Properties of solutions of a class of planar elliptic operators with degeneracies

da Silva, P.; Meziani, A.

doi:10.1090/S0002-9939-2011-10826-8

Properties of solutions of a class of planar elliptic operators with degeneracies
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by P. L. Dattori da Silva and A. Meziani PDF

Proc. Amer. Math. Soc. 139 (2011), 3937-3949 Request permission

Abstract:

In this paper we investigate properties of solutions of first and second order elliptic equations that degenerate along a simple closed curve in $\mathbb {R}^2$. These equations are generated by a $\mathbb {C}$-valued vector field $L$. To the vector field $L$, we associate the second order operator $\mathbb {P}=\mathrm {Re}\left [L\overline {L}+p L \right ]$, where $p$ is a $\mathbb {C}$-valued function. We establish a one-to-one correspondence between the solutions of the equation $\mathbb {P}u=0$ and those of an associated first order equation of type $Lw=Aw+B\overline {w}$.

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