= Solution
The <Cauchy problem for a partial differential equation> here prescribes both the value $u|_\Gamma=g$ and the <normal derivative> $\partial_nu|_\Gamma=h$, together with $\Delta u=0$. Only one of these traces would be boundary data for a usual elliptic boundary problem, rather than full <Cauchy data>.
Every real hypersurface is a <non-characteristic hypersurface> for the <Laplace equation>. In local <real analytic> coordinates flattening the <real analytic> hypersurface $\Gamma$, the coefficient of the second transverse derivative is nonzero: its principal coefficient is the squared length of the conormal. The equation can therefore be solved for that second derivative. The <normal derivative> data determine the transverse first derivative, because the coefficient relating them is nonzero and the tangential first derivatives are already determined by $g$.
The coefficients, flattened <Cauchy data>, and coordinate change are all <real analytic>. \b[The <Cauchy-Kovalevskaya theorem> applies], giving a unique local <real analytic> solution around each point of $\Gamma$. This is a local existence assertion, not a claim of stable dependence in arbitrary <Sobolev space> norms.
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