Viscosity Solutions of Balanced Quasi-Monotone Fully Nonlinear Weakly Coupled Systems

We introduce so called balanced quasi-monotone systems. These are systems $F(x,r,p,X)=(F_1(x,r,p,X),\ldots,F_m(x,r,p,X))$, where $x$ belongs to a domain $\Omega$, $r=u(x)\in\mathbb{R}^m$, $p=Du(x)$ and $X=D^2u(x)$, that can be arranged into two categories that are mutually competitive but internally cooperative. More precisely, for all $i\neq j$ in the set $\{1,2,\ldots,m\}$, $F_j$ is monotone non-decreasing (non-increasing) in $r_i$ if and only if $F_i$ is monotone non-decreasing (non-increasing) in $r_j$ and $F_j$ is a monotone function in $r_i$. We prove the existence and uniqueness of viscosity solutions to systems of this type. For uniqueness we need to require that $F_j$ is monotone increasing in $r_j$, at an at least linear rate. This should be compared to the quasi-monotone systems studied by Ishii and Koike, where they assume that $F(x,r,p,X)\ge F(x,s,p,X)$ if $r\le s$.Comment: 10 page

Optimal Control of the Obstacle Problem in a Perforated Domain

Abstract We study the problem of optimally controlling the solution of the obstacle problem in a domain perforated by small periodically distributed holes. The solution is controlled by the choice of a perforated obstacle which is to be chosen in such a fashion that the solution is close to a given profile and the obstacle is not too irregular. We prove existence, uniqueness and stability of an optimal obstacle and derive necessary and sufficient conditions for optimality. When the number of holes increase indefinitely we determine the limit of the sequence of optimal obstacles and solutions. This limit depends strongly on the rate at which the size of the holes shrink

Application of uniform distribution to homogenization of a thin obstacle problem with p-Laplacian

In this paper we study the homogenization of p-Laplacian with thin obstacle in a perforated domain. The obstacle is defined on the intersection between a hyperplane and a periodic perforation. We construct the family of correctors for this problem and show that the solutions for the epsilon-problem converge to a solution of a minimization problem of similar form but with an extra term involving the mean capacity of the obstacle. The novelty of our approach is based on the employment of quasi-uniform convergence. As an application we obtain Poincare's inequality for perforated domains.QC 20140919. Updated from accepted to published.</p

A proposed set of metrics for standardized outcome reporting in the management of low back pain

Background and purpose - Outcome measurement has been shown to improve performance in several fields of healthcare. This understanding has driven a growing interest in value-based healthcare, where value is defined as outcomes achieved per money spent. While low back pain (LBP) constitutes an enormous burden of disease, no universal set of metrics has yet been accepted to measure and compare outcomes. Here, we aim to define such a set. Patients and methods - An international group of 22 specialists in several disciplines of spine care was assembled to review literature and select LBP outcome metrics through a 6-round modified Delphi process. The scope of the outcome set was degenerative lumbar conditions. Results - Patient-reported metrics include numerical pain scales, lumbar-related function using the Oswestry disability index, health-related quality of life using the EQ-5D-3L questionnaire, and questions assessing work status and analgesic use. Specific common and serious complications are included. Recommended follow-up intervals include 6, 12, and 24 months after initiating treatment, with optional follow-up at 3 months and 5 years. Metrics for risk stratification are selected based on preexisting tools. Interpretation - The outcome measures recommended here are structured around specific etiologies of LBP, span a patient's entire cycle of care, and allow for risk adjustment. Thus, when implemented, this set can be expected to facilitate meaningful comparisons and ultimately provide a continuous feedback loop, enabling ongoing improvements in quality of care. Much work lies ahead in implementation, revision, and validation of this set, but it is an essential first step toward establishing a community of LBP providers focused on maximizing the value of the care we deliver