In this paper we study Mumford–Shah-type functionals associated with doubling metric measures or strong A∞ weights in the setting of the perimeter theory in the sense of Ambrosio and Miranda in metric spaces. We prove an existence theorem in a suitably defined class of special BV functions.

We investigate some properties of solutions of the Degasperis–Procesi equation, which is an approximation to the incompressible Euler equation in shallow water theory. Sufficient conditions for wave breaking are found both on an infinite line and in a periodic domain by the method of characteristics. Moreover, we show that the solution enjoys the same decay property as the initial data. Finally, the weak and strong limits, respectively, of the solution as the dispersive parameter goes to zero are investigated.

Existence and uniqueness theorems are established for dual trigonometric equations having right-hand sides that are given functions of bounded variation. The first equation in each pair has coefficients, say {Jn(n + h)} or (jn(n + h – ½)}, and the second equation coefficients {jn)}, where h is a nonnegative constant. A potential problem involving mixed boundary conditions of first and third kind is associated with each dual series. The potential problem is analysed using a stepwise perturbation procedure involving solutions in powers of h. The analysis demonstrates that the present dual series problem can be resolved if the dual series problem associated with the case h = 0 is solvable, the latter being a result obtained earlier.

The blow-up behaviour of radially symmetric classical solutions to the quasilinear parabolic equation

is analysed assuming k(u) and Q(u) are small perturbations of {k, Q} ≡ {1, up},p > 1. Moreover, it is proved that the asymptotic behaviour near blowup of solutions to the semilinear equation ut = Δu + up, and in particular the final-time profile, is stable with respect to small quasilinear perturbations of the elliptic operator.

Certain representations of the alternating group of degree 6 over the field of 2 elements are shown to be completely reducible.

By looking for critical points of functionals defined in some subspaces of , invariant under some subgroups of O (N), we prove the existence of many positive non-radial solutions for the following semilinear elliptic problem involving critical Sobolev exponent on an annulus, where 2* − 1 := (N + 2)/(N − 2) (N ≥ 4), the domain is an annulus and f : R+ × R+ → R is a C1 function, which is a subcritical perturbation.

This paper examines the Cauchy problem for a viscoelastic model with relaxation

with discontinuous, large initial data, where ½ ≦ μ <1, δ > 0 are constants. We first give a definition of admissible (or entropic) solutions to the system. Under this definition, we prove the existence, uniqueness and continuous dependence of the global admissible solution for the system. Our methods are essentially due to Kruzkov, and the requirement that f(u) is not badly degenerate (more precisely, meas {x: f″(x) = 0} = 0), needed previously when considering the global existence problem for the same system, is removed.

In this paper we prove the existence of classical solutions for all t ≧ 0 for parabolic equations u′ + A(t)u = –f(u, ∇y, …, ∇2m–2u) of arbitrary order. 2m is the order of the elliptic principal part. f must satisfy some monotonicity and growth conditions. Moreover, similar results are also valid for semilinear elliptic equations.

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

An initial–boundary-value problem for a class of wave equations with nonlinear damping and source terms in a bounded domain is considered. We establish the non-existence result of global solutions with the initial energy controlled above by a critical value via the method introduced in a work by Autuori et al. in 2010. This improves the 2009 result of Liu and Wang.

In this paper, we consider the chemotaxis-Navier-Stokes model with realistic boundary conditions matching the experiments of Hillesdon, Kessler et al. in a two-dimensional periodic strip domain. For the lower boundary, we impose the usual homogeneous Neumann-Neumann-Dirichlet boundary condition. While, for the upper boundary, since it is open to the atmosphere, we consider three kinds of different mixed non-homogeneous boundary conditions, that is, (i) Neumann-Dirichlet-Navier slip boundary condition; (ii) Zero flux-Dirichlet-Navier slip boundary condition; (iii) Zero flux-Robin-Navier slip boundary condition. For boundary conditions (i) and (iii), the existence and uniqueness of global classical solutions for any initial data and any large chemotactic sensitivity coefficient is established, and for boundary condition (ii), the existence and uniqueness of global classical solutions for any initial data and small chemotactic sensitivity coefficient is proved.

This paper investigates the effect on the structure of a band of imposing conditions on the congruence or right congruence lattice of the band. Bands whose congruence lattices are distributive or Boolean are characterized, as are bands whose right congruence lattices are Boolean.

An analogue of the group algebra of a finite group is defined for any finite projective plane, in the case where the characteristic of the scalar field divides the order of the plane.

In this note we show that certain perturbations, involving rank-one stabilising operators, which correspond to small delays in some feedback control problems, shift a part of the point spectrum of the unperturbed operator into the right-half plane.

We establish the persistence of the asymptotic stability of a linear equation $v'=A(t)v$ in a Banach space under sufficiently small perturbations, when the linear equation admits a non-uniform exponential contraction or a non-uniform exponential dichotomy. Moreover, we obtain optimal estimates for the decay of solutions of the perturbed equation, that in general may depend on the initial time.

By considering the norm of a locally univalent function given by

we obtain such norm estimates for an operator of functions involving a convolution structure of convex univalent functions with a subclass of convex functions (defined by subordination). We also obtain some inequalities concerning this norm for functions under a certain fractional integral operator. Some implications of our results are briefly pointed out in the concluding section.

Throughout this article, the near-ring N is assumed to be zero symmetric and to satisfy the right distributive law. We construct an ideal called the socle-ideal of N which is antiradical in the sense that it is a direct sum of minimal left ideals and annihilates one or more of the radicals of N. We then use this socle-ideal to obtain a decomposition theorem for the s-radical JS(N) of N.

We present a new test for studying asphericity and diagrammatic reducibility of group presentations. Our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails. We use this criterion to generalize results of J. Howie and S.M. Gersten on asphericity of LOTs and of Adian presentations, and derive new results on solvability of equations over groups. We also use our methods to investigate a conjecture of S.V. Ivanov related to Kaplansky's problem on zero divisors: we strengthen Ivanov's result for locally indicable groups and prove a weak version of the conjecture.

We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous articles of Carlier–Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet–Choné model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity.

We study the right-definite separated half-linear Sturm–Liouville eigenvalue problems. It is proved that the $n$th real eigenvalue of the problem depends smoothly on the equation, but may have jump discontinuities with respect to the boundary condition. Formulae are found for the derivatives of the $n$th real eigenvalue with respect to all parameters: the endpoints, the boundary condition and the coefficient functions, whenever they exist. Monotone properties and a comparison result for real eigenvalues are deduced as consequences. The generalized Prüfer transformation and the implicit function theorem in Banach spaces play key roles in the proofs.