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.

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, set We consider the eigenvalue-eigenvector problem where 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.

We consider the class of nonlinear eigenvalue problems where 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.

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.

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.

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 prove that the Cauchy problem for the one-dimensional parabolic equations , with initial data in Hs(R), cannot be solved by an iterative scheme based on the Duhamel formula for s < −1 if (k, d) = (2, 0) and s < sc(k, d) = ½ − (2 − d)/(k − 1) otherwise. This exactly completes the positive results on the Cauchy problem in Hs(R) for these equations and shows the particularity of the case (k, d) = (2, 0), for which we prove that the critical space Hsc(R) = H−3/2(R), by standard scaling arguments, cannot be reached. Our results also hold in the periodic setting.

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]).

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 problem where 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).

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.

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 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.

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).