A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain $\Omega\subset R^2$ the functional is $I_{\epsilon}(u)=1/2\int_{\Omega} \epsilon^{-1}|1-|Du|^2|^2+\epsilon|D^2 u|^2$ where $u$ belongs to the subset of functions in $W^{2,2}_{0}(\Omega)$ whose gradient (in the sense of trace) satisfies $Du(x)\cdot \eta_x=1$ where $\eta_x$ is the inward pointing unit normal to $\partial \Omega$ at $x$. In Jabin, Otto, Perthame characterized a class of functions which includes all limits of sequences $u_n\in W^{2,2}_0(\Omega)$ with $I_{\epsilon_n}(u_n)\to 0$ as $\epsilon_n\to 0$. A corollary to their work is that if there exists such a sequence $(u_n)$ for a bounded domain $\Omega$, then $\Omega$ must be a ball and (up to change of sign) $u:=\lim_{n\to \infty} u_n =\mathrm{dist}(\cdot,\partial\Omega)$. Recently we provided a quantitative generalization of this corollary over the space of convex domains using `compensated compactness' inspired calculations originating from the proof of coercivity of $I_{\epsilon}$ by DeSimone, Muller, Kohn, Otto. In this note we use methods of regularity theory and ODE to provide a sharper estimate and a much simpler proof for the case where $\Omega=B_1(0)$ without the requiring the trace condition on $Du$.Comment: 16 pages, 1 figur

Differential inclusions, non-absolutely convergent integrals and the first theorem of complex analysis

In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies $C^{\infty}$ regularity. As mentioned in Ahlfors's standard textbook there have been a number of studies proving this theorem without use of complex integration but at the cost of considerably more complexity. In this note we will use the theory of non-absolutely convergent integrals to firstly give a very short proof of this result without complex integration and secondly (in combination with some elements of the theory of elliptic regularity) provide a far reaching generalization

Null Lagrangian Measures in subspaces, compensated compactness and conservation laws

Compensated compactness is an important method used to solve nonlinear PDEs. A simple formulation of a compensated compactness problem is to ask for conditions on a set $\mathcal{K}\subset M^{m\times n}$ such that $$\lim_{n\rightarrow \infty} \mathrm{dist}(Du_n,\mathcal{K})\overset{L^p}{\rightarrow} 0\; \Rightarrow \{Du_{n}\}_{n}\text{ is precompact.}$$ Let $M_1,M_2,\dots, M_q$ denote the set of minors of $M^{m\times n}$. A sufficient condition for this is that any measure $\mu$ supported on $\mathcal{K}$ satisfying $$\int M_k(X) d\mu (X)=M_k\left(\int X d\mu (X)\right)\text{ for }k=1,2,\dots, q$$ is a Dirac measure. We call measures that satisfy the above equation "Null Lagrangian Measures" and we denote the set of Null Lagrangian Measures supported on $\mathcal{K}$ by $\mathcal{M}^{pc}(\mathcal{K})$. For general $m,n$, a necessary and sufficient condition for triviality of $\mathcal{M}^{pc}(\mathcal{K})$ was an open question even in the case where $\mathcal{K}$ is a linear subspace of $M^{m\times n}$. We answer this question and provide a necessary and sufficient condition for any linear subspace $\mathcal{K}\subset M^{m\times n}$. The ideas also allow us to show that for any $d\in \left\{1,2,3\right\}$, $d$-dimensional subspaces $\mathcal{K}\subset M^{m\times n}$ support non-trivial Null Lagrangian Measures if and only if $\mathcal{K}$ has Rank-$1$ connections. This is known to be false for $d\ge 4$. Using the ideas developed we are able to answer (up to first order) a question of Kirchheim, M\"{u}ller and Sverak on the Null Lagrangian measures arising in the study of a (one) entropy solution of a $2\times 2$ system of conservation laws that arises in elasticity.Comment: The results are significantly extended from previous versions 1,

On indecomposable sets with applications

In this note we show the characteristic function of every indecomposable set $F$ in the plane is $BV$ equivalent to the characteristic function a closed set $\mathbb{F}$, i.e. $||\mathbb{1}_{F}-\mathbb{1}_{\mathbb{F}}||_{BV(\mathbb{R}^2)}=0$. We show by example this is false in dimension three and above. As a corollary to this result we show that for every $\epsilon>0$ a set of finite perimeter $S$ can be approximated by a closed subset $\mathbb{S}_{\epsilon}$ with finitely many indecomposable components and with the property that $H^1(\partial^M \mathbb{S}_{\epsilon}\backslash \partial^M S)=0$ and $||\mathbb{1}_{S}-\mathbb{1}_{\mathbb{S}_{\epsilon}}||_{BV(\mathbb{R}^2)}<\epsilon$. We apply this corollary to give a short proof that locally quasiminimizing sets in the plane are $BV_l$ extension domains.Comment: 20 page

La collection "Poétique" : terre d'accueil

peer reviewe

Timing of antiretroviral therapy in Cambodian hospital after diagnosis of tuberculosis: impact of revised WHO guidelines

OBJECTIVE: To determine if implementation of 2010 World Health Organization (WHO) guidelines on antiretroviral therapy (ART) initiation reduced delay from tuberculosis diagnosis to initiation of ART in a Cambodian urban hospital. METHODS: A retrospective cohort study was conducted in a nongovernmental hospital in Phnom Penh that followed new WHO guidelines in patients with human immunodeficiency virus (HIV) and tuberculosis. All ART-naïve, HIV-positive patients initiated on antituberculosis treatment over the 18 months before and after guideline implementation were included. A competing risk regression model was used. FINDINGS: After implementation of the 2010 WHO guidelines, 190 HIV-positive patients with tuberculosis were identified: 53% males; median age, 38 years; median baseline CD4+ T-lymphocyte (CD4+ cell) count, 43 cells/µL. Before implementation, 262 patients were identified; 56% males; median age, 36 years; median baseline CD4+ cell count, 59 cells/µL. With baseline CD4+ cell counts ≤ 50 cells/µL, median delay to ART declined from 5.8 weeks (interquartile range, IQR: 3.7–9.0) before to 3.0 weeks (IQR: 2.1–4.4) after implementation (P < 0.001); with baseline CD4+ cell counts > 50 cells/µL, delay dropped from 7.0 (IQR: 5.3–11.3) to 3.6 (IQR: 2.9–5.3) weeks (P < 0.001). The probability of ART initiation within 4 and 8 weeks after tuberculosis diagnosis rose from 23% and 65%, respectively, before implementation, to 62% and 90% after implementation. A non-significant increase in 6-month retention and antiretroviral substitution was seen after implementation. CONCLUSION: Implementation of 2010 WHO recommendations in a routine clinical setting shortens delay to ART. Larger studies with longer follow-up are needed to assess impact on patient outcomes