Abstract view
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
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
