Communications in Contemporary Mathematics

A new approach is developed for the solvability of nonlocal problems in Hilbert spaces associated to nonlinear differential equations. It is based on a joint combination of the degree theory with the approximation solvability method and the bounding functions technique. No compactness or condensivity condition on the nonlinearities is assumed. Some applications of the abstract result to the study of nonlocal problems for integro-differential equations and systems of integro-differential equations are then showed. A generalization of the result by using nonsmooth bounding functions is given.

We provide sufficient conditions for the existence of a uniparametric family of entire spacelike graphs with prescribed mean curvature in a Friedmann–Lemaître–Robertson–Walker spacetime with flat fiber. The proof is based on the analysis of the associated homogeneous Dirichlet problem on a Euclidean ball together with suitable bounds for the gradient which permit the prolongability of the solution to the whole space.

The Beauville–Fujiki relation for a compact Hyperkähler manifold $X$ of dimension $2k$ allows to equip the symmetric power ${\mathrm{\text{Sym}}}^k{H}^2(X)$ with a symmetric bilinear form induced by the Beauville–Bogomolov form. We study some of its properties and compare it to the form given by the Poincaré pairing. The construction generalizes to a definition for an induced symmetric bilinear form on the symmetric power of any free module equipped with a symmetric bilinear form. We point out how the situation is related to the theory of orthogonal polynomials in several variables. Finally, we construct a basis of homogeneous polynomials that are orthogonal when integrated over the unit sphere ${𝕊}^d$, or equivalently, over ${ℝ}^{d+1}$ with a Gaussian kernel.

We consider the notion of strong nonuniform spectrum for a nonautonomous dynamics with discrete time obtained from a sequence of matrices, which is defined in terms of the existence of strong nonuniform exponential dichotomies with an arbitrarily small nonuniform part. The latter exponential dichotomies are ubiquitous in the context of ergodic theory and correspond to have both lower and upper bounds along the stable and unstable directions, besides possibly a nonuniform conditional stability although with an arbitrarily small exponential dependence on the initial time. Moreover, we consider arbitrary growth rates instead of only the usual exponential rates. We give a complete characterization of the possible strong nonuniform spectra and for a Lyapunov regular trajectory, we show that the spectrum is the set of Lyapunov exponents. In addition, we provide explicit examples of nonautonomous dynamics for all possible strong nonuniform spectra. A remarkable consequence of our results is that for a sequence of matrices $A_m$, either $e^{-a}{A}_m$ does not admit a strong exponential dichotomy for any $a\in ℝ$, or if $e^{-a}{A}_m$ admits an exponential dichotomy for some $a\in ℝ$, then it also admits a strong exponential dichotomy for that $a$. We emphasize that this result is not in the literature even in the special case of uniform exponential dichotomies.

We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded vertex algebra under a certain finiteness condition and a condition related to the horizontal gradings. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory for such strongly graded generalized modules developed by Huang, Lepowsky and Zhang.

As a generalization of semi-invariant submersions, we introduce conformal semi-invariant submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations which arise from the definition of a conformal submersion and show that there are certain product structures on the total space of a conformal semi-invariant submersion. Moreover, we also check the harmonicity of such submersions and find necessary and sufficient conditions of a conformal semi-invariant submersion to be totally geodesic.

In this paper, we are concerned with the zero Dirichlet boundary value problem associated to the quasilinear elliptic equation

Given a vertex Lie algebra $\mathcal{ℒ}$ equipped with an action by automorphisms of a cyclic group $\mathrm{\Gamma }$, we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over “local” Lie algebras $L{(\mathcal{ℒ})}_{z_i}$ assigned to marked points $z_i$, by the action of a “global” Lie algebra $L_{\{z_i\}}^{\mathrm{\Gamma }}(\mathcal{ℒ})$ of $\mathrm{\Gamma }$-equivariant functions. On the other hand, the universal enveloping vertex algebra $𝕍(\mathcal{ℒ})$ of $\mathcal{ℒ}$ is itself a vertex Lie algebra with an induced action of $\mathrm{\Gamma }$. This gives “big” analogs of the Lie algebras above. From these we construct the space of “big” cyclotomic coinvariants, i.e. coinvariants with respect to $L_{\{z_i\}}^{\mathrm{\Gamma }}(𝕍(\mathcal{ℒ}))$. We prove that these two definitions of cyclotomic coinvariants in fact coincide, provided the origin is included as a marked point. As a corollary, we prove a result on the functoriality of cyclotomic coinvariants which we require for the solution of cyclotomic Gaudin models in [B. Vicedo and C. Young, Cyclotomic Gaudin models: Construction and Bethe ansatz, preprint (2014); arXiv:1409.6937]. At the origin, which is fixed by $\mathrm{\Gamma }$, one must assign a module over the stable subalgebra $L{(\mathcal{ℒ})}^{\mathrm{\Gamma }}$ of $L(\mathcal{ℒ})$. This module becomes a $𝕍(\mathcal{ℒ})$-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.

For a broad class of integral functionals defined on the space of $n$-dimensional convex bodies, we establish necessary and sufficient conditions for monotonicity, and necessary conditions for the validity of a Brunn–Minkowski type inequality. In particular, we prove that a Brunn–Minkowski type inequality implies monotonicity, and that a general Brunn–Minkowski type inequality is equivalent to the functional being a mixed volume.

In this paper, we construct for every $\alpha >0$ and $\lambda \in ℂ$ a class of initial values $u_0$ for which there exists a local solution of the nonlinear Schrödinger equation $i{u}_t+\mathrm{\Delta }u+\lambda |u{|}^{\alpha }u=0$ on ${ℝ}^N$ with the initial condition $u(0,x)=u_0$. Moreover, we construct for every $\alpha >\frac{2}{N}$ a class of (arbitrarily large) initial values for which there exists a global solution that scatters as $t\to \infty$.

We prove existence, non-degeneracy, and exponential decay at infinity of a non-trivial solution to Emden's equation $-\mathrm{\Delta }u=|u{|}^3$ on an unbounded $L$-shaped domain, subject to Dirichlet boundary conditions. Besides the direct value of this result, we also regard this solution as a building block for solutions on expanding bounded domains with corners, to be established in future work. Our proof makes heavy use of computer assistance. Starting from a numerical approximate solution, we use a fixed-point argument to prove existence of a near-by exact solution. The eigenvalue bounds established in the course of this proof also imply non-degeneracy of the solution.