Existence of solutions for singular critical semilinear elliptic equation

This paper is devoted to the existence of solutions for a singular critical semilinear elliptic equation. Some existence and multiplicity results are obtained by using mountain pass arguments and analysis techniques. The results of Ding and Tang (2007) and Kang (2007) and related are improved.     

A class of stochastic differential equations with super-linear growth and non-Lipschitz coefficients

The purpose of this paper is to study some properties of solutions to one-dimensional as well as multidimensional stochastic differential equations (SDEs in short) with super-linear growth and non-Lipschitz conditions on the coefficients. Taking inspiration from [K. Bahlali, E.H. Essaky, M. Hassani, and E. Pardoux Existence, uniqueness and stability of backward stochastic differential equation with locally monotone coefficient, C.R.A.S. Paris. 335(9) (2002), pp. 757–762; K. Bahlali, E. H. Essaky, and H. Hassani, Multidimensional BSDEs with super-linear growth coefficients: Application to degenerate systems of semilinear PDEs, C. R. Acad. Sci. Paris, Ser. I. 348 (2010), pp. 677-682; K. Bahlali, E. H. Essaky, and H. Hassani, p-Integrable solutions to multidimensional BSDEs and degenerate systems of PDEs with logarithmic nonlinearities, (2010). Available at arXiv:1007.2388v1 [math.PR]], we introduce a new local condition which ensures the pathwise uniqueness, as well as the non-contact property. We moreover show that the solution produces a stochastic flow of continuous maps and satisfies a large deviations principle of Freidlin–Wentzell type. Our conditions on the coefficients go beyond the existing ones in the literature. For instance, the coefficients are not assumed uniformly continuous and therefore cannot satisfy the classical Osgood condition. The drift coefficient could not be locally monotone and the diffusion is neither locally Lipschitz nor uniformly elliptic. Our conditions on the coefficients are, in some sense, near the best possible. Our results are sharp and mainly based on Gronwall lemma and the localization of the time parameter in concatenated intervals.     

Sasakian流形上的Schur引理

根据截面曲率给出维数≥5的Sasakian流形是Sasakian空间形式的一个充分必要条件,它是Schur引理的延伸.     

Reduced limits for nonlinear equations with measures

We consider equations (E) −Δu+g(u)=μ in smooth bounded domains Ω⊂RN, where g is a continuous nondecreasing function and μ is a finite measure in Ω. Given a bounded sequence of measures (μk), assume that for each k?1 there exists a solution uk of (E) with datum μk and zero boundary data. We show that if uk→u# in L¹(Ω), then u# is a solution of (E) relative to some finite measure μ#. We call μ# the reduced limit of (μk). This reduced limit has the remarkable property that it does not depend on the boundary data, but only on (μk) and on g. For power nonlinearities g(t)=|t|^q−1t, ∀t∈R, we show that if (μk) is nonnegative and bounded in W−2,q(Ω), then μ and μ# are absolutely continuous with respect to each other; we then produce an example where μ#≠μ.     

Integrability of Solutions of Nonlinear Elliptic Equations with Right-Hand Sides from Classes Close to L 1

We establish a number of results on the integrability of the entropy and the weak solutions of the Dirichlet problem for nonlinear elliptic equations of second order depending on the integrability properties of the right-hand sides of these equations.     

Sparse partition universal graphs for graphs of bounded degree

In 1983, Chvátal, Trotter and the two senior authors proved that for any Δ there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph KN with N?Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Δ. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and edges, with N=⌈B^′n⌉ for some constant B^′ that depends only on Δ. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Δ is . Our approach is based on random graphs; in fact, we show that the classical Erd?s–Rényi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Δ.The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed.     

Cea's error estimate for strongly monotone variational inequalities

Céa's approximation lemma is extended to variational inequalities which are defined by strongly monotone operators in closed convex subsets of linear normed spaces. This abstract error estimate is applied to the finite element discretization of a nonlinear elliptic two-sided obstacle problem providing an asymptotic error estimate for a smooth enough solution.     

Markov and Artin Normal Form Theorem for Braid Groups∗

In this article, we will present the results of Artin–Markov on braid groups by using the Gröbner–Shirshov basis. As a consequence, we can reobtain the normal form of Artin–Markov–Ivanovsky as an easy corollary.     

Extensions of Jacobson's Lemma

Jacobson's Lemma, relating different kinds of non singularity of ca ? 1 and ac ? 1, extends to ba ? 1 whenever aca = aba.