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A Reflection Principle and an Orthogonal Decomposition Concernig Hypoelliptic Equations
Author(s) -
Witte Jörg
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200204)237:1<169::aid-mana169>3.0.co;2-8
Subject(s) - hypoelliptic operator , mathematics , reflection (computer programming) , decomposition , reflection principle (wiener process) , mathematical analysis , pure mathematics , algebra over a field , partial differential equation , method of characteristics , computer science , chemistry , knowledge management , innovation diffusion , diffusion process , geometric brownian motion , programming language , organic chemistry
A jump relation for a boundary integral representation of solutions of hypoelliptic equations is described by a reflection principle. An orthogonal decomposition of L 2 can be proved by the jump relation. In the orthogonal complement of the space of regular functions, i.e. the space of solutions of the homogeneous equation, the inhomogeneous adjoint equation has a solution with homogeneous boundary values. As a conclusion, one obtains Sobolev's regularity theorem. Furthermore it will be proved that the existence of the orthogonal decomposition and Sobolev's regularity theorem are equivalent. Theorems of Runge's type will be proved in order to determine countable dense subsets of the space of regular functions.