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


Published:20120908
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 nonelliptic 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 nonzero $\xi\in\mathbb{R}^n$. 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.
MSC Classifications: 
35A01, 35A02, 35D30, 35J70, 35H20 show english descriptions
Existence problems: global existence, local existence, nonexistence Uniqueness problems: global uniqueness, local uniqueness, nonuniqueness Weak solutions Degenerate elliptic equations Subelliptic equations
35A01  Existence problems: global existence, local existence, nonexistence 35A02  Uniqueness problems: global uniqueness, local uniqueness, nonuniqueness 35D30  Weak solutions 35J70  Degenerate elliptic equations 35H20  Subelliptic equations
