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# Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients

Published:2012-09-08
Printed: Dec 2012
• Scott Rodney,
Cape Breton University, Nova Scotia Canada
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## Abstract

This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form \begin{align*} \nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &= f+{\bf T'g} \text{ in }\Theta \\ u&=\varphi\text{ on }\partial \Theta. \end{align*} The principal part $\xi'P(x)\xi$ of the above equation is assumed to be comparable to a quadratic form ${\mathcal Q}(x,\xi) = \xi'Q(x)\xi$ that may vanish for non-zero $\xi\in\mathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(\Theta)=W^{1,2}(\Theta,Q)$ and $QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in previous works. Sawyer and Wheeden give a regularity theory for a subset of the class of equations dealt with here.
 Keywords: degenerate quadratic forms, linear equations, rough coefficients, subelliptic, weak solutions
 MSC Classifications: 35A01 - Existence problems: global existence, local existence, non-existence 35A02 - Uniqueness problems: global uniqueness, local uniqueness, non-uniqueness 35D30 - Weak solutions 35J70 - Degenerate elliptic equations 35H20 - Subelliptic equations