In this paper maximum principles are employed to relate solutions of certain classes of nonlinear elliptic problems to solutions of the associated torsion problem. By this method a number of new isoperimetric inequalities are derived. In special cases solutions of the nonlinear problems are also related to solutions of the clamped membrane problem.

We give a unified approach to strong maximum principles for a large class of nonlocal operators of order s ∈ (0, 1) that includes the Dirichlet, the Neumann Restricted (or Regional) and the Neumann Semirestricted Laplacians.

Conditions are obtained which ensure that the maximal operator generated by the formally self-adjoint second-order differential expression

in L2(Rn), n ≥ 1 has the Dirichlet and separation properties.

A fundamental comparison theorem is established for general semilinear parabolic systems via the notions of sectorial operators, analytic semigroups and the application of the Tychonoff Fixed Point Theorem. Based on this result, we establish a maximum principle for systems of general parabolic operators and general comparison theorems for parabolic systems with quasimonotone or mixed quasimonotone nonlinearities. These results cover and extend most currently used forms of maximum principles and comparison theorems. A global existence theorem for parabolic systems is derived as an application which, in particular, gives rise to some global existence results for Fujita type systems and certain generalisations.

H-measures were recently introduced by Luc Tartar as a tool which might provide better understanding of propagating oscillations. Independently, Patrick Gerard introduced the same objects under the name of microlocal defect measures. Partial differential equations of mathematical physics can often be written in the form of a symmetric system:

where Ak and B are matrix functions, while u is an unknown vector function, and f a known vector function. In this work we prove a general propagation theorem for H-measures associated to symmetric systems. This result, combined with the localisation property, is then used to obtain more precise results on the behaviour of H-measures associated to the wave equation, Maxwell's and Dirac's systems, and second-order equations in two variables.

We study the problem of the well-posedness for the abstract Cauchy problem associated to the non-autonomous one-dimensional wave equation utt = A(t)u with general Wentzell boundary conditions Here A(t)u := (a(x, t)ux)x, a(x, t) ≥ ε > 0 in [0, 1] × [0, + ∞) and βj(t) > 0, γj(t) ≥ 0, (γ0(t), γ1(t)) ≠ (0,0). Under suitable regularity conditions on a, βj, γj we prove the well-posedness in a suitable (energy) Hilbert space

The equation to be considered is

where pi(t), 0≦i≦n, and q(t) are continuous and positive on some half-line [a, ∞). It is known that (*) always has “strictly monotone” nonoscillatory solutions defined on [a, ∞), so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of (*). In this paper sufficient conditions are given for this situation to hold for (*). The structure of the solution space of (*) is also studied.

Variational methods are used to prove the multiplicity of positive solutions for the following singular elliptic equation:

where 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary ∂ Ω, λ > 0 , , 0 ≤ s < 2, 2*(s)=2(N−s)/(N−2) and f and g are continuous functions on , that change sign on Ω.

We consider asymptotic problems for coupled equations modelling interactions between particles and a viscous fluid. The particles are driven by a Vlasov-like equation, involving the velocity of the fluid. We obtain, as certain parameters tend to 0, hydrodynamic equations for the macroscopic density and the velocity.

This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.

In this paper, a class of Boolean rings containing the class discussed in papers by Seever (1968) and Faires (1976), is defined in such a way that an extension of the classical Vitali–Hahn–Saks theorem holds for exhausting additive set functions. Some new compact topological spaces K for which C(K) is a Grothendieck space are constructed and a Nikodym type theorem is deduced from it. The Boolean algebras of Seever and Faires and those we study here are defined by ‘interpolation properties’ between disjoint sequences in the algebra. We give an example at the end of the paper that illustrates the difficulties arising when we try to find a larger class of Boolean algebras, defined in terms of such properties, for which the Vitali–Hanh–Saks theorem holds.

W1,p-gradient Young measures supported in the set Q2(K) of two-dimensional K-quasiconformal matrices are studied. We prove that these Young measures can be generated by gradients of K-quasiregular mappings. This leads, for example, to the 0-1 law for quasiregular W1,p-gradient Young measures and other quasiregular properties such as higher integrability.

For a semilinear second order differential equation on (0, ∞), conditions are given for the bifurcation and asymptotic bifurcation in Lp of solutions to the Neumann problem. Bifurcation occurs at the lowest point of the spectrum of the linearised problem. Under stronger hypotheses, there is a global branch of solutions. These results imply similar conclusions for the same equation on R with appropriate symmetry.

A characterisation is obtained of all the regularly solvable operators and their adjoints generated by a general differential expression in . The domains of these operators are described in terms of boundary conditions involving the solutions of Mu = λwu and the adjoint equation . The results include those of Sun Jiong [15] concerning self-adjoint realisations of a symmetric M when the minimal operator has equal deficiency indices: if the deficiency indices are unequal the maximal symmetric operators are determined by the results herein. Another special case concerns the J -self-adjoint operators, where J denotes complex conjugation, and for this we recover the results of Zai-jiu Shang in [16].

I Fear that most, on reading the above title, will think it is either a printer's or an author's error, and that the title ought to have been “The Sun as a Fog Disperser.” We are familiar with the sun's power of clearing away fogs; we so often see fogs clear away as the sun rises, that we seem to have shut our eyes to another side of the question, which is, that under certain conditions the very opposite of this happens, the air thickening and fogging as the sun rises. Some years ago it was noticed at Falkirk, especially during the winter months, that on many mornings on which the air was clear before sunrise it gradually thickened to a dense haze, or to a dense fog, while in pure country air the observer never noticed similar changes. The question naturally suggested itself, what was the cause of this difference between the air at Falkirk and that of the country? A series of meteorological observations was therefore begun to ascertain whether this occasional fogging after sunrise was merely an accident, or if it had any connection with the sun; that is, to ascertain whether its appearance was what we call fortuitous, or if it appeared according to any law in nature.

We prove the following: suppose that a and b are complex numbers, with a non-zero, and that n is an integer not less than 5. Then, if F is a family of functions meromorphic in a plane domain Dsuch that, for each f in F, the equation

has no solutions in D, then F is normal in D.

In this paper, we prove that solutions of the anisotropic Allen–Cahn equation in doubleobstacle form

where A is a strictly convex function, homogeneous of degree two, converge to an anisotropic mean-curvature flow

when this equation admits a smooth solution in ℝn. Here VN and R respectively denote the normal velocity and the second fundamental form of the interface, and More precisely, we show that the Hausdorff-distance between the zero-level set of φ and the interface of the above anisotropic mean-curvature flow is of order O(ε2).

This paper deals with optimal control problems governed by quasilinear parabolic equations in divergence form, whose cost functional is of Lagrangian type. Our aim is to prove the existence of solutions and derive some optimality conditions. To attain this second objective, we accomplish the sensitivity analysis of the state equation with respect to the control, proving that, under some assumptions, this relation is Gâteaux differentiable. Finally, a regularising procedure along with Ekeland's variational principle allow us to treat some other problems for which this differentiability property cannot be stated.

Poincaré's theorem on the sum of the indices of a plane autonomous differential equation at its critical points inside a periodic orbit is here extended to the periodic orbits of higher-dimensional equations under certain conditions. Under the same conditions, higher-dimensional extensions are obtained of some theorems of Bendixson and others which exclude periodic orbits from regions in which the differential equation haspositive divergence. The application of these results to feedback control equations is also considered.

It is shown that the Cesàro averaging operatorℜα > – 1, satisfies an inequality which immediately implies that it is bounded on certain Hardy spaces including Hp, 0 < p < ∞. This answers an open question of Stempak, who introduced these operators and obtained their boundedness on Hp, 0 < p ≦ 2, for ℜα ≧ 0. The operator which is conjugate to on H2 is also shown to be bounded on Hp for 1 < p < ∞ and ℜα = – 1. This extends a result of Stempak who obtained this boundedness for 2 ≦ p≦ ∞ and ℜα ≧:0.