Cauchy problem

Consider a smooth hypersurface Sinmathbb{R}^n having a continuous, non-tangential direction field described by

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unitary vectors hat{mathbf{d}(mathbf{x})},mathbf{x}in S, i.e.

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:hat{mathbf{d}(mathbf{x})}cdothat{mathbf{n}(mathbf{x})} eq0, orallmathbf{x}in S

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

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

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: u(mathbf{x})=f_0(mathbf{x}), mathbf{x}in S

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:rac{part^m u(mathbf{x})}{part d^m}=f_m(mathbf{x}), m=1,ldots,kappa-1

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where f_m are given functions defined on surface S (Cauchy surface).

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