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Search: MSC category 35A01
( Existence problems: global existence, local existence, nonexistence )
1. CJM 2012 (vol 64 pp. 1395)
 Rodney, Scott

Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients
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.
Keywords:degenerate quadratic forms, linear equations, rough coefficients, subelliptic, weak solutions Categories:35A01, 35A02, 35D30, 35J70, 35H20 
