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How to find a measure from its potential. (English) Zbl 1160.31004

The authors consider the problem of finding a measure from the given values of its logarithmic potential on the support. The case studied is when the support is contained in a rectifiable curve and the measure is absolutely continuous with respect to the arclength on the curve. The generalized Laplacian is expressed by a sum of normal derivatives of the potential.
It is well known that generally, finding a measure from its potential often leads to another closely related problem of solving a singular integral equation with Cauchy kernels. The theory of such equations is well developed for smooth curves. In this paper, the author generalizes this theory to the class of Ahlfors regular curves and arcs and characterizes the bounded solutions on arcs.

MSC:

31A25 Boundary value and inverse problems for harmonic functions in two dimensions
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
31A10 Integral representations, integral operators, integral equations methods in two dimensions
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