It is well known from the work of Noether that every variational symmetry of an integral functional gives rise to a corresponding conservation law. In this paper, we prove that each such conservation law arises directly as the Euler-Lagrange equation for the functional on taking suitable variations around a minimizer.

This paper is concerned with the existence of stationary solutions for some cross-diffusion systems with small parameters. Using a singular perturbation method, we prove the existence of stationary solutions with boundary and interior layers, which extends the results of Fife and Ito to the cross-diffusion cases.

We consider solutions of Lagrangian variational problems with linear constraints on the derivative. More precisely, given a smooth distribution D ⊂ TM on M and a time-dependent Lagrangian L defined on D, we consider an action functional L defined on the set ΩPQ(M, D) of horizontal curves in M connecting two fixed submanifolds P, Q ⊂ M. Under suitable assumptions, the set ΩPQ(M, D) has the structure of a smooth Banach manifold and we can thus study the critical points of L. If the Lagrangian L satisfies an appropriate hyper-regularity condition, we associate to it a degenerate Hamiltonian H on TM* using a general notion of Legendre transform for maps on vector bundles. We prove that the solutions of the Hamilton equations of H are precisely the critical points of L. In the particular case where L is given by the quadratic form corresponding to a positive-definite metric on D, we obtain the well-known characterization of the normal geodesics in sub-Riemannian geometry (see [8]). By adding a potential energy term to L, we obtain again the equations of motion for the Vakonomic mechanics with non-holonomic constraints (see [6]).

We consider the class of nonlinear eigenvalue problemswhere yp* = |y|p sgn y, pi > 0 and p0p1 … pn−1 = r, with various boundary conditions. We prove the existence of eigenvalues and study the zero properties and structure of the corresponding eigenfunctions.

Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain Ω ⊂ Rn, n ≥ 2, and a, b ∈ L∞(Ω). If the equation Lu = au+ − bu− + λu (where λ ∈ R and u±(x) = max{±u(x), 0}) has a non-trivial solution u, then λ is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are ‘simple’.

We also consider the semilinear problemwhere f : Ω × R → R is a Carathéodory function such that, for a.e. x ∈ Ω,and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

A finite combinatorial inverse semigroup Θ of moderate size is presented such that the variety of combinatorial inverse semigroups generated by Θ possesses the following properties. The lattice of all subvarieties of this variety has the cardinality of the continuum. Moreover, this semigroup Θ, and hence also the variety it generates and its subvarieties, all have E-unitary covers over any non-trivial variety of groups. This indicates that the mentioned uncountable sublattice appears quite near the bottom of the lattice of all varieties of combinatorial inverse semigroups.

Let Ai, i = 1, …, m, be a set of Ni × Ni−1 strictly totally positive (STP) matrices, with N0 = Nm = N. For a vector x = (x1, …, xN) ∈ RN and arbitrary p > 0, setWe consider the eigenvalue-eigenvector problemwhere p1 … pm−1 = r. We prove an analogue of the classical Gantmacher-Krein theorem for the eigenvalue-eigenvector structure of STP matrices in the case where pi ≥ 1 for each i, plus various extensions thereof.

In this paper, we are concerned with discrete Schauder estimates for solutions of fully nonlinear elliptic difference equations. Our estimates are discrete versions of second derivative Hölder estimates of Evans, Krylov and Safonov for fully nonlinear elliptic partial differential equations. They extend previous results of Holtby for the special case of functions of pure second-order differences on cubic meshes. As with Holtby's work, the fundamental ingredients are the pointwise estimates of Kuo-Trudinger for linear difference schemes on general meshes.

We obtain the existence and decay rates of the classical solution to the initial-value problem of a non-uniformly parabolic equation. Our method is to set up two equivalent sequences of the successive approximations. One converges to a weak solution of the initial-value problem; the other shows that the weak solution is the classical solution for t > 0. Moreover, we show how bounds of the derivatives to the classical solution depend explicitly on the interval with compact support in (0, ∞). Then we study decay rates of this classical solution.

This paper is devoted to the study of semilinear degenerate elliptic boundary-value problems arising in combustion theory that obey a general Arrhenius equation and a general Newton law of heat exchange. We prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless rate of heat production.

This paper is devoted to the study of the existence and uniqueness of almost-periodic solutions for elliptic and parabolic partial differential equations in unbounded domains. This kind of investigation had originally been motivated by the study of the so-called boundary layers, whose behaviour is crucial in the framework of periodic homogenization.

Consider the functional I(u) = ∫Ω ‖Du|n − L det Du| dx, whose energy well consists of matrices satisfying |ξ|n = L det ξ. We show that the relaxations of this functional in various Sobolev spaces are significantly different. We also make several remarks concerning various p-growth semiconvex hulls of the energy-well set and prove an attainment result for a special Hamilton-Jacobi equation, |Du|n = L det Du, in the so-called grand Sobolev space W1,q)(Ω; Rn), with q = nL/(L + 1).

We examine how symmetry and computer algebra can assist in solving the Cauchy problem for Pfaffian systems. We use recent results on integrating Frobenius integrable distributions via solvable symmetry structures to develop two techniques that when used in conjunction with symmetry determination software DIMSYM, allow us to solve the Cauchy problem for the special situation when there exists a one-dimensional Cauchy characteristic space. We also illustrate how our work can assist in extracting local solutions of a certain class of first and second order non-linear partial differential equations.

We prove the existence of energy equilibria of the di-block copolymer problem in the unit disk. They consist of concentrically layered micro-domains rich in one of the two monomer building units. We construct them by solving the proper singular limit of the free energy functional. The same limit also explains how under a dynamic law of the free energy, circular interfaces of non-equilibria may move to the origin and vanish, or collapse to each other, thereby reducing the number of layers.

An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.

Dorsey et al. [8] have constructed formal solutions for the half-space Ginzburg–Landau model, when κ is small. Dorsey et al. deduce a formal expansion for the superheating field in powers of κ½ up to order 4. In this paper, we show how the formal construction gives a natural way for constructing a subsolution for the Ginzburg–Landau system. We improve the result obtained by Bolley & Helffer [2], and take a step in the proof of the Parr Formula [13], getting two terms in the lower bound for the superheating field as κ → 0.

We consider the two-dimensional Rayleigh–Taylor problem for the dynamics of the free interface Γ between two layers of immiscible viscous liquids. For a slow flow model (which corresponds to the case of a small relative jump of density) and under sufficiently wide assumptions on the geometry of Γ, we analyze the time dynamics of Γ. In particular, we prove that its increase in time t is bounded by an exponential function with exponent independent of Γ.

The Wiener index is analysed for random recursive trees and random binary search trees in uniform probabilistic models. We obtain expectations, asymptotics for the variances, and limit laws for this parameter. The limit distributions are characterized as the projections of bivariate measures that satisfy certain fixed point equations. Covariances, asymptotic correlations, and bivariate limit laws for the Wiener index and the internal path length are given.

We show that recognizing the K3-freeness and K4-freeness of graphs is hard, respectively, for two-player nondeterministic communication protocols using exponentially many partitions and for nondeterministic syntactic read-r times branching programs.

The key ingredient is a generalization of a colouring lemma, due to Papadimitriou and Sipser, which says that for every balanced red—blue colouring of the edges of the complete n-vertex graph there is a set of εn2 triangles, none of which is monochromatic, such that no triangle can be formed by picking edges from different triangles. We extend this lemma to exponentially many colourings and to partial colourings.

A top to random shuffle of a deck of cards is performed by taking the top card off of the deck and replacing it in a randomly chosen position of the deck. We find approximations of the relative entropy of a deck of n cards after m successive top to random shuffles. Initially the relative entropy decays linearly and for larger m it decays geometrically at a rate that alters abruptly at m = n log n. It converges to an explicitly given expression when m = [n log n+cn] for a constant c.