Logarithmically-concave moment measures I

We discuss a certain Riemannian metric, related to the toric Kahler-Einstein equation, that is associated in a linearly-invariant manner with a given log-concave measure in R^n. We use this metric in order to bound the second derivatives of the solution to the toric Kahler-Einstein equation, and in order to obtain spectral-gap estimates similar to those of Payne and Weinberger.Comment: 27 page

Global W2,pW^{2,p} estimates for solutions to the linearized Monge--Amp\`ere equations

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Amp\`ere equations under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin's global $W^{2,p}$ estimates for the Monge-Amp\`ere equations.Comment: v2: presentation improve

Free boundary problems for Tumor Growth: a Viscosity solutions approach

The mathematical modeling of tumor growth leads to singular stiff pressure law limits for porous medium equations with a source term. Such asymptotic problems give rise to free boundaries, which, in the absence of active motion, are generalized Hele-Shaw flows. In this note we use viscosity solutions methods to study limits for porous medium-type equations with active motion. We prove the uniform convergence of the density under fairly general assumptions on the initial data, thus improving existing results. We also obtain some additional information/regularity about the propagating interfaces, which, in view of the discontinuities, can nucleate and, thus, change topological type. The main tool is the construction of local, smooth, radial solutions which serve as barriers for the existence and uniqueness results as well as to quantify the speed of propagation of the free boundary propagation

Obesity and fracture risk.

Obesity and osteoporosis are two common diseases with an increasing prevalence and a high impact on morbidity and mortality. Obese women have always been considered protected against osteoporosis and osteoporotic fractures. However, several recent studies have challenged the widespread belief that obesity is protective against fracture and have suggested that obesity is a risk factor for certain fractures. Fat and bone are linked by many pathways, which ultimately serve the function of providing a skeleton appropriate to the mass of adipose tissue it is carrying. Leptin, adiponectin, adipocytic estrogens and insulin/amylin are involved in this connection. However, excessive body fat, and particularly abdominal fat, produces inflammatory cytokines which may stimulate bone resorption and reduce bone strength. This review aimed to examine the literature data on the relationships of BMI and fat mass with factures in adult and elderly subjects. Even though the more recent studies have shown conflicting results, there is growing evidence that obesity, and particularly severe obesity, may be related to an increased risk of fracture at different skeletal sites which is partially independent from BMD. Moreover, the relationship between obesity and fracture appears to be markedly influenced by ethnicity, gender and fat distribution. Even though the incidence and the pathogenesis of fracture in obese individuals has not yet been clearly defined, the growing evidence that obesity may be related to an increased risk of fracture has important public health implications and emphasizes the need to develop effective strategies to reduce fracture risk in obese subject

Divergent effects of obesity on fragility fractures

Obesity was commonly thought to be advantageous for maintaining healthy bones due to the higher bone mineral density observed in overweight individuals. However, several recent studies have challenged the widespread belief that obesity is protective against fracture and have suggested that obesity is a risk factor for certain fractures. The effect of obesity on fracture risk is site-dependent, the risk being increased for some fractures (humerus, ankle, upper arm) and decreased for others (hip, pelvis, wrist). Moreover, the relationship between obesity and fracture may also vary by sex, age, and ethnicity. Risk factors for fracture in obese individuals appear to be similar to those in nonobese populations, although patterns of falling are particularly important in the obese. Research is needed to determine if and how visceral fat and metabolic complications of obesity (type 2 diabetes mellitus, insulin resistance, chronic inflammation, etc) are causally associated with bone status and fragility fracture risk. Vitamin D deficiency and hypogonadism may also influence fracture risk in obese individuals. Fracture algorithms such as FRAX® might be expected to underestimate fracture probability. Studies specifically designed to evaluate the antifracture efficacy of different drugs in obese patients are not available; however, literature data may suggest that in obese patients higher doses of the bisphosphonates might be required in order to maintain efficacy against nonvertebral fractures. Therefore, the search for better methods for the identification of fragility fracture risk in the growing population of adult and elderly subjects with obesity might be considered a clinical priority which could improve the prevention of fracture in obese individual

Geometric approach to nonvariational singular elliptic equations

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter $(\gamma -1)$, for $0 < \gamma < 1$, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases. We establish existence as well sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and obtain fine geometric-measure properties of the free boundary $\mathfrak{F} = \partial \{u > 0 \}$. In particular we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region $\{u > 0 \}$ and $\mathcal{H}^{n-1}$ a.e. weak differentiability property of $\mathfrak{F}$.Comment: Paper from D. Araujo's Ph.D. thesis, distinguished at the 2013 Carlos Gutierrez prize for best thesis, Archive for Rational Mechanics and Analysis 201

Porous medium equation with nonlocal pressure

We provide a rather complete description of the results obtained so far on the nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$, which describes a flow through a porous medium driven by a nonlocal pressure. We consider constant parameters $m>1$ and $0<s<1$, we assume that the solutions are non-negative, and the problem is posed in the whole space. We present a theory of existence of solutions, results on uniqueness, and relation to other models. As new results of this paper, we prove the existence of self-similar solutions in the range when $N=1$ and $m>2$, and the asymptotic behavior of solutions when $N=1$. The cases $m = 1$ and $m = 2$ were rather well known.Comment: 24 pages, 2 figure

C1,αC^{1,\alpha} regularity of solutions of degenerate fully non-linear elliptic equations

In the present paper, a class of fully non-linear elliptic equations are considered, which are degenerate as the gradient becomes small. H\"older estimates obtained by the first author (2011) are combined with new Lipschitz estimates obtained through the Ishii-Lions method in order to get $C^{1,\alpha}$ estimates for solutions of these equations.Comment: Submitte

Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations

The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an interface (Imbert, 2008) on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations. In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces. For parabolic integro-differential equations, players choose smooth functions on the whole space

Component-resolved diagnosis of hazelnut allergy in children

Hazelnuts commonly elicit allergic reactions starting from childhood and adolescence, with a rare resolution over time. The definite diagnosis of a hazelnut allergy relies on an oral food challenge. The role of component resolved diagnostics in reducing the need for oral food challenges in the diagnosis of hazelnut allergies is still debated. Therefore, three electronic databases were systematically searched for studies on the diagnostic accuracy of specific-IgE (sIgE) on hazelnut proteins for identifying children with a hazelnut allergy. Studies regarding IgE testing on at least one hazelnut allergen component in children whose final diagnosis was determined by oral food challenges or a suggestive history of serious symptoms due to a hazelnut allergy were included. Study quality was assessed by the Quality Assessment of Diagnostic Accuracy Studies-2 tool. Eight studies enrolling 757 children, were identified. Overall, sensitivity, specificity, area under the curve and diagnostic odd ratio of Cor a 1 sIgE were lower than those of Cor a 9 and Cor a 14 sIge. When the test results were positive, the post-test probability of a hazelnut allergy was 34% for Cor a 1 sIgE, 60% for Cor a9 sIgE and 73% for Cor a 14 sIgE. When the test results were negative, the post-test probability of a hazelnut allergy was 55% for Cor a 1 sIgE, 16% for Cor a9 sIgE and 14% for Cor a 14 sIgE. Measurement of IgE levels to Cor a 9 and Cor a 14 might have the potential to improve specificity in detecting clinically tolerant children among hazelnut-sensitized ones, reducing the need to perform oral food challenges