Traveling Wave Solutions of Advection-Diffusion Equations with Nonlinear Diffusion

We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds c {\in}]c*, +{\infty}, where c* > 0 is explicitly computed but may not be optimal. We also prove that a free boundary hy- persurface separates a region where u = 0 and a region where u > 0, and that this free boundary can be globally parametrized as a Lipschitz continuous graph under some additional non-degeneracy hypothesis; we investigate solutions which are, in the region u > 0, planar and linear at infinity in the propagation direction, with slope equal to the propagation speed.Comment: 40 pages, 1 figur

Nonlinear porous medium flow with fractional potential pressure

We study a porous medium equation, with nonlocal diffusion effects given by an inverse fractional Laplacian operator. We pose the problem in n-dimensional space for all t>0 with bounded and compactly supported initial data, and prove existence of a weak and bounded solution that propagates with finite speed, a property that is nor shared by other fractional diffusion models.Comment: 32 pages, Late

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

A logarithmic epiperimetric inequality for the obstacle problem

For the general obstacle problem, we prove by direct methods an epiperimetric inequality at regular and singular points, thus answering a question of Weiss (Invent. Math., 138 (1999), 23--50). In particular at singular points we introduce a new tool, which we call logarithmic epiperimetric inequality, which yields an explicit logarithmic modulus of continuity on the $C^1$ regularity of the singular set, thus improving previous results of Caffarelli and Monneau

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

Report on advances for pediatricians in 2018: allergy, cardiology, critical care, endocrinology, hereditary metabolic diseases, gastroenterology, infectious diseases, neonatology, nutrition, respiratory tract disorders and surgery.

This review reported notable advances in pediatrics that have been published in 2018. We have highlighted progresses in allergy, cardiology, critical care, endocrinology, hereditary metabolic diseases, gastroenterology, infectious diseases, neonatology, nutrition, respiratory tract disorders and surgery. Many studies have informed on epidemiologic observations. Promising outcomes in prevention, diagnosis and treatment have been reported. We think that advances realized in 2018 can now be utilized to ameliorate patient car

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

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