Cauchy–Pompeiu-Type Formulas for $\bar{\partial }$ on Affine Algebraic Riemann Surfaces and Some Applications

Abstract

We present explicit solution formulas $f =\hat{ R}\varphi$and u=R_λffor the equations $\bar{\partial }f = \varphi$and $(\partial+ \lambda \mathrm{d}{z}_{1})u = f -{\mathcal{H}}_{\lambda }f$on an affine algebraic curve V⊂ℂ². Here ${\mathcal{H}}_{\lambda }f$denotes the projection of $f \in {\tilde{W}}_{1,0}^{1,\tilde{p}}(V )$to the subspace of pseudoholomorphic (1,0)-forms on V: $\bar{\partial }{\mathcal{H}}_{\lambda }f =\bar{ \lambda }\mathrm{d}\bar{{z}}_{1} \wedge {\mathcal{H}}_{\lambda }f$. These formulas can be interpreted as explicit versions and refinements of the Hodge–Riemann decomposition on Riemann surfaces. The main application consists in the construction of the Faddeev–Green function for $\bar{\partial }(\partial+ \lambda \mathrm{d}{z}_{1})$on Vas the kernel of the operator ${R}_{\lambda } \circ \hat{ R}$. This Faddeev–Green function is the main tool for the solution of the inverse conductivity problem on bordered Riemann surfaces X⊂V, that is, for the reconstruction of the conductivity function σ in the equation d(σd^c U)=0 from the Dirichlet-to-Neumann mapping U|x _bX ↦σd^c U{|} _bX . The case V=ℂwas treated by R. Novikov [N1]. In Sect. 4we give a correction to the paper [HM], in which the case of a general algebraic curve Vwas first considered.