Nonanalytic solutions of certain linear PDEs

Author:
E. C. Zachmanoglou

Journal:
Trans. Amer. Math. Soc. **277** (1983), 805-814

MSC:
Primary 35A07; Secondary 35B65

MathSciNet review:
694389

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Abstract: It is shown that if is a linear partial differential operator with analytic coefficients, and if is an analytic submanifold of codimensions in , which is partially characteristic with respect to and satisfies certain additional conditions, then one can find, in a neighborhood of any point of , solutions of the equation which are flat or singular precisely on . The additional condition requires that a nonhomogeneous Laplace equation in two variables possesses a solution with a strong extremum at the origin. The right side of this nonhomogeneous equation is a homogeneous polynomial in two variables with coefficients being repeated Poisson brackets of the real and imaginary parts of the principal symbol of .

**[1]**M. S. Baouendi, F. Trevés, and E. C. Zachmanoglou,*Flat solutions and singular solutions of homogeneous linear partial differential equations with analytic coefficients*, Duke Math. J.**46**(1979), no. 2, 409–440. MR**534059****[2]**Lars Hörmander (ed.),*Seminar on Singularities of Solutions of Linear Partial Differential Equations*, Annals of Mathematics Studies, vol. 91, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. Held at the Institute for Advanced Study, Princeton, N.J., 1977/78. MR**547013****[3]**E. C. Zachmanoglou,*Manifolds with arbitrary submanifolds of nonanalyticity of solutions of linear PDEs*, Comm. Partial Differential Equations**5**(1980), no. 3, 225–243. MR**562543**, 10.1080/03605308008820139**[4]**E. C. Zachmanoglou,*Manifolds of nonanalyticity of solutions of certain linear PDEs*, Trans. Amer. Math. Soc.**266**(1981), no. 2, 573–582. MR**617552**, 10.1090/S0002-9947-1981-0617552-1

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1983-0694389-0

Article copyright:
© Copyright 1983
American Mathematical Society