We study, via an asymptotic analysis, an elliptic eigenvalue problem in a 1D–1D multidomain and in a 1D–2D multidomain filled with anisotropic material. The corresponding isotropic cases were considered in a previous work by Gaudiello and Sili.

We study some properties of space-like submanifolds in Minkowski n-space, whose points are all umbilic with respect to some normal field. As a consequence of these and some results contained in a paper by Asperti and Dajczer, we obtain that being ν-umbilic with respect to a parallel light-like normal field implies conformal flatness for submanifolds of dimension n − 2 ≥ 3. In the case of surfaces, we relate the umbilicity condition to that of total semi-umbilicity (degeneracy of the curvature ellipse at every point). Moreover, if the considered normal field is parallel, we show that it is everywhere time-like, space-like or light-like if and only if the surface is included in a hyperbolic 3-space, a de Sitter 3-space or a three-dimensional light cone, respectively. We also give characterizations of total semi-umbilicity for surfaces contained in hyperbolic 4-space, de Sitter 4-space and four-dimensional light cone.

We present a comprehensive study of interpolation inequalities for periodic functions with zero mean, including the existence of and the asymptotic expansions for the extremals, best constants, various remainder terms, etc. Most attention is paid to the critical (logarithmic) Sobolev inequality in the two-dimensional case, although a number of results concerning the best constants in the algebraic case and different space dimensions are also obtained.

An adequate semigroup S is called ample if ea = a(ea)* and ae = (ae)†a for all a ∈ S and e ∈ E(S). Inverse semigroups are exactly those ample semigroups that are regular. After obtaining some characterizations of finite ample semigroups, it is proved that semigroup algebras of finite ample semigroups have generalized triangular matrix representations. As applications, the structure of the radicals of semigroup algebras of finite ample semigroups is obtained. In particular, it is determined when semigroup algebras of finite ample semigroup are semiprimitive.

The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ν), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b:= b(a) ∈ (γ, ∞), and furthermore, b(·): (0, γ) → (γ, ∞) is continuous and bijective

§ 65. Referring to § 63, we must, for the present, as time presses, leave detailed interpretation of the curves of fig. 17: merely remarking that, according to § 44, if δ=0, (which means that J is an integer), the disturbance, d, is infinitely great; of which the dynamical meaning is clear in (70) of § 39.

In a former paper on the same subject the writer pointed out that the sequence used by D. Bernoulli for approximating to the greatest root of an algebraic equation could be further utilised in such a way as to give all the roots. It is suggested in the present paper that there is really no need to compute a first Bernoullian sequence at all, but that by the theory of dual symmetric functions the coefficients in the given equation may be used with equal convenience. In a practical respect this simplifies the technique of root-evaluation.

New proofs are given for some main results in the theory of linear-quadratic optimal control.

In this paper, we study the multiplicity of positive solutions for the following semilinear elliptic equation:

\begin{alignat*}{2} -\Delta u+\lambda u&=f(x)u^{p-1}+h(x)u^{q-1} &\quad&\text{in }\mathbb{R}^{N}, \\ u&>0 &&\text{in }\mathbb{R}^{N}, \\ u&\in H^{1}(\mathbb{R}^{N}), \end{alignat*}

where $1\leq q<2<p<2^{\ast}$ ($2^{\ast}=2N/(N-2)$ if $N\geq3$ and $2^{\ast}=\infty$ if $N=1,2$), $\lambda>0,h\in L^{2/(2-q)}(\mathbb{R}^{N})\setminus\{0\}$ is non-negative and $f\in C(\mathbb{R}^{N})$. We will show how the shape of the graph of $f(x)$ affects the number of positive solutions.

The inequality (0·1) below is naturally associated with the equation −(pu′)′ + qu = λu. By assuming that one end-point of the interval (a, b) is regular and the other limit-point for this equation, Everitt characterized the best constant K in tems of spectral properties of the equation. This paper sketches a theory for more general inequalities (0·2), (0·3) similarly related to the equation Su = λTu. Here S and T are ordinary, symmetric differential expressions. A characterization of the best constants in (0·2), (0·3) is given which generalises that of Everitt.

For the case when S is of order 1 and T is multiplication by a positive function, all possible inequalities are given together with the best constants and cases of equality. Furthermore, an example is given of a valid inequality (0·1) on an interval with both end-points regular for the corresponding differential equation. This contradicts a conjecture by Everitt and Evans. Finally, the general theory for the left-definite inequality (0·3) is specialised to the case when S is a Sturm-Liouville expression. A family of examples is given for which the best constants can be explicitly calculated.

Shortly after Juhn, Faulkner and Gustavson (1931) had shown that natural differentials in the growth‐rate of individual feathers on one and the same fowl determined, in part, the threshold of their response to œstrin, the faster the growth‐rate the greater being the amount of œstrin required to produce female feathering, Lillie (1931) extended their conclusions to a re‐examination of the cases of lateral gynandromorphism in birds recorded in the literature up to that time. He pointed out that this condition seemed always to be associated with lateral size asymmetry, and argued therefore that, since the male side was the larger and presumably the faster growing, to account for lateral gynandromorphism it was necessary only to postulate that the ovary of the bird should be malfunctioning, producing a sub‐normal amount of œstrin, sufficient to evoke henny‐feathering on the smaller, more slowly growing side, but insufficient to do so on the larger, faster growing side.

We investigate the asymptotic behaviour as $k\to+\infty$ of sequences $(u_k)_{k\in\mathbb{N}}\in C^4(\varOmega)$ of solutions of the equations $\Delta^2u_k=V_k\mathrm{e}^{4u_k}$ on $\varOmega$, where $\varOmega$ is a bounded domain of $\mathbb{R}^4$ and $\lim_{k\to+\infty}V_k=1$ in $C^0_{\mathrm{loc}}(\varOmega)$. The corresponding two-dimensional problem was studied by Brézis and Merle and Li and Shafrir, who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Robert and Struwe in 2006, such a quantization does not hold in dimension $4$ for the problem in its full generality. We prove here that, under a natural hypothesis on $\Delta u_k$, we recover such a quantization as in dimension $2$.

Global existence of uniformly bounded BV entropy solutions to a 2 × 2 quasilinear system with relaxation, arising in viscoelasticity, is established by using special fractional step versions of Godunov and Glimm schemes for arbitrarily large initial data. Thanks to the uniform estimates obtained, we prove the convergence of solutions to the corresponding equilibrium limit as the relaxation parameter tends to zero.

This paper concerns the reaction-diffusion equation ut = uxx + u2(1 − u). Previous numerical solutions of this equation have demonstrated various different types of wave front solutions, generated by different initial conditions. In this paper, the authors use a phase-plane form of comparison theorems for partial differential equations (PDEs) to confirm analytically these numerical results. In particular, they show that initial conditions with an exponentially decaying tail evolve to the unique exponentially decaying travelling wave, while initial conditions with algebraically decaying tails evolve either to an algebraically decaying travelling wave, or to the exponentially decaying wave, or to a perpetually accelerating wave, dependent upon the exact form of the decay of the initial conditions. We then focus on the case of accelerating waves and investigate their form in more detail, by approximating the full equation in this case with a hyperbolic PDE, which we solve using the method of characteristics. We use this approximate solution to derive a leading-order approximation to the wave speed.

We review some recent results concerning tug-of-war games and their relation to some well-known partial differential equations (PDEs). In particular, we will show that solutions to certain PDEs can be obtained as limits of values of tug-of-war games when the parameter that controls the length of the possible movements goes to zero. Since the equations being studied are nonlinear and are not in divergence form, we will make extensive use of the concept of viscosity solutions.

For m > 0 and p ∈ (1, (N + 2)/(N − 2)), we show the uniqueness and a linearized non-degeneracy of solutions for the following problem:

Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions that are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that SBDp functions are approximately continuous -almost everywhere away from the jump set. On the negative side, we construct a function that is BD but not in BV and has distributional strain consisting only of a jump part, and one that has a distributional strain consisting of only a Cantor part.

A new formula is obtained for the number of subgroups of given index in the modular group. The formula is used to prove a recent conjecture on the parity of the number of subgroups.

The periodic quasigeostrophic equations are a coupled system of a second order elliptic equation for a streamfunction and first order hyperbolic equations for the relative potential vorticity and surface potential temperatures, on a three-dimensional domain which is periodic in both horizontal spatial co-ordinates. Such equations are used in both numerical and theoretical studies in meteorology and oceanography. In this paper Schauder estimates and a Schauder fixed point theorem are used to prove the existence and uniqueness of strong, that is classical, solutions of the periodic quasigeostrophic equations for a finite interval of time, which is inversely proportional to the sum of the norms of the initial vorticity and surface temperatures.

The weighted Lq − Lr-estimates for the Stokes flow are given in half-spaces. Furthermore, the weighted decays for the first and second spatial derivatives of the Navier-Stokes flows are also established, where the unboundedness of the projection operator is overcome by employing a decomposition for the convection term. The main results in this paper are inspired by the works of Bae and Jin.