Global gradient estimates for degenerate parabolic equations in nonsmooth domains

This paper studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates for the gradient. The results extend to the parabolic systems as well. The higher integrability estimates provide a useful tool in several applications.      

The number of lattice paths below a cyclically shifting boundary

We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result can be viewed as an extension of well-known enumerative formulae concerning lattice paths dominated by lines of integer slope (e.g. the generalized ballot theorem). Its proof is bijective, involving a classical “reflection” argument. Moreover, a straightforward refinement of our bijection allows for the counting of paths with a specified number of corners. We also show how the result can be applied to give elegant derivations for the number of lattice walks under certain periodic boundaries. In particular, we recover known expressions concerning paths dominated by a line of half-integer slope, and some new and old formulae for paths lying under special “staircases.”     

Solutions and Multiple Solutions for p(x)-Laplacian Equations with Nonlinear Boundary Condition

The authors study the p(x)-Laplacian equations with nonlinear boundary condition.By using the variational method,under appropriate assumptions on the perturbation terms f1(x,u),f2(x,u) and h1(x),h2(x),such that the associated functional satisfies the "mountain pass lemma" and "fountain theorem" respectively,the existence and multiplicity of solutions are obtained.The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.     

具Hardy-Sobolev界指数的非齐次椭圆方程的正解

讨论-类具Hardy-Sobolev临界指数的非齐次半线性椭圆方程,通过应用Lions集中紧性原理建立了S_μ(Q)的极小函数,再结合Ekeland变分原理、山路引理和Nehari流形的分析方法证明了方程在适当条件下正解的存在性与多重性.     

Properly coloured copies and rainbow copies of large graphs with small maximum degree

Let G be a graph on n vertices with maximum degree Δ. We use the Lovász local lemma to show the following two results about colourings χ of the edges of the complete graph K_n. If for each vertex v of K_n the colouring χ assigns each colour to at most (n ‐ 2)/(22.4Δ²) edges emanating from v, then there is a copy of G in K_n which is properly edge‐coloured by χ. This improves on a result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4), 409–433, 2003]. On the other hand, if χ assigns each colour to at most n/(51Δ²) edges of K_n, then there is a copy of G in K_n such that each edge of G receives a different colour from χ. This proves a conjecture of Frieze and Krivelevich [Electron. J. Comb. 15(1), R59, 2008]. Our proofs rely on a framework developed by Lu and Székely [Electron. J. Comb. 14(1), R63, 2007] for applying the local lemma to random injections. In order to improve the constants in our results we use a version of the local lemma due to Bissacot, Fernández, Procacci, and Scoppola [preprint, arXiv:0910.1824]. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 425–436, 2012     

Weak quasi‐randomness for uniform hypergraphs

We study quasi‐random properties of k‐uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung‐Graham‐Wilson theorem for quasi‐random graphs. Moreover, let K_k be the complete graph on k vertices and M(k) the line graph of the graph of the k‐dimensional hypercube. We will show that the pair of graphs (K_k,M(k)) has the property that if the number of copies of both K_k and M(k) in another graph G are as expected in the random graph of density d, then G is quasi‐random (in the sense of the Chung‐Graham‐Wilson theorem) with density close to d. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Some Univalence Conditions for a General Integral Operator

The authors consider the classes of the univalent functions denoted by SH(β),SP and SP(α,β).On these classes,the univalence conditions for a general integral operator are studied.