Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem

We study the second-order nonlinear differential equation \(u'' + a(t) g(u) = 0\), where \(g\) is a continuously differentiable function of constant sign defined on an open interval \(I\subseteq {\mathbb R}\) and \(a(t)\) is a sign-changing weight function. We look for solutions \(u(t)\) of the differential equation such that \(u(t)\in I,\) satisfying the Neumann boundary conditions. Special examples, considered in our model, are the equations with singularity, for \(I = {\mathbb R}^+_0\) and \(g(u) \sim - u^{-\sigma },\) as well as the case of exponential nonlinearities, for \(I = {\mathbb R}\) and \(g(u) \sim \exp (u)\). The proofs are obtained by passing to an equivalent equation of the form \(x'' = f(x)(x')^2 + a(t)\).     

Left annihilator of generalized derivations on Lie ideals in prime rings

Let \(R\) be a prime ring, \(L\) a noncentral Lie ideal of \(R\), \(F\) a generalized derivation with associated nonzero derivation \(d\) of \(R\). If \(a\in R\) such that \(a(d(u)^{l_1} F(u)^{l_2} d(u)^{l_3} F(u)^{l_4} \ldots F(u)^{l_k})^{n}=0\) for all \(u\in L\), where \(l_1,l_2,\ldots ,l_k\) are fixed non negative integers not all are zero and \(n\) is a fixed integer, then either \(a=0\) or \(R\) satisfies \(s_4\), the standard identity in four variables.     

Existence and Uniqueness of Invariant Measures for Stochastic Reaction–Diffusion Equations in Unbounded Domains

In this paper, we investigate the long-time behavior of stochastic reaction–diffusion equations of the type \(\text {d}u = (Au + f(u))\text {d}t + \sigma (u) \text {d}W(t)\), where \(A\) is an elliptic operator, \(f\) and \(\sigma \) are nonlinear maps and \(W\) is an infinite-dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function \(f\) possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure, which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper, we expand the existing classes of nonlinear functions \(f\) and \(\sigma \) and elliptic operators \(A\) for which the invariant measure exists, in particular in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if \(A\) is the Shrödinger-type operator \(A = \frac{1}{\rho }(\text {div} \rho \nabla u)\) where \(\rho = \text {e}^{-|x|^2}\) is the Gaussian weight.     

Hardy Inequalities for Finsler <Emphasis Type="Italic">p</Emphasis>-Laplacian in the Exterior Domain

The Finsler p-Laplacian is the class of nonlinear differential operators given by

$$\begin{aligned} \Delta _{H,p}u:= \text {div}(H(\nabla u)^{p-1}\nabla _{\eta } H(\nabla u)) \end{aligned}$$

where \(1<p<\infty \) and \(H:\mathbf {R}^N\rightarrow [0,\infty )\) is in \(C^2(\mathbf {R}^N\backslash \{0\})\) and is positively homogeneous of degree 1. Under some additional constraints on H, we derive the Hardy inequality for Finsler p-Laplacian in exterior domain for \(1<p\le N\). We also provide an improved version of Hardy inequality for the case \(p=2\).

     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

Estimates for large solutions of a quasilinear elliptic equation

Let \(\Omega \) be a bounded smooth domain of \(R^{n}\). We study the asymptotic behaviour of the solutions to the equation \(\triangle u-|Du|^{q}=f(u)\) in \(\Omega , 1<q<2,\) which satisfy the boundary condition \(u(x)\rightarrow \infty \) as \(x\rightarrow \partial \Omega \). These solutions are called large or blowup solutions. Near the boundary we give lower and upper bounds for the ratio \(\psi (u)/\delta \), where \(\psi (u) = \int _{u}^{\infty }1/\sqrt{2F}dt\), \(F'=f\), \(\delta =dist(x,\partial \Omega )\) or for the ratio \(u/\delta ^{(2-q)/(1-q)}\). When in particular the ratio \(f/F^{q/2}\)is regular at infinity, we find again known results (Bandle and Giarrusso, in Adv Diff Equ 1, 133–150, 1996; Giarrusso, in Comptes Rendus de l’Acad Sci 331, 777–782 2000).     

Classifying First Extremal Points for a Fractional Boundary Value Problem with a Fractional Boundary Condition

Let \(n\in \mathbb {N}\), \(n\ge 2\), \(\beta >0\) fixed, and \(0<b\le \beta \). For \(n-1<\alpha \le n\), we look to classify extremal points for the fractional differential equation \(D_{0^+}^{\alpha }u+p(t) u=0\), satisfying the boundary conditions \(u^{(i)}(0)=0\), \(i=0,\ldots ,n-2\), \(D_{0^+}^\gamma u(b)=0\), where p(t) is a continuous nonnegative function on \([0,\beta ]\) which does not vanish identically on any nondegenerate compact subinterval of \([0,\beta ]\). Using the theory of Krein and Rutman, first extremal points of this boundary value problem are classified. As an application, the results are applied, along with a fixed-point theorem, to show the existence of a solution of a nonlinear fractional boundary value problem.     

Existence and Uniqueness of Renormalized Solutions to Nonlinear Parabolic Equations with Lower Order Term and Diffuse Measure Data

Here we give an existence and uniqueness result of a renormalized solution for a class of nonlinear parabolic equations \(\displaystyle {\partial b(u) \over \partial t} - \mathrm{div}(a(x,t,\nabla u))+\mathrm{div}(\Phi (x,t, u))=\mu \), where the right side is a measure data, b is a strictly increasing \(C^1\)-function, \(- \mathrm{div}(a(x,t,\nabla u))\) is a Leray–Lions type operator with growth \(|\nabla u|^{p-1}\) in \(\nabla u\) and \(\Phi (x,t, u)\) is a nonlinear lower order term.     

Elimination of extremal index zeroes from generic paths of closed $$$$-forms

Let \( \alpha \) be a Morse closed \( 1 \)-form of a smooth \( n \)-dimensional manifold \( M \). The zeroes of \( \alpha \) of index \( 0 \) or \( n \) are called centers. It is known that every non-vanishing de Rham cohomology class \( u \) contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the last statement: every generic path \( (\alpha _t)_{ t\in [0,1] }\) of closed \( 1 \)-forms in a fixed class \( u\ne 0 \) such that \( \alpha _0,\alpha _1 \) have no centers, can be modified relatively to its extremities to another such path \( (\beta _t)_{t \in [0,1]} \) having no center at all.     

A potential well for an integral functional with singular Lagrangian

This paper concerns a functional of the form

$$\begin{aligned} \Phi (u)=\int _\Omega L(x,u(x),\nabla u(x))\, dx \end{aligned}$$

on the Sobolev space \(H_0^1(\Omega )\) where \(\Omega \) is a bounded open subset of \({\mathbb {R}}^N\) with \(N\ge 3\) and \(0\in \Omega \). The hypotheses on L ensure that \(u\equiv 0\) is a critical point of \(\Phi \), but allow the Lagrangian to be singular at \(x=0\). It is shown that, under these assumptions, the usual conditions associated with Jacobi (positive definiteness of the second variation of \(\Phi \) at \(u\equiv 0\)), Legendre (ellipticity at \(u\equiv 0\)) and Weierstrass [strict convexity of \(L(x,s,\xi )\) with respect to \(\xi \)] from the calculus of variations are not sufficient ensure that \(u\equiv 0\) is a local minimum of \(\Phi \). Using recent criteria for the existence of a potential well of a \(C^1\)-functional on a real Hilbert space, conditions implying that \(u\equiv 0\) lies in a potential well of \(\Phi \) are established. They are shown to be sharp in some cases.

     

A Heroin Epidemic Model: Very General Non Linear Incidence,Treat-Age,and Global Stability

The aim of this work is to establish the existence of multi-peak solutions for the following class of quasilinear problems

$$ - \mbox{div} \bigl(\epsilon^{2}\phi\bigl(\epsilon|\nabla u|\bigr)\nabla u \bigr) + V(x)\phi\bigl(\vert u\vert\bigr)u = f(u)\quad\mbox{in } \mathbb{R}^{N}, $$

where \(\epsilon\) is a positive parameter, \(N\geq2\), \(V\), \(f\) are continuous functions satisfying some technical conditions and \(\phi\) is a \(C^{1}\)-function.

     

The $$$$-Affine Caacity

In this paper, the \(p\)-affine capacity is introduced for \(1<p<n\) and then developed to discover the upper and lower isocapacitary inequalities that strengthen optimally both the Maz’ya \(p\)-isocapacitary inequality and the Lutwak–Yang–Zhang \(L_p\) affine isoperimetric inequality over the \(n\)-dimensional Euclidean space \({\mathbb {R}}^{n}\).     

Short-Time Nonlinear Effects in the Exciton-Polariton System

In the exciton-polariton system, a linear dispersive photon field is coupled to a nonlinear exciton field. Short-time analysis of the lossless system shows that, when the photon field is excited, the time required for that field to exhibit nonlinear effects is longer than the time required for the nonlinear Schrödinger equation, in which the photon field itself is nonlinear. When the initial condition is scaled by \(\epsilon ^\alpha \), it is found that the relative error committed by omitting the nonlinear term in the exciton-polariton system remains within \(\epsilon \) for all times up to \(t=C\epsilon ^\beta \), where \(\beta =(1-\alpha (p-1))/(p+2)\). This is in contrast to \(\beta =1-\alpha (p-1)\) for the nonlinear Schrödinger equation. The result is proved for solutions in \(H^s(\mathbb {R}^n)\) for \(s>n/2\). Numerical computations indicate that the results are sharp and also hold in \(L^2(\mathbb {R}^n)\).     

Remarks on curvature dimension conditions on graphs

We show a connection between the \(CDE'\) inequality introduced in Horn et al. (Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for nonnegative curvature graphs. arXiv:1411.5087v2, 2014) and the \(CD\psi \) inequality established in Münch (Li–Yau inequality on finite graphs via non-linear curvature dimension conditions. arXiv:1412.3340v1, 2014). In particular, we introduce a \(CD_\psi ^\varphi \) inequality as a slight generalization of \(CD\psi \) which turns out to be equivalent to \(CDE'\) with appropriate choices of \(\varphi \) and \(\psi \). We use this to prove that the \(CDE'\) inequality implies the classical CD inequality on graphs, and that the \(CDE'\) inequality with curvature bound zero holds on Ricci-flat graphs.     

Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions,

$$\begin{aligned} Q^\Omega _\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text { on } \partial \Omega , \end{aligned}$$

in a class of bounded smooth domains \(\Omega \in \mathbb {R}^\nu \); here \(n\) is the outward unit normal and \(\alpha >0\) is a constant. We show that for each \(j\in \mathbb {N}\) and \(\alpha \rightarrow +\infty \), the \(j\)th eigenvalue \(E_j(Q^\Omega _\alpha )\) has the asymptotics

$$\begin{aligned} E_j(Q^\Omega _\alpha )=-\alpha ^2 -(\nu -1)H_\mathrm {max}(\Omega )\,\alpha +{\mathcal O}(\alpha ^{2/3}), \end{aligned}$$

where \(H_\mathrm {max}(\Omega )\) is the maximum mean curvature at \(\partial \Omega \). The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of \(H_\mathrm {max}\). In particular, we show that the ball is the strict minimizer of \(H_\mathrm {max}\) among the smooth star-shaped domains of a given volume, which leads to the following result: if \(B\) is a ball and \(\Omega \) is any other star-shaped smooth domain of the same volume, then for any fixed \(j\in \mathbb {N}\) we have \(E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )\) for large \(\alpha \). An open question concerning a larger class of domains is formulated.

     

On Locally Gabriel Geometric Graphs

Let \(P\) be a set of \(n\) points in the plane. A geometric graph \(G\) on \(P\) is said to be locally Gabriel if for every edge \((u,v)\) in \(G\), the Euclidean disk with the segment joining \(u\) and \(v\) as diameter does not contain any points of \(P\) that are neighbors of \(u\) or \(v\) in \(G\). A locally Gabriel graph(LGG) is a generalization of Gabriel graph and is motivated by applications in wireless networks. Unlike a Gabriel graph, there is no unique LGG on a given point set since no edge in a LGG is necessarily included or excluded. Thus the edge set of the graph can be customized to optimize certain network parameters depending on the application. The unit distance graph(UDG), introduced by Erdos, is also a LGG. In this paper, we show the following combinatorial bounds on edge complexity and independent sets of LGG: (i) For any \(n\), there exists LGG with \(\Omega (n^{5/4})\) edges. This improves upon the previous best bound of \(\Omega (n^{1+\frac{1}{\log \log n}})\). (ii) For various subclasses of convex point sets, we show tight linear bounds on the maximum edge complexity of LGG. (iii) For any LGG on any \(n\) point set, there exists an independent set of size \(\Omega (\sqrt{n}\log n)\).     

The Gray images of $$$$ constacyclic codes over $$F_{2^m}[u]/\langle u^{k} \rangle $$F2m[u]/⟨uk⟩

Let \(R_{k}\) denote the polynomial residue ring \(F_{2^m}[u]/\langle u^{k} \rangle \), where \(2^{j-1}+1\le k\le 2^{j}\) for some positive integer \(j\). Motivated by the work in [1], we introduce a new Gray map from \(R_{k}\) to \(F_{2^m}^{2^{j}}\). It is proved that the Gray image of a linear \((1+u)\) constacyclic code of an arbitrary length \(N\) over \(R_{k}\) is a distance invariant linear cyclic code of length \(2^{j}N\) over \(F_{2^m}\). Moreover, the generator polynomial of the Gray image of such a constacyclic code is determined, and some optimal linear cyclic codes over \(F_{2}\) and \(F_{4}\) are constructed under this Gray map.     

The solution gap of the Brezis–Nirenberg problem on the hyperbolic space

We consider the positive solutions of the nonlinear eigenvalue problem \(-\Delta _{\mathbb {H}^n} u = \lambda u + u^p, \) with \(p=\frac{n+2}{n-2}\) and \(u \in H_0^1(\Omega ),\) where \(\Omega \) is a geodesic ball of radius \(\theta _1\) on \(\mathbb {H}^n.\) For radial solutions, this equation can be written as an ordinary differential equation having n as a parameter. In this setting, the problem can be extended to consider real values of n. We show that if \(2<n<4\) this problem has a unique positive solution if and only if \(\lambda \in \left( n(n-2)/4 +L^*\,,\, \lambda _1\right) .\) Here \(L^*\) is the first positive value of \(L = -\ell (\ell +1)\) for which a suitably defined associated Legendre function \(P_{\ell }^{-\alpha }(\cosh \theta ) >0\) if \(0 < \theta <\theta _1\) and \(P_{\ell }^{-\alpha }(\cosh \theta _1)=0,\) with \(\alpha = (2-n)/2\).     

On Weak Invariance Principles for Partial Sums

Given a sequence of random functionals \(\bigl \{X_k(u)\bigr \}_{k \in \mathbb {Z}}\), \(u \in \mathbf{I}^d\), \(d \ge 1\), the normalized partial sums \(\check{S}_{nt}(u) = n^{-1/2}\bigl (X_1(u) + \cdots + X_{\lfloor n t \rfloor }(u)\bigr )\), \(t \in [0,1]\) and its polygonal version \({S}_{nt}(u)\) are considered under a weak dependence assumption and \(p > 2\) moments. Weak invariance principles in the space of continuous functions and càdlàg functions are established. A particular emphasis is put on the process \(\check{S}_{nt}(\widehat{\theta })\), where \(\widehat{\theta } \xrightarrow {\mathbb {P}} \theta \), and weaker moment conditions (\(p = 2\) if \(d = 1\)) are assumed.     

Gagliardo-Nirenberg inequalities for differential forms in Heisenberg groups

The \(L^1\)-Sobolev inequality states that for compactly supported functions u on the Euclidean n-space, the \(L^{n/(n-1)}\)-norm of a compactly supported function is controlled by the \(L^1\)-norm of its gradient. The generalization to differential forms (due to Lanzani and Stein and Bourgain and Brezis) is recent, and states that a the \(L^{n/(n-1)}\)-norm of a compactly supported differential h-form is controlled by the \(L^1\)-norm of its exterior differential du and its exterior codifferential \(\delta u\) (in special cases the \(L^1\)-norm must be replaced by the \(\mathcal H^1\)-Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms.