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

Rodney, Scott

Canadian journal of mathematics, Tome 64 (2012) no. 6, pp. 1395-1414

Résumé

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.

Détail

DOI : 10.4153/CJM-2012-029-1

Mots-clés : 35A01, 35A02, 35D30, 35J70, 35H20, degenerate quadratic forms, linear equations, rough coefficients, subelliptic, weak solutions.