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On first and second order planar elliptic equations with degeneracies. (English) Zbl 1250.35114
Mem. Am. Math. Soc. 1019, iii, 77 p. (2012).
In this monograph, the author studies the properties of solutions of first and second order equations in the plane. These equations are generated by a complex vector field \(X\) that is elliptic everywhere except along a simple closed curve. The technique used by the author consists to give a thorough study of the operator \(\mathcal{L}\) defined by a unique vector field conjugated to \(X\). The main properties of solutions to \(\mathcal{L}u=0\) or the nonhomogeneous equation are established in a neighborhood of the degeneracy curve through integral and series representations. An application to a class of second order elliptic operators with punctual singularity is given.

MSC:
35J70 Degenerate elliptic equations
35F05 Linear first-order PDEs
35G20 Nonlinear higher-order PDEs
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