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

Singular solutions of fully nonlinear elliptic equations and applications

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragm\'en-Lindel\"of result as well as a principle of positive singularities in certain Lipschitz domains.Comment: 41 pages, 2 figure

Stability of flows associated to gradient vector fields and convergence of iterated transport maps

In this paper we address the problem of stability of flows associated to a sequence of vector fields under minimal regularity requirements on the limit vector field, that is supposed to be a gradient. We apply this stability result to show the convergence of iterated compositions of optimal transport maps arising in the implicit time discretization (with respect to the Wasserstein distance) of nonlinear evolution equations of a diffusion type. Finally, we use these convergence results to study the gradient flow of a particular class of polyconvex functionals recently considered by Gangbo, Evans ans Savin. We solve some open problems raised in their paper and obtain existence and uniqueness of solutions under weaker regularity requirements and with no upper bound on the jacobian determinant of the initial datum

Nonlinear Parabolic Equations arising in Mathematical Finance

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387

Mucosal Expression of Type 2 and Type 17 Immune Response Genes Distinguishes Ulcerative Colitis From Colon-Only Crohn's Disease in Treatment-Naive Pediatric Patients

Background & Aims There is controversy regarding the role of the type 2 immune response in the pathogenesis of ulcerative colitis (UC)?few data are available from treatment-naive patients. We investigated whether genes associated with a type 2 immune response in the intestinal mucosa are up-regulated in treatment-naive pediatric patients with UC compared with patients with Crohn's disease (CD)-associated colitis or without inflammatory bowel disease (IBD), and whether expression levels are associated with clinical outcomes. Methods We used a real-time reverse-transcription quantitative polymerase chain reaction array to analyze messenger RNA (mRNA) expression patterns in rectal mucosal samples from 138 treatment-naive pediatric patients with IBD and macroscopic rectal disease, as well as those from 49 children without IBD (controls), enrolled in a multicenter prospective observational study from 2008 to 2012. Results were validated in real-time reverse-transcription quantitative polymerase chain reaction analyses of rectal RNA from an independent cohort of 34 pediatric patients with IBD and macroscopic rectal disease and 17 controls from Cincinnati Children's Hospital Medical Center. Results We measured significant increases in mRNAs associated with a type 2 immune response (interleukin [IL]5 gene, IL13, and IL13RA2) and a type 17 immune response (IL17A and IL23) in mucosal samples from patients with UC compared with patients with colon-only CD. In a regression model, increased expression of IL5 and IL17A mRNAs distinguished patients with UC from patients with colon-only CD (P =.001; area under the receiver operating characteristic curve, 0.72). We identified a gene expression pattern in rectal tissues of patients with UC, characterized by detection of IL13 mRNA, that predicted clinical response to therapy after 6 months (odds ratio [OR], 6.469; 95% confidence interval [CI], 1.553?26.94), clinical response after 12 months (OR, 6.125; 95% CI, 1.330?28.22), and remission after 12 months (OR, 5.333; 95% CI, 1.132?25.12). Conclusions In an analysis of rectal tissues from treatment-naive pediatric patients with IBD, we observed activation of a type 2 immune response during the early course of UC. We were able to distinguish patients with UC from those with colon-only CD based on increased mucosal expression of genes that mediate type 2 and type 17 immune responses. Increased expression at diagnosis of genes that mediate a type 2 immune response is associated with response to therapy and remission in pediatric patients with UC

Variation in care in the management of children with Crohn's disease: Data from a multicenter inception cohort study

Background: Variation in care is common in medical practice. Reducing variation in care is shown to improve quality and increase favorable outcomes in chronic diseases. We sought to identify factors associated with variation in care in children with newly diagnosed Crohn's disease (CD). Methods: Prospectively collected data from a 28-site multicenter inception CD cohort were analyzed for variations in diagnostic modalities, treatment, and follow-up monitoring practices, along with complicated disease outcomes over 3 years in 1046 children. Generalized linear mixed effects models were used to investigate the intercenter variations in each outcome variable. Results: The mean age at diagnosis was 12 years, and 25.9% were nonwhite. The number of participants ranged from 5 to 112 per site. No variation existed in the initial diagnostic approach. When medication exposure was analyzed, steroid exposure varied from 28.6% to 96.9% (P 0.99). Use of immunomodulators (IMs) varied among centers both within 90 days (P < 0.01) and during 3 years of follow-up (P < 0.01). A significant variation was seen at the geographic level with follow-up small bowel imaging and colonoscopy surveillance after initial therapy. Conclusions: Intercenter variation in care was seen with the initial use of steroids and anti-TNF, but there was no difference in total 3-year exposure to these drugs. Variation in the initiation and long-term use of IMs was significant among sites, but further research with objective measures is needed to explain this variation of care. Small bowel imaging or repeat colonoscopy in CD patients was not uniformly performed across sites. As our data show the widespread existence of variation in care and disease monitoring at geographic levels among pediatric CD patients, future implementation of various practice strategies may help reduce the variation in care