Higher-Order Linearization and Regularity in Nonlinear Homogenization.

We prove large-scale C ∞ regularity for solutions of nonlinear elliptic equations with random coefficients, thereby obtaining a version of the statement of Hilbert's 19th problem in the context of homogenization. The analysis proceeds by iteratively improving three statements together: (i) the regularity of the homogenized Lagrangian  L ¯ , (ii) the commutation of higher-order linearization and homogenization, and (iii) large-scale C 0 , 1 -type regularity for higher-order linearization errors. We consequently obtain a quantitative estimate on the scaling of linearization errors, a Liouville-type theorem describing the polynomially-growing solutions of the system of higher-order linearized equations, and an explicit (heterogenous analogue of the) Taylor series for an arbitrary solution of the nonlinear equations-with the remainder term optimally controlled. These results give a complete generalization to the nonlinear setting of the large-scale regularity theory in homogenization for linear elliptic equations.

Guide to nonlinear potential estimates.

One of the basic achievements in nonlinear potential theory is that the typical linear pointwise estimates via fundamental solutions find a precise analog in the case of nonlinear equations. We give a comprehensive account of this fact and prove new unifying families of potential estimates. We also describe new fine properties of solutions to measure data problems.

Brain metabolic alterations in adolescents and young adults with fetal alcohol spectrum disorders.

BACKGROUND: Prenatal alcohol exposure affects brain structure and function. This study examined brain metabolism using magnetic resonance spectroscopy (MRS) and searched for regions of specific vulnerability in adolescents and young adults prenatally exposed to alcohol. METHODS: Ten adolescents and young adults with confirmed heavy prenatal alcohol exposure and a diagnosis within the fetal alcohol spectrum disorders (FASD) were included. Three of them had fetal alcohol syndrome (FAS), 3 had partial FAS (PFAS), and 4 had alcohol-related neurobehavioral disorder (ARND). The control group consisted of 10 adolescents matched for age, sex, head circumference, handedness, and body mass. Exclusionary criteria were learning disorders and prenatal alcohol exposure. Three-dimensional (1)H magnetic resonance spectroscopic imaging ((1)H MRSI) was performed in the cerebrum and cerebellum. Metabolite ratios N-acetylaspartate/choline (NAA/Cho), NAA/creatine (Cr) and Cho/Cr, and absolute metabolite intensities were calculated for several anatomic regions. RESULTS: In patients with FASD, lower NAA/Cho and/or NAA/Cr compared with controls were found in parietal and frontal cortices, frontal white matter, corpus callosum, thalamus, and cerebellar dentate nucleus. There was an increase in the absolute intensity of the glial markers Cho and Cr but no change in the neuronal marker NAA. CONCLUSIONS: Our results suggest that prenatal alcohol exposure alters brain metabolism in a long-standing or permanent manner in multiple brain areas. These changes are in accordance with previous findings from structural and functional studies. Metabolic alterations represent changes in the glial cell pool rather than in the neurons.

Preliminary findings of proton magnetic resonance spectroscopy in occipital cortex during sleep deprivation.

Proton magnetic resonance spectroscopy ((1)H MRS) has revealed biochemical alterations in various psychiatric disorders. Changes in brain metabolites may be caused not only by the disease's progression or response to treatment, but also by physiological variability. The aim of this study was to use (1)H MRS to assess the effects of specific short-term physiological states on major metabolites. Eight healthy women underwent (1)H MRS at the beginning and end of a 40-h period of sleep deprivation. The ratios of N-acetyl-aspartate (NAA), total creatine (tCr), and choline-containing compounds (Cho) to water (H(2)O) were determined from the occipital cortex during both baseline and photic stimulation conditions. During sleep deprivation, NAA/H(2)O decreased by 7% and Cho/H(2)O by 12%. Photic stimulation had no effect on the measured metabolites in the alert state, but in the sleep-deprived state the level of Cho/H(2)O increased during neuronal activation. The results suggest that NAA/H(2)O and Cho/H(2)O may depend on the state of alertness.