Large solutions of semilinear elliptic equations under the Keller–Osserman condition

AbstractWe consider the equation Δu=p(x)f(u) where p is a nonnegative nontrivial continuous function and f is continuous and nondecreasing on [0,∞), satisfies f(0)=0, f(s)>0 for s>0 and the Keller–Osserman condition ∫1∞(F(s))−1/2ds=∞ where F(s)=∫0sf(t)dt. We establish conditions on the function p that are necessary and sufficient for the existence of positive solutions, bounded and unbounded, of the given equation

A Necessary and Sufficient Condition for Uniqueness of Solutions of Singular Differential Inequalities

The author proves that the abstract differential inequality ‖ u ′ ( t ) − A ( t ) u ( t ) ‖ 2 ≤ γ [ ω ( t ) + ∫ 0 t ω ( η ) d η ] in which the linear operator A ( t ) = M ( t ) + N ( t ) , M symmetric and N antisymmetric, is in general unbounded, ω ( t ) = t − 2 ψ ( t ) ‖ u ( t ) ‖ 2 + ‖ M ( t ) u ( t ) ‖ ‖ u ( t ) ‖ and γ is a positive constant has a nontrivial solution near t = 0 which vanishes at t = 0 if and only if ∫ 0 1 t − 1 ψ ( t ) d t = ∞ . The author also shows that the second order differential inequality ‖ u ″ ( t ) − A ( t ) u ( t ) ‖ 2 ≤ γ [ μ ( t ) + ∫ 0 t μ ( η ) d η ] in which μ ( t ) = t − 4 ψ 0 ( t ) ‖ u ( t ) ‖ 2 + t − 2 ψ 1 ( t ) ‖ u ′ ( t ) ‖ 2 has a nontrivial solution near t = 0 such that u ( 0 ) = u ′ ( 0 ) = 0 if and only if either ∫ 0 1 t − 1 ψ 0 ( t ) d t = ∞ or ∫ 0 1 t − 1 ψ 1 ( t ) d t = ∞ . Some mild restrictions are placed on the operators M and N . These results extend earlier uniqueness theorems of Hile and Protter

Large Solutions of Semilinear Elliptic Equations with Nonlinear Gradient Terms

We show that large positive solutions exist for the equation ( P ± ) : Δ u ± | ∇ u | q = p ( x ) u γ in Ω ⫅ R N ( N ≥ 3 ) for appropriate choices of γ \u3e 1 , q \u3e 0 in which the domain Ω is either bounded or equal to R N . The nonnegative function p is continuous and may vanish on large parts of Ω . If Ω = R N , then p must satisfy a decay condition as | x | → ∞ . For ( P + ) , the decay condition is simply ∫ 0 ∞ t ϕ ( t ) d t \u3c ∞ , where ϕ ( t ) = max | x | = t p ( x ) . For ( P − ) , we require that t 2 + β ϕ ( t ) be bounded above for some positive β . Furthermore, we show that the given conditions on γ and p are nearly optimal for equation ( P + ) in that no large solutions exist if either γ ≤ 1 or the function p has compact support in Ω

Anisotropic Nonlinear Diffusion with Absorption: Existence and Extinction

The authors prove that the nonlinear parabolic partial differential equation ∂ u ∂ t = ∑ i , j = 1 n ∂ 2 ∂ x i ∂ x j φ i j ( u ) − f ( u ) with homogeneous Dirichlet boundary conditions and a nonnegative initial condition has a nonnegative generalized solution u . They also give necessary and sufficient conditions on the constitutive functions φ i j and f which ensure the existence of a time t 0 \u3e 0 for which u vanishes for all t ≥ t 0

A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities

The author proves that the abstract differential inequality ‖u′(t)−A(t)u(t)‖2≤γ[ω(t)+∫0tω(η)dη] in which the linear operator A(t)=M(t)+N(t), M symmetric and N antisymmetric, is in general unbounded, ω(t)=t−2ψ(t)‖u(t)‖2+‖M(t)u(t)‖‖u(t)‖ and γ is a positive constant has a nontrivial solution near t=0 which vanishes at t=0 if and only if ∫01t−1ψ(t)dt=∞. The author also shows that the second order differential inequality ‖u″(t)−A(t)u(t)‖2≤γ[μ(t)+∫0tμ(η)dη] in which μ(t)=t−4ψ0(t)‖u(t)‖2+t−2ψ1(t)‖u′(t)‖2 has a nontrivial solution near t=0 such that u(0)=u′(0)=0 if and only if either ∫01t−1ψ0(t)dt=∞ or ∫01t−1ψ1(t)dt=∞. Some mild restrictions are placed on the operators M and N. These results extend earlier uniqueness theorems of Hile and Protter

Subaru and Keck Observations of the Peculiar Type Ia Supernova 2006gz at Late Phases

Recently, a few peculiar Type Ia supernovae (SNe) that show exceptionally large peak luminosity have been discovered. Their luminosity requires more than 1 Msun of 56Ni ejected during the explosion, suggesting that they might have originated from super-Chandrasekhar mass white dwarfs. However, the nature of these objects is not yet well understood. In particular, no data have been taken at late phases, about one year after the explosion. We report on Subaru and Keck optical spectroscopic and photometric observations of the SN Ia 2006gz, which had been classified as being one of these "overluminous" SNe Ia. The late-time behavior is distinctly different from that of normal SNe Ia, reinforcing the argument that SN 2006gz belongs to a different subclass than normal SNe Ia. However, the peculiar features found at late times are not readily connected to a large amount of 56Ni; the SN is faint, and it lacks [Fe II] and [Fe III] emission. If the bulk of the radioactive energy escapes the SN ejecta as visual light, as is the case in normal SNe Ia, the mass of 56Ni does not exceed ~ 0.3 Msun. We discuss several possibilities to remedy the problem. With the limited observations, however, we are unable to conclusively identify which process is responsible. An interesting possibility is that the bulk of the emission might be shifted to longer wavelengths, unlike the case in other SNe Ia, which might be related to dense C-rich regions as indicated by the early-phase data. Alternatively, it might be the case that SN 2006gz, though peculiar, was actually not substantially overluminous at early times.Comment: 8 pages, 6 figures, 4 tables. Accepted for publication in The Astrophysical Journa

Designing all-graphene nanojunctions by covalent functionalization

We investigated theoretically the effect of covalent edge functionalization, with organic functional groups, on the electronic properties of graphene nanostructures and nano-junctions. Our analysis shows that functionalization can be designed to tune electron affinities and ionization potentials of graphene flakes, and to control the energy alignment of frontier orbitals in nanometer-wide graphene junctions. The stability of the proposed mechanism is discussed with respect to the functional groups, their number as well as the width of graphene nanostructures. The results of our work indicate that different level alignments can be obtained and engineered in order to realize stable all-graphene nanodevices

An algorithm to identify patients with treated type 2 diabetes using medico-administrative data

<p>Abstract</p> <p>Background</p> <p>National authorities have to follow the evolution of diabetes to implement public health policies. An algorithm was developed to identify patients with treated type 2 diabetes and estimate its annual prevalence in Luxembourg using health insurance claims when no diagnosis code is available.</p> <p>Methods</p> <p>The DIABECOLUX algorithm was based on patients' age as well as type and number of hypoglycemic agents reimbursed between 1995 and 2006. Algorithm validation was performed using the results of a national study based on medical data. Sensitivity, specificity and predictive values were estimated.</p> <p>Results</p> <p>The sensitivity of the DIABECOLUX algorithm was found superior to 98.2%. Between 2000 and 2006, 22,178 patients were treated for diabetes in Luxembourg, among whom 21,068 for type 2 diabetes (95%). The prevalence was estimated at 3.79% in 2006 and followed an increasing linear trend during the period. In 2005, the prevalence was low for young age classes and increased rapidly from 40 to 70 for male and 80 for female, reaching a peak of, respectively 17.0% and 14.3% before decreasing.</p> <p>Conclusions</p> <p>The DIABECOLUX algorithm is relevant to identify treated type 2 diabetes patients. It is reproducible and should be transferable to every country using medico-administrative databases not including diagnosis codes. Although undiagnosed patients and others with lifestyle recommendations only were not considered in this study, this algorithm is a cheap and easy-to-use tool to inform health authorities. Further studies will use this tool with the aim of improving the quality of health care dedicated to diabetic patients in Luxembourg.</p