We study curvature structures of compact hypersurfaces in the unit sphere Sn+1(1) with two distinct principal curvatures. First of all, we prove that the Riemannian product is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies where n(n − 1)r is the scalar curvature of hypersurfaces and c2 = (n − 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product is the only compact hypersurface with non-zero mean curvature in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies This gives a partial answer for the problem proposed by Cheng.

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

We study the existence and multiplicity of positive solutions for the Dirichlet problem

where λ > 0, 1 < q < 2, p = 2* = 2N/(N − 2), 0 ε Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary ∂Ω and f is a non-negative continuous function on . Assuming that f satisfies some hypothesis, we prove that the equation admits at least three positive solutions for sufficiently small λ.

In this paper, assume that $q$ is a positive continuous function in $\mathbb{R}^{N}$ satisfying suitable conditions. We prove that the Dirichlet problem $-\Delta u+u=q(z)|u|^{p-2}u$ in an exterior domain admits at least two positive solutions and a solution which changes sign.

The existence and multiplicity of T-periodic solutions to a class of differential equations with attractive singularities at the origin are investigated in the paper. The approach is based on a new method of construction of strict upper and lower functions. The multiplicity results of Ambrosetti–Prodi type are established using a priori estimates and certain properties of topological degree.

We completely characterize the validity of the inequality $\| u \|_{Y(\mathbb R)} \leq C \| \nabla^{m} u \|_{X(\mathbb R)}$, where X and Y are rearrangement-invariant spaces, by reducing it to a considerably simpler one-dimensional inequality. Furthermore, we fully describe the optimal rearrangement-invariant space on either side of the inequality when the space on the other side is fixed. We also solve the same problem within the environment in which the competing spaces are Orlicz spaces. A variety of examples involving customary function spaces suitable for applications is also provided.

We provide sufficient conditions for the non-existence, existence and uniqueness of limit cycles surrounding a focus of a quadratic polynomial differential system in the plane.

We prove the partial Hölder continuity for minimizers of quasiconvex functionals

\begin{equation*} \mathcal{F}({\bf u}) \colon =\int_{\Omega} f(x,{\bf u},D{\bf u})\,\textrm{d} x, \end{equation*}

$f$

$x$

${\bf u}$

In this paper, we study negative classical solutions and stable solutions of the following k-Hessian equation

$$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$

We are concerned with the semi-classical states for the Choquard equation

$$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v\in H^1({\mathbb R}^N),$$

The creation and propagation of oscillations in a model for the dynamics of fine structure under viscoelastic damping is studied. It is shown that oscillations in the velocity ut are lost immediately as time evolves, while oscillations in the initial strain ux cannot be created, and they persist for all time if initially present. Uniqueness of generalized solutions (Young measures) is obtained, and a characterization of these Young measures is provided in the case of periodic modulated initial data.

Till within recent years the epiphysis cerebri has been generally regarded as a mesial outgrowth from the roof of the thalamencephalon in Vertebrates. The researches of Béraneck (5), Dendy (11), Hill (17), and Locy (19), however, tend to demonstrate the fact that this structure arises in the form of two bilateral outgrowths; while Gaskell (12) has drawn attention to its bilateral nature in Ammocœtes. Some observations which the author made on the development of the epiphysis in Amphibia (8 and 9) were found to agree in the main with those of the above-mentioned workers. The present research was therefore undertaken with the view of corroborating the results which had been obtained in the Amphibia, and it was found that these received support in the case of the chick.

In a recent paper, Professor E. T. Whittaker discussed the problem of defining, in a general riemannian space-time, the concept of spatial distance between material particles. It is the object of this paper to give an alternative definition, and to compare the new formula with that of Whittaker.

In this paper, we consider an n-dimensional semilinear equation of parabolic type with a discontinuous source term arising from combustion theory. We prove local existence for a classical solution having a ‘regular’ free boundary. In this regard, the free boundary is a surface through which the discontinuous source term exhibits a switch-like behaviour. We specify conditions under which this solution and its free boundary are global in time; moreover, we exhibit a special domain for which, for t tending to infinity, such a global-in-time solution converges, together with its free boundary, to the solution of the stationary problem and to its regular free boundary (which is proved to exist), respectively. We also prove uniqueness and continuous dependence theorems.

We define spaces of functions on an interval naturally related to functional differential equations of retarded type in the sense that exponential solutions of such equations form a uniform basis (or Riesz basis) in these spaces. Embedding theorems relating these spaces to the Sobolev spaces are established using a result on Carleson's measure defined on a half plane.

Conjectures are given for Hilbert series related to polynomial invariants of finite general linear groups: one for invariants mod Frobenius powers of the irrelevant ideal and one for cofixed spaces of polynomials.

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

We consider internal gravity waves in a stratified fluid layer with rigid horizontal boundaries and periodic boundary conditions on the sides at constant temperature with a small constant viscosity, modelled using the incompressible Navier-Stokes equations. Using operator-theoretic methods to study the damping rates of internal waves we prove there are non-oscillatory wave modes with arbitrarily small damping rates. We provide an asymptotic approximation for these non-oscillatory modes. Additionally, we find that the eigenvalues for damped oscillations are in an explicitly describable half-ring.

The Brezis–Nirenberg equation and the scalar field equation on the three-dimensional unit ball are studied. Under the Robin condition, we show the existence and uniqueness of radial solutions in a unified way. In particular, it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value.

Given differential expressions τ1; τ2, …, τn— not necessarily symmetric—which are regular on [0,∞), we investigate the relationship between the number of linearly independent L2(0,∞) solutions of the equations τjy = 0 and of the product equation (τ1τ2 … τn)y = 0. Our results extend those recently obtained in [15, 16, 17] for the special case τJ = τ for j = 1, …, n and τ is symmetric. In particular they include the classification results of Everitt and Giertz [4,5,6] for this special case when τ is a real second-order symmetric expression.