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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 non-elliptic 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 non-zero $\xi\in\mathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(\Theta)=W^{1,2}(\Theta,Q)$ 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 solutionsCategories:35A01, 35A02, 35D30, 35J70, 35H20

2. CJM 2009 (vol 61 pp. 721)

Calin, Ovidiu; Chang, Der-Chen; Markina, Irina
 SubRiemannian Geometry on the Sphere $\mathbb{S}^3$ We discuss the subRiemannian geometry induced by two noncommutative vector fields which are left invariant on the Lie group $\mathbb{S}^3$. Keywords:noncommutative Lie group, quaternion group, subRiemannian geodesic, horizontal distribution, connectivity theorem, holonomic constraintCategories:53C17, 53C22, 35H20

3. CJM 2004 (vol 56 pp. 590)

Ni, Yilong
 The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary We study the Riemannian Laplace-Beltrami operator $L$ on a Riemannian manifold with Heisenberg group $H_1$ as boundary. We calculate the heat kernel and Green's function for $L$, and give global and small time estimates of the heat kernel. A class of hypersurfaces in this manifold can be regarded as approximations of $H_1$. We also restrict $L$ to each hypersurface and calculate the corresponding heat kernel and Green's function. We will see that the heat kernel and Green's function converge to the heat kernel and Green's function on the boundary. Categories:35H20, 58J99, 53C17

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