Cauchy problem and analytic continuation for systems of first order elliptic equations with analytic coefficients

Yu, Chung Ling

doi:10.1090/S0002-9947-1973-0326162-0

Cauchy problem and analytic continuation for systems of first order elliptic equations with analytic coefficients
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Trans. Amer. Math. Soc. 185 (1973), 429-443 Request permission

Abstract:

Let a, b, c, d, f, g be analytic functions of two real variables x, y in the $z = x + iy$ plane. Consider the elliptic equation (M) $\partial u/\partial x - \partial v/\partial y = au + bv + f,\partial u/\partial y + \partial v/\partial x = cu + dv + g$. The following areas will be investigated: (1) the integral respresentations for solutions of (M) to the boundary $\partial G$ of a simply connected domain G; (2) reflection principles for (M) under nonlinear analytic boundary conditions; (3) the sufficient conditions for the nonexistence and analytic continuation for the solutions of the Cauchy problem for (M).

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