Vol. 24, No. 3, 1968

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Reflection laws of systems of second order elliptic differential equations in two independent variables with constant coefficients

James McLean Sloss

Vol. 24 (1968), No. 3, 541–575

In this paper we shall consider the reflection of solutions of systems of equations

uxx + uyy + Aux +Buy + Cu = 0,

where u = (u1,u2,,un)T,A,B,C are constant, pairwise commutative n × n matrices, across an analytic arc κ on which the solutions satisfy n analytic linear differential boundary conditions. If the boundary conditions have coefficients which are analyiic in a specific preassigned geometrical region cantaining κ, then we shall show that the solution of (1.1) satisfying such boundary conditions can be extended across κ, provided certain inequalities are satisfied. Moreover, the region into which u can be extended will depend only on the analytic arc κ, the original region, and the coefficients of the boundary conditions; i.e., we shall have reflection “in the large” and the region will not be restricted by the equation.

Mathematical Subject Classification
Primary: 35.46
Received: 9 November 1966
Published: 1 March 1968
James McLean Sloss