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

Published online by Cambridge University Press:  20 November 2018

Scott Rodney
Scott Rodney*
Affiliation:
Cape Breton University, Nova Scotia email: scott.rodney@gmail.com
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Abstract

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This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form

$${\nabla }'P\left( x \right)\nabla u+\text{HR}u+\text{{S}'G}u\text{+ F}u\text{=}\text{f}\,\text{+}\,\text{{T}'g}\,\text{in}\,\Theta$$

$$u=\varphi \,\,on\,\,\partial \Theta$$

The principal part ${\xi }'P\left( x \right)\xi$ of the above equation is assumed to be comparable to a quadratic form $\mathcal{Q}\left( x,\xi \right)={\xi }'Q\left( x \right)\xi$ 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 $Q{{H}^{1}}\left( \Theta \right)={{W}^{1,2}}\left( \Theta ,Q \right)$ and $QH_{0}^{1}\left( \Theta \right)=W_{0}^{1,2}\left( \Theta ,Q \right)$ as defined in previous works. E.T. Sawyer and R.L. Wheeden (2010) have given a regularity theory for a subset of the class of equations dealt with here.

Keywords

35A0135A0235D3035J7035H20degenerate quadratic formslinear equationsrough coefficientssubellipticweak solutions.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 6 , 01 December 2012 , pp. 1395 - 1414
Copyright
Copyright © Canadian Mathematical Society 2012

References

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