A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type

We consider the operators $$\nabla_X\cdot(A(X)\nabla_X),\ \nabla_X\cdot(A(X)\nabla_X)-\partial_t,\ \nabla_X\cdot(A(X)\nabla_X)+X\cdot\nabla_Y-\partial_t,$$ where $X\in \Omega$, $(X,t)\in \Omega\times \mathbb R$ and $(X,Y,t)\in \Omega\times \mathbb R^m\times \mathbb R$, respectively, and where $\Omega\subset\mathbb R^m$ is a (unbounded) Lipschitz domain with defining function $\psi:\mathbb R^{m-1}\to\mathbb R$ being Lipschitz with constant bounded by $M$. Assume that the elliptic measure associated to the first of these operators is mutually absolutely continuous with respect to the surface measure $\mathrm{d} \sigma(X)$, and that the corresponding Radon-Nikodym derivative or Poisson kernel satisfies a scale invariant reverse H\"older inequality in $L^p$, for some fixed $p$, $1<p<\infty$, with constants depending only on the constants of $A$, $m$ and the Lipschitz constant of $\psi$, $M$. Under this assumption we prove that then the same conclusions are also true for the parabolic measures associated to the second and third operator with $\mathrm{d} \sigma(X)$ replaced by the surface measures $\mathrm{d} \sigma(X)\mathrm{d} t$ and $\mathrm{d} \sigma(X)\mathrm{d} Y\mathrm{d} t$, respectively. This structural theorem allows us to reprove several results previously established in the literature as well as to deduce new results in, for example, the context of homogenization for operators of Kolmogorov type. Our proof of the structural theorem is based on recent results established by the authors concerning boundary Harnack inequalities for operators of Kolmogorov type in divergence form with bounded, measurable and uniformly elliptic coefficients

Tort, Social Security, and No-Fault Schemes: Lessons from Real-World Experiments

Background Anthropometric measurements are useful in clinical practice since they are non-invasive and cheap. Previous studies suggest that sagittal abdominal diameter (SAD) may be a better measure of visceral fat depots. The aim of this study was to prospectively explore and compare how laboratory and anthropometric risk markers predicted subclinical organ damage in 255 patients, with type 2 diabetes, after four years. Methods Baseline investigations were performed in 2006 and were repeated at follow-up in 2010. Carotid intima-media thickness (IMT) was evaluated by ultrasonography and aortic pulse wave velocity (PWV) was measured with applanation tonometry over the carotid and femoral arteries at baseline and at follow-up in a cohort of subjects with type 2 diabetes aged 55–65 years old. Results There were significant correlations between apolipoprotein B (apoB) (r = 0.144, p = 0.03), C - reactive protein (CRP) (r = 0.172, p = 0.009) at baseline and IMT measured at follow-up. After adjustment for sex, age, treatment with statins and Hba1c, the associations remained statistically significant. HbA1c, total cholesterol or LDL-cholesterol did not correlate to IMT at follow-up. Baseline body mass index (BMI) (r = 0.130, p = 0.049), waist circumference (WC) (r = 0.147, p = 0.027) and sagittal Abdominal Diameter (SAD) (r = 0.184, p = 0.007) correlated to PWV at follow-up. Challenged with sex, SBP and HbA1c, the association between SAD, not WC nor BMI, and PWV remained statistically significant (p = 0.036). In a stepwise linear regression, entering both SAD and WC, the association between SAD and PWV was stronger than the association between WC and PWV. Conclusions We conclude that apoB and CRP, but not LDL-cholesterol predicted subclinical atherosclerosis. Furthermore, SAD was more independent in predicting arterial stiffness over time, compared with WC, in middle-aged men and women with type 2 diabetes.Funding Agencies|Medical Research Council of Southeast Sweden||Center for Medical Image Science and Visualization (CMIV)||Linkoping University||Futurum||King Gustaf V and Queen Victoria Freemason Foundation||GE Healthcare||Swedish Heart-Lung Foundation||Swedish Research Council Grant|12661|</p

Lethal and non-lethal effects of multiple indigenous predators on the invasive golden apple snail (Pomacea canaliculata)

1. We investigated the individual and combined effects of two predators (the climbing perch, Anabas testudineus, and the wetland crab, Esanthelphusa nimoafi) indigenous to wetlands in Laos, on the behaviour and survival of the invasive South American golden apple snail (Pomacea canaliculata). The snail is considered a pest, consuming large amounts of rice and other aquatic vegetation in the region. 2. Snail avoidance reactions to released predator chemical cues were investigated in aquaria while the effects of predators on a mixed snail population were studied in field enclosures that contained native aquatic plants (Salvinia cucullata, Ludwigia adscendens and Ipomoea aquatica). 3. In the aquaria experiment, neonate (2-3 mm) and medium-sized snails (8-10 mm) responded to fish chemical cues by going to the surface, whereas adult snails (35-40 mm) went to the bottom. In contrast, no size class of snails reacted to chemical cues released by crabs. 4. In the field experiment, fish reduced the abundance of neonate snails, and crabs reduced the abundance of all size classes. The effect of the combined predators could not be predicted from the mortality rate observed in single predator treatments. The survival of neonate and medium-sized snails was greater and of adults less than expected. The presence of predators did not affect egg production. Snails consumed significant amounts of plants despite the presence of predators. 5. Our findings suggest that some indigenous Asian predators have lethal and sublethal effects on P. canaliculata that depend on snail size and predator type. When in the presence of several predators the response of snails to one predator may either increase or decrease the vulnerability of snails to the others

Secretory leukocyte protease inhibitor in punch biopsies from human colonic mucosa.

Secretory leukocyte protease inhibitor (SLPI) is a well-known protease inhibitor. Its function is thought to be protease/protease-inhibitor balance. Free proteolytic activity, mainly pancreatic elastase, anionic trypsin and granulocytic elastase, has been demonstrated in faecal extracts from patients with ulcerative colitis. We wanted to verify that SLPI is actually secreted from normal human colonic mucosa. Also, we wanted to ascertain whether studies of SLPI secretion based on punch biopsies were dependent on biopsy area or on biopsy circumference. Normal colonic mucosa was sampled during surgery for colonic cancer. A total of 36 samples from four patients were used. Mucosa preparation was carried out using a punch biopsy technique, and samples of 3, 4 and 6 mm diameter were used. All media contained SLPI at varying concentrations. When expressed in terms of the sample area, the secretion per millimetre-squared seemed to decrease with increasing area. When calculated as secretion per circumference, secretion seemed to be constant. In conclusion, SLPI was secreted from normal human colonic mucosa. The SLPI secretion seemed dependent on the circumference of the biopsy rather than on the area of the biopsy

Green's functions for parabolic systems of second order in time-varying domains

We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and 1/2-H\"older continuous in the time variable, under the assumption that weak solutions of the system satisfy an interior H\"older continuity estimate. We also derive global pointwise estimates for Green's function in such time-varying domains under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local H\"older continuity estimate. In particular, our results apply to complex perturbations of a single real equation.Comment: 25 pages, 0 figur

The presence of elafin, SLPI, IL1-RA and STNFalpha RI in head and neck squamous cell carcinomas and their relation to the degree of tumour differentiation.

Biopsy samples of head and neck carcinomas were investigated with regard to elafin, secretory leukocyte protease inhibitor (SLPI), interleukin 1-receptor antagonist [(IL)1-RA] and soluble tumour necrosis factor alpha receptor antagonist (STNFalpha RI). SLPI and elafin are serine protease inhibitors produced in the serous cells of the upper respiratory airways and in the keratinocytes, respectively. We have now found the presence of elafin and SLPI in squamous cell carcinomas of the upper respiratory tract (tonsillar, hypopharyngeal, tongue, mouth floor, gingival and laryngeal cancer). Significantly higher amounts of SLPI and elafin are present in well-differentiated and moderately differentiated tumours than in poorly differentiated tumours (p < 0.0001 and p < 0.0015). Tumour necrosis factor-alpha and IL-1beta have been shown to stimulate the production of SLPI and elafin. Since these cytokines can both be difficult to detect, we chose to study their inhibitors, STNFalpha RI and IL1-RA, instead. IL1-RA was expressed in highly differentiated tumours as well as in poorly differentiated ones. No significant difference was seen between the groups. STNFalpha RI was only found in very small amounts, sparsely distributed in the tumours, and was not related to the degree of differentiation