Abstract
Boundary value problems are formulated on infinite-genus surfaces. These are solved for a variety of boundary conditions. The symbol calculus for differential operators is developed further for solution of parabolic differential equations at infinite genus.
Highlights
The theory of partial differential equations with given boundary conditions has developed from series solutions and integral transforms to an operator calculus (Hormander, 1985)
The matrix algebra formed from the symbols representing a set of operators for a pseudodifferential equation defined at the interior and the boundary can be used to evaluate the inverse for elliptic boundary value problems (Boutet de Monvel, 1971)
It had been proven that an isomorphism existed between a normed space of solutions to elliptic boundary value problems with these boundary conditions in the complex plane on the real line to a normed space of solutions to a corresponding parabolic differential equation (Agronovich and Vishik, 1964)
Summary
The theory of partial differential equations with given boundary conditions has developed from series solutions and integral transforms to an operator calculus (Hormander, 1985). The path to ideal boundary will be parameterized by a coordinate t tending to infinity, and the mapping from infinity to the origin can have an image that is a discrete set or the real line When it is not the real line, by the uniformization of surfaces of genus g ≥ 2, there is a formalism based on the automorphic functions defined on the entire upper half plane instead of a fundamental region for a Fuchsian group, and boundary conditions may be specified on the real line.
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