Monotonicity of the reflected Bessel transition density on the diagonal

For α∈R$\alpha \in \mathbb{R}$, let p^R(t,x,x)$p^R(t,x,x)$ denote the diagonal of the transition density of the α$\alpha$-Bessel process in (0,1]$(0,1]$, killed at 0 and reflected at 1. As a function of x$x$, if either α≥3$\alpha \ge 3$ or α=1$\alpha =1$, then for t>0$t>0$, the diagonal is nondecreasing. This monotonicity property fails if 1≠α<3$1\ne \alpha <3$.     

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.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.     

Breaking the chain

We consider the motion of a Brownian particle in R$\mathbb{R}$, moving between a particle fixed at the origin and another moving deterministically away at slow speed ε>0$\epsilon >0$. The middle particle interacts with its neighbours via a potential of finite range b>0$b>0$, with a unique minimum at a>0$a>0$, where b<2a$b<2a$. We say that the chain of particles breaks on the left- or right-hand side when the middle particle is at a distance greater than b$b$ from its left or right neighbour, respectively. We study the asymptotic location of the first break of the chain in the limit of small noise, in the case where ε=ε(σ)$\epsilon =\epsilon \left(\sigma \right)$ and σ>0$\sigma >0$ is the noise intensity.     

Boundary value problems in spaces of distributions on smooth and polygonal domains

We study boundary value problems of the form -Δu=f$-\mathrm{\Delta }u=f$ on Ω$\Omega$ and Bu=g$\mathit{Bu}=g$ on the boundary ∂Ω$\mathrm{\partial }\Omega$, with either Dirichlet or Neumann boundary conditions, where Ω$\Omega$ is a smooth bounded domain in Rⁿ${\mathbb{R}}^n$ and the data f,g$f,g$ are distributions  . This problem has to be first properly reformulated and, for practical applications, it is of crucial importance to obtain the continuity of the solution u$u$ in terms of f and g  . For f=0$f=0$, taking advantage of the fact that u$u$ is harmonic on Ω$\Omega$, we provide four formulations of this boundary value problem (one using nontangential limits of harmonic functions, one using Green functions, one using the Dirichlet-to-Neumann map, and a variational one); we show that these four formulations are equivalent. We provide a similar analysis for f≠0$f\ne 0$ and discuss the roles of f and g, which turn to be somewhat interchangeable in the low regularity case. The weak formulation is more convenient for numerical approximation, whereas the nontangential limits definition is closer to the intuition and easier to check in concrete situations. We extend the weak formulation to polygonal domains using weighted Sobolev spaces. We also point out some new phenomena for the “concentrated loads” at the vertices in the polygonal case.     

Critical exponents for non-simultaneous blow-up in a localized parabolic system

The paper deals with the radially symmetric solutions of _ut=Δu+u^m(x,t)vⁿ(0,t)$u_t=\Delta u+u^m(x,t)v^n(0,t)$, _vt=Δv+u^p(0,t)v^q(x,t)$v_t=\Delta v+u^p(0,t)v^q(x,t)$, subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u$u$ (v$v$) blows up alone if and only if m>p+1$m>p+1$ (q>n+1$q>n+1$), which means that any blow-up is simultaneous if and only if m≤p+1$m\le p+1$, q≤n+1$q\le n+1$. (ii) Any blow-up is u$u$ (v$v$) blowing up with v$v$ (u$u$) remaining bounded if and only if m>p+1$m>p+1$, q≤n+1$q\le n+1$ (m≤p+1$m\le p+1$, q>n+1$q>n+1$). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1$m>p+1$, q>n+1$q>n+1$. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model.     

Asymptotic analysis of the p-Laplacian flow in an exterior domain

We consider the Dirichlet problem for the p  -Laplacian evolution equation, ut=Δpu$u_t={\mathrm{\Delta }}_{p}u$, where p>2$p>2$, posed in an exterior domain in RN${\mathbf{R}}^N$, with zero Dirichlet boundary condition and with integrable and nonnegative initial data. We are interested in describing the influence of the holes of the domain on the large time behaviour of the solutions. Such behaviour varies depending on the relative values of N and p  . We must distinguish between the behaviour near infinity of space (outer analysis), and near the holes (inner analysis). We obtain that the outer analysis is given in all cases by certain self-similar solutions and the inner analysis is given by quasi-stationary states. Logarithmic corrections to exact self-similarity appear in the critical case N=p$N=p$, which is mathematically more interesting. In this first paper we treat only the cases N>p$N>p$ and N=p$N=p$, the case N<p$N<p$ will be considered in a companion work.     

Existence and multiplicity of nonnegative solutions for semilinear elliptic problems involving nonlinearities indefinite in sign

Let r,s∈]1,2[$r,s\in ]1,2[$ and λ,μ∈]0,+∞[$\lambda ,\mu \in ]0,+\infty [$. In this paper, we deal with the existence and multiplicity of nonnegative and nonzero solutions of the Dirichlet problem with 0$0$ boundary data for the semilinear elliptic equation −Δu=λu^s−1−u^r−1$-\Delta u=\lambda u^{s-1}-u^{r-1}$ in Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, where N≥2$N\ge 2$. We prove that there exists a positive constant Λ$\Lambda$ such that the above problem has at least two solutions, at least one solution or no solution according to whether λ>Λ$\lambda >\Lambda$, λ=Λ$\lambda =\Lambda$ or λ<Λ$\lambda <\Lambda$. In particular, a result by Hernandéz, Macebo and Vega is improved and, for the semilinear case, a result by Díaz and Hernandéz is partially extended to higher dimensions. Finally, an answer to a conjecture, recently stated by the author, is also given.     

Phase transition in equilibrium fluctuations of symmetric slowed exclusion

We analyze the equilibrium fluctuations of density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is αn^−β$\alpha n^{-\beta }$, with α>0$\alpha >0$, β∈[0,+∞]$\beta \in [0,+\infty ]$ and n$n$ is the scaling parameter. Depending on the regime of β$\beta$, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value β=1$\beta =1$, starting a tagged particle near the slow bond, we obtain a family of Gaussian processes indexed in α$\alpha$, interpolating a fractional Brownian motion of Hurst exponent 1/4$1/4$ and the degenerate process equal to zero.     

Convergence of invariant measures for singular stochastic diffusion equations

It is proved that the solutions to the singular stochastic p$p$-Laplace equation, p∈(1,2)$p\in (1,2)$ and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1)$r\in (0,1)$ on a bounded open domain Λ⊂R^d$\Lambda \subset {\mathbb{R}}^d$ with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p$p$ and r$r$ respectively (in the Hilbert spaces L²(Λ)$L^2\left(\Lambda \right)$, H⁻¹(Λ)$H^{-1}\left(\Lambda \right)$ respectively). The highly singular limit case p=1$p=1$ is treated with the help of stochastic evolution variational inequalities, where P$\mathbb{P}$-a.s. convergence, uniformly in time, is established.     

On a family of differential operators with the coupling parameter in the boundary condition

We study a family of differential operators _Lα${\mathbf{L}}_{\alpha }$ in two variables, depending on the coupling parameter α?0$\alpha ?0$ that appears only in the boundary conditions. Our main concern is the spectral properties of _Lα${\mathbf{L}}_{\alpha }$, which turn out to be quite different for α<1$\alpha <1$ and for α>1$\alpha >1$. In particular, _Lα${\mathbf{L}}_{\alpha }$ has a unique self-adjoint realization for α<1$\alpha <1$ and many such realizations for α>1$\alpha >1$. In the more difficult case α>1$\alpha >1$ an analysis of non-elliptic pseudodifferential operators in dimension one is involved.     

Approximate martingale estimating functions for stochastic differential equations with small noises

An approximate martingale estimating function with an eigenfunction is proposed for an estimation problem about an unknown drift parameter for a one-dimensional diffusion process with small perturbed parameter ε$\epsilon$ from discrete time observations at n$n$ regularly spaced time points k/n$k/n$, k=0,1,…,n$k=0,1,\dots ,n$. We show asymptotic efficiency of an M$M$-estimator derived from the approximate martingale estimating function as ε→0$\epsilon \to 0$ and n→∞$n\to \infty$ simultaneously.     

A note on distance-regular graphs with a small number of vertices compared to the valency

In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α>2$\alpha >2$, there are finitely many distance-regular graphs Γ$\Gamma$ with valency k$k$, diameter D≥3$D\ge 3$ and v$v$ vertices satisfying v≤αk$v\le \alpha k$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k≥3$k\ge 3$, diameter D≥3$D\ge 3$ and _c2≥εk$c_2\ge \epsilon k$ for a given 0<ε<1$0<\epsilon <1$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite).     

Differential inequalities and equations in Banach spaces with a cone

A class of second-order abstract dissipative evolution differential operators D$D$ with 0∈kerD$0\in kerD$ is shown for which the fact that a non-zero t?u(t)$t?u\left(t\right)$ belongs to a cone and −Du$-Du$ to a dual cone may hold only on time intervals whose length is less than or equal to a defined number. Then oscillatory functions are dealt with in the framework of Banach spaces with a cone and conditions for the existence of a uniform oscillatory time for solutions of the equation Du=0$Du=0$ are given.