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

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

2011-07-30

arxiv.org/abs/1108.0035v1

Mathematics

Analysis of PDEs

Scientific paper

This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form {eqnarray} \nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &=& f+{\bf T'g} \textrm{in}\Theta u&=&\phi\textrm{on}\partial \Theta.{eqnarray} The principal part $\xi'P(x)\xi$ of the above equation is assumed to be comparable to a quadratic form ${\cal Q}(x,\xi) = \xi'Q(x)\xi$ 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 $QH^1(\Theta)=W^{1,2}(\Omega,Q)$ and $QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in recent work of E. Sawyer and R. L. Wheeden. The aforementioned authors in referenced work give a regularity theory for a subset of the class of equations dealt with here.

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