(\mathbf {x} ),m=1,\ldots ,\kappa -1$

where $f_{m}$ are given functions defined on surface $S$ (Cauchy Surface).

See also

Cauchy boundary condition