A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144

Veterans walk to beat back pain: study rationale, design and protocol of a randomized trial of a pedometer-based Internet mediated intervention for patients with chronic low back pain

<p>Abstract</p> <p>Background</p> <p>Chronic back pain is a significant problem worldwide and may be especially prevalent among patients receiving care in the U.S. Department of Veterans Affairs healthcare system. Back pain affects adults at all ages and is associated with disability, lost workplace productivity, functional limitations and social isolation. Exercise is one of the most effective strategies for managing chronic back pain. Yet, there are few clinical programs that use low cost approaches to help patients with chronic back pain initiate and maintain an exercise program.</p> <p>Methods/Design</p> <p>We describe the design and rationale of a randomized controlled trial to assess the efficacy of a pedometer-based Internet mediated intervention for patients with chronic back pain. The intervention uses an enhanced pedometer, website and e-community to assist these patients with initiating and maintaining a regular walking program with the primary aim of reducing pain-related disability and functional interference. The study specific aims are: 1) To determine whether a pedometer-based Internet-mediated intervention reduces pain-related functional interference among patients with chronic back pain in the short term and over a 12-month timeframe. 2) To assess the effect of the intervention on walking (measured by step counts), quality of life, pain intensity, pain related fear and self-efficacy for exercise. 3) To identify factors associated with a sustained increase in walking over a 12-month timeframe among patients randomized to the intervention.</p> <p>Discussion</p> <p>Exercise is an integral part of managing chronic back pain but to be effective requires that patients actively participate in the management process. This intervention is designed to increase activity levels, improve functional status and make exercise programs more accessible for a broad range of patients with chronic back pain.</p> <p>Trial Registration Number</p> <p>NCT00694018</p