A Sobolev inequality with reciprocal weights

We give a Sobolev inequality with the weight K(x)$K\left(x\right)$ belonging to the class _A2∩_Gn$A_2\cap G_n$ for the function |u|^t${\left|u\right|}^t$ and the weight K(x)⁻¹$K{\left(x\right)}^{-1}$ for |∇u|²${\left|\nabla u\right|}^2$. The constant in the relevant inequality is seen to depend on the _Gn$G_n$ and _A2$A_2$ constants of the weight.     

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

In this note we derive a maximum principle for an appropriate functional combination of u(x)$u(\mathbf{x})$ and |∇u|²${|\mathrm{\nabla }u|}^2$, where u(x)$u(\mathbf{x})$ is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in R^N+1${\mathbb{R}}^{N+1}$.     

A structure theorem for small sumsets in nonabelian groups

Let G$G$ be an arbitrary finite group and let S$S$ and T$T$ be two subsets such that |S|≥2$\left|S\right|\ge 2$, |T|≥2$\left|T\right|\ge 2$, and |TS|≤|T|+|S|−1≤|G|−2$\left|TS\right|\le \left|T\right|+\left|S\right|-1\le \left|G\right|-2$. We show that if |S|≤|G|−4|G|^1/2$\left|S\right|\le \left|G\right|-4{\left|G\right|}^{1/2}$ then either S$S$ is a geometric progression or there exists a non-trivial subgroup H$H$ such that either |HS|≤|S|+|H|−1$\left|HS\right|\le \left|S\right|+\left|H\right|-1$ or |SH|≤|S|+|H|−1$\left|SH\right|\le \left|S\right|+\left|H\right|-1$. This extends to the nonabelian case classical results for abelian groups. When we remove the hypothesis |S|≤|G|−4|G|^1/2$\left|S\right|\le \left|G\right|-4{\left|G\right|}^{1/2}$ we show the existence of counterexamples to the above characterization whose structure is described precisely.     

Traveling wave solutions of the heat flow of director fields

We consider the simplest possible heat equation for director fields, ut=Δu+|∇u|²u$u_t=\mathrm{\Delta }u+{|\mathrm{\nabla }u|}^{2}u$ (|u|=1$|u|=1$), and construct axially symmetric traveling wave solutions defined in an infinitely long cylinder. The traveling waves have a point singularity of topological degree 0 or 1.     

Boundedness and blowup for nonlinear degenerate parabolic equations

The author deals with the quasilinear parabolic equation _ut=[u^α+g(u)]Δu+bu^α+1+f(u,∇u)$u_t=[u^{\alpha }+g\left(u\right)]\Delta u+b{u}^{\alpha +1}+f(u,\nabla u)$ with Dirichlet boundary conditions in a bounded domain Ω$\Omega$, where f$f$ and g$g$ are lower-order terms. He shows that, under suitable conditions on f$f$ and g$g$, whether the solution is bounded or blows up in a finite time depends only on the first eigenvalue of −Δ$-\Delta$ in Ω$\Omega$ with Dirichlet boundary condition. For some special cases, the result is sharp.     

Boundedness of extremal solutions in dimension 4

In this paper we establish the boundedness of the extremal solution u^∗$u^{\ast }$ in dimension N=4$N=4$ of the semilinear elliptic equation −Δu=λf(u)$-\Delta u=\lambda f\left(u\right)$, in a general smooth bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, with Dirichlet data u_|∂Ω=0$u{|}_{\partial \Omega }=0$, where f$f$ is a C¹$C^1$ positive, nondecreasing and convex function in [0,∞)$[0,\infty )$ such that f(s)/s→∞$f\left(s\right)/s\to \infty$ as s→∞$s\to \infty$.     

Bernstein–Nagumo conditions and solutions to nonlinear differential inequalities

For Ω$\Omega$, an open bounded subset of R^N${\mathbb{R}}^N$ with smooth boundary and 1<p<∞$1<p<\infty$, we establish W^1,p(Ω)$W^{1,p}\left(\Omega \right)$a priori bounds and prove the compactness of solution sets to differential inequalities of the form

|divA(x,∇u)|≤F(x,u,∇u),$\left|\text{div}A(x,\nabla u)\right|\le F(x,u,\nabla u)\text{,}$

which are bounded in L^∞(Ω)$L^{\infty }\left(\Omega \right)$. The main point in this work is that the nonlinear term F$F$ may depend on ∇u$\nabla u$ and may grow as fast as a power of order p$p$ in this variable. Such growth conditions have been used extensively in the study of boundary value problems for nonlinear ordinary differential equations and are known as Bernstein–Nagumo growth conditions. In addition, we use these results to establish a sub-supersolution theorem.     

Local and global properties of solutions of quasilinear Hamilton–Jacobi equations

We study some properties of the solutions of (E) −_Δpu+|∇u|^q=0$-{\mathrm{\Delta }}_{p}u+{|\mathrm{\nabla }u|}^q=0$ in a domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, mostly when p≥q>p−1$p\ge q>p-1$. We give a universal a priori estimate of the gradient of the solutions with respect to the distance to the boundary. We give a full classification of the isolated singularities of the nonnegative solutions of (E), a partial classification of isolated singularities of the negative solutions. We prove a general removability result expressed in terms of some Bessel capacity of the removable set. We extend our estimates to equations on complete noncompact manifolds satisfying a lower bound estimate on the Ricci curvature, and derive some Liouville type theorems.     

Global existence of weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain

We consider two-dimensional mixed problems in an exterior domain for a semilinear strongly damped wave equation with a power-type nonlinearity |u|^p${\left|u\right|}^p$. If the initial data have a small weighted energy, we shall derive a global existence and energy decay results in the case when the power p$p$ of the nonlinear term satisfies p>6$p>6$.     

Cycles embedding on folded hypercubes with faulty nodes

Let F_Fv$F{F}_v$ be the set of faulty nodes in an n$n$-dimensional folded hypercube F_Qn$F{Q}_n$ with |F_Fv|≤n−2$\left|F{F}_v\right|\le n-2$. In this paper, we show that if n≥3$n\ge 3$, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every even length from 4$4$ to 2ⁿ−2|F_Fv|$2^n-2\left|F{F}_v\right|$, and if n≥2$n\ge 2$ and n$n$ is even, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every odd length from n+1$n+1$ to 2ⁿ−2|F_Fv|−1$2^n-2\left|F{F}_v\right|-1$.     

On a multi-dimensional transport equation with nonlocal velocity

We study a multi-dimensional nonlocal active scalar equation of the form _ut+v⋅∇u=0$u_t+v\cdot \mathrm{\nabla }u=0$ in R⁺×R^d${\mathbb{R}}^{+}\times {\mathbb{R}}^d$, where v=Λ^−2+α∇u$v={\Lambda }^{-2+\alpha }\mathrm{\nabla }u$ with Λ=(−Δ)^1/2$\Lambda ={(-\mathrm{\Delta })}^{1/2}$. We show that when α∈(0,2]$\alpha \in (0,2]$ certain radial solutions develop gradient blowup in finite time. In the case when α=0$\alpha =0$, the equations are globally well-posed with arbitrary initial data in suitable Sobolev spaces.     

Counting minimal semi-Sturmian words

A finite Sturmian   word w$w$ is a balanced word over the binary alphabet {a,b}$\{a,b\}$, that is, for all subwords u$u$ and v$v$ of w$w$ of equal length, |_|u|a−_|v|a|≤1$|{\left|u\right|}_a-{\left|v\right|}_a|\le 1$, where _|u|a${\left|u\right|}_a$ and _|v|a${\left|v\right|}_a$ denote the number of occurrences of the letter a$a$ in u$u$ and v$v$, respectively. There are several other characterizations, some leading to efficient algorithms for testing whether a finite word is Sturmian. These algorithms find important applications in areas such as pattern recognition, image processing, and computer graphics. Recently, Blanchet-Sadri and Lensmire considered finite semi-Sturmian words of minimal length and provided an algorithm for generating all of them using techniques from graph theory. In this paper, we exploit their approach in order to count the number of minimal semi-Sturmian words. We also present some other results that come from applying this graph theoretical framework to subword complexity.     

Crossing number additivity over edge cuts

Consider a graph G$G$ with a minimal edge cut F$F$ and let _G1$G_1$, _G2$G_2$ be the two (augmented) components of G−F$G-F$. A long-open question asks under which conditions the crossing number of G$G$ is (greater than or) equal to the sum of the crossing numbers of _G1$G_1$ and _G2$G_2$—which would allow us to consider those graphs separately. It is known that crossing number is additive for |F|∈{0,1,2}$\left|F\right|\in \{0,1,2\}$ and that there exist graphs violating this property with |F|≥4$\left|F\right|\ge 4$. In this paper, we show that crossing number is additive for |F|=3$\left|F\right|=3$, thus closing the final gap in the question.     

A boundary blow-up elliptic problem with an inhomogeneous term

By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h$h$ affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)$△u=b\left(x\right)g\left(u\right)+\lambda h\left(x\right)$, u>0$u>0$ in Ω$\Omega$, u_|∂Ω=∞$u{|}_{\partial \Omega }=\infty$, where Ω$\Omega$ is a bounded domain with smooth boundary in R^N${\mathbb{R}}^N$, λ>0$\lambda >0$, g∈C¹[0,∞)$g\in C^1[0,\infty )$ is increasing on [0,∞)$[0,\infty )$, g(0)=0$g\left(0\right)=0$, g^′$g^{\prime }$ is regularly varying at infinity with positive index ρ$\rho$, the weight b$b$, which is non-trivial and non-negative in Ω$\Omega$, may be vanishing on the boundary, and the inhomogeneous term h$h$ is non-negative in Ω$\Omega$ and may be singular on the boundary.     

Asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term

This paper deals with the asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term |u|^β−1u(β≥3)${\left|u\right|}^{\beta -1}u(\beta \ge 3)$. First, we establish an upper bound for the difference between the solution of our equation and the heat equation in L²$L^2$ space. Then, we optimize the upper bound of decay for the solutions and obtain their algebraic lower bound by using Fourier Splitting method.     

A Strong Maximum Principle for parabolic equations with the p-Laplacian

We prove the Strong Maximum Principle (SMP) under suitable assumptions for a class of quasilinear parabolic problems with the p  -Laplacian, p>1$p>1$, on bounded cylindrical domains of R^N+1${\mathbb{R}}^{N+1}$,

_∂tu−_Δpu−λ|u|^p−2u≥0,${\partial }_{t}u-{\mathrm{\Delta }}_{p}u-\lambda {|u|}^{p-2}u\ge 0,$

with nonnegative initial–boundary conditions and λ≤0$\lambda \le 0$, and we give some counterexamples to the SMP if some of our assumptions are violated. We show that the Hopf Maximum Principle holds for 1<p<2$1<p<2$, and give a counterexample to it for p>2$p>2$. Also the Weak Maximum Principle for λ≤_λ1$\lambda \le {\lambda }_1$ is established.