Hajnal and Szemerédi proved that if G is a finite graph with maximum degree $\Delta $ , then for every integer $k \geq \Delta +1$ , G has a proper colouring with k colours in which every two colour classes differ in size at most by $1$ ; such colourings are called equitable. We obtain an analogue of this result for infinite graphs in the Borel setting. Specifically, we show that if G is an aperiodic Borel graph of finite maximum degree $\Delta $ , then for each $k \geq \Delta + 1$ , G has a Borel proper k-colouring in which every two colour classes are related by an element of the Borel full semigroup of G. In particular, such colourings are equitable with respect to every G-invariant probability measure. We also establish a measurable version of a result of Kostochka and Nakprasit on equitable $\Delta $ -colourings of graphs with small average degree. Namely, we prove that if $\Delta \geq 3$ , G does not contain a clique on $\Delta + 1$ vertices and $\mu $ is an atomless G-invariant probability measure such that the average degree of G with respect to $\mu $ is at most $\Delta /5$ , then G has a $\mu $ -equitable $\Delta $ -colouring. As steps toward the proof of this result, we establish measurable and list-colouring extensions of a strengthening of Brooks’ theorem due to Kostochka and Nakprasit.

In this paper we study the small-scale equidistribution property of random waves whose coefficients are determined by an unfair coin. That is, the coefficients take value $+1$ with probability p and $-1$ with probability $1-p$ . Random waves whose coefficients are associated with a fair coin are known to equidistribute down to the wavelength scale. We obtain explicit requirements on the deviation from the fair ( $p=0.5$ ) coin to retain equidistribution.

In this article, we study the observability (or equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the ‘degenerated directions’ of the subelliptic structure.

First, for any $\gamma \geq 1$ , we establish a resolvent estimate for the Baouendi–Grushin-type operator $\Delta _{\gamma }=\partial _x^2+\left \lvert x\right \rvert ^{2\gamma }\partial _y^2$ , which has step $\gamma +1$ . We then derive consequences for the observability of the Schrödinger-type equation $i\partial _tu-\left (-\Delta _{\gamma }\right )^{s}u=0$ , where $s\in \mathbb N$ . We identify three different cases: depending on the value of the ratio $(\gamma +1)/s$ , observability may hold in arbitrarily small time or only for sufficiently large times or may even fail for any time.

As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial _tu+\left (-\Delta _{\gamma }\right )^su=0$ and establish a decay rate for the damped wave equation associated with $\Delta _{\gamma }$ .

We derive and numerically implement various asymptotic approximations for the lowest or principal eigenvalue of the Laplacian with a periodic arrangement of localised traps of small \[\mathcal{O}(\varepsilon )\] spatial extent that are centred at the lattice points of an arbitrary Bravais lattice in \[{\mathbb{R}^2}\]. The expansion of this principal eigenvalue proceeds in powers of \[\nu \equiv - 1/\log (\varepsilon {d_c})\], where dc is the logarithmic capacitance of the trap set. An explicit three-term approximation for this principal eigenvalue is derived using strong localised perturbation theory, with the coefficients in this series evaluated numerically by using an explicit formula for the source-neutral periodic Green’s function and its regular part. Moreover, a transcendental equation for an improved approximation to the principal eigenvalue, which effectively sums all the logarithmic terms in powers of v, is derived in terms of the regular part of the periodic Helmholtz Green’s function. By using an Ewald summation technique to first obtain a rapidly converging infinite series representation for this regular part, a simple Newton iteration scheme on the transcendental equation is implemented to numerically evaluate the improved ‘log-summed’ approximation to the principal eigenvalue. From a numerical computation of the PDE eigenvalue problem defined on the fundamental Wigner–Seitz (WS) cell for the lattice, it is shown that the three-term asymptotic approximation for the principal eigenvalue agrees well with the numerical result only for a rather small trap radius. In contrast, the log-summed asymptotic result provides a very close approximation to the principal eigenvalue even when the trap radius is only moderately small. For a circular trap, the first few transcendental correction terms that further improves the log-summed approximation for the principal eigenvalue are derived. Finally, it is shown numerically that, amongst all Bravais lattices with a fixed area of the primitive cell, the principal eigenvalue is maximised for a regular hexagonal arrangement of traps.

We consider two natural gradings on the space of symmetric functions: by degree and by length. We introduce a differential operator T that leaves the components of this double grading invariant and exhibit a basis of bihomogeneous symmetric functions in which this operator is triangular. This allows us to compute the eigenvalues of T, which turn out to be nonnegative integers.

Let $\Sigma $ be a compact surface with boundary. For a given conformal class c on $\Sigma $ the functional $\sigma _k^*(\Sigma ,c)$ is defined as the supremum of the kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior of this functional on the moduli space of conformal classes on $\Sigma $ . A precise formula for the limit of $\sigma _k^*(\Sigma ,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander–Nadirashvili invariants of closed manifolds defined as $\inf _{c}\sigma _k^*(\Sigma ,c)$ , where the infimum is taken over all conformal classes c on $\Sigma $ . We show that these quantities are equal to $2\pi k$ for any surface with boundary. As an application of our techniques we obtain new estimates on the kth normalized Steklov eigenvalue of a nonorientable surface in terms of its genus and the number of boundary components.

By means of a counter-example, we show that the Reilly theorem for the upper bound of the first non-trivial eigenvalue of the Laplace operator of a compact submanifold of Euclidean space (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) does not work for a (codimension ⩾2) compact spacelike submanifold of Lorentz–Minkowski spacetime. In the search of an alternative result, it should be noted that the original technique in (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) is not applicable for a compact spacelike submanifold of Lorentz–Minkowski spacetime. In this paper, a new technique, based on an integral formula on a compact spacelike section of the light cone in Lorentz–Minkowski spacetime is developed. The technique is genuine in our setting, that is, it cannot be extended to another semi-Euclidean spaces of higher index. As a consequence, a family of upper bounds for the first eigenvalue of the Laplace operator of a compact spacelike submanifold of Lorentz–Minkowski spacetime is obtained. The equality for one of these inequalities is geometrically characterized. Indeed, the eigenvalue achieves one of these upper bounds if and only if the compact spacelike submanifold lies minimally in a hypersphere of certain spacelike hyperplane. On the way, the Reilly original result is reproved if a compact submanifold of a Euclidean space is naturally seen as a compact spacelike submanifold of Lorentz–Minkowski spacetime through a spacelike hyperplane.

We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein–Gordon and fractional Schrödinger equations. Our estimates extend those of Frank–Sabin in the case of the wave and Klein–Gordon equations, and generalize work of Frank et al. and Frank–Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results.

The main novelty of this paper is our derivation and use of estimates for weighted oscillatory integrals, which we combine with an approach due to Frank and Sabin. Our weighted oscillatory integral estimates are, in a certain sense, rather delicate endpoint versions of known dispersive estimates with power-type weights of the form $|\xi |^{-\lambda }$ or $(1 + |\xi |^2)^{-\lambda /2}$, where $\lambda \in \mathbb {R}$. We achieve optimal decay rates by considering such weights with appropriate $\lambda \in \mathbb {C}$. For the wave and Klein–Gordon equations, our weighted oscillatory integral estimates are new. For the fractional Schrödinger equation, our results overlap with prior work of Kenig–Ponce–Vega in a certain regime. Our contribution to the theory of weighted oscillatory integrals has also been influenced by earlier work of Carbery–Ziesler, Cowling et al., and Sogge–Stein.

Finally, we provide some applications of our new Strichartz estimates for orthonormal families of data to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces.

We show that for any n divisible by 3, almost all order-n Steiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almost resolvable).

In this paper we give sufficient conditions to obtain continuity results of solutions for the so called ϕ-Laplacian Δϕ with respect to domain perturbations. We point out that this kind of results can be extended to a more general class of operators including, for instance, nonlocal nonstandard growth type operators.

In this article we consider the ergodic risk-sensitive control problem for a large class of multidimensional controlled diffusions on the whole space. We study the minimization and maximization problems under either a blanket stability hypothesis, or a near-monotone assumption on the running cost. We establish the convergence of the policy improvement algorithm for these models. We also present a more general result concerning the region of attraction of the equilibrium of the algorithm.

In this paper, we look at a linear system of ordinary differential equations as derived from the two-dimensional Ginzburg–Landau equation. In two cases, it is known that this system admits bounded solutions coming from the invariance of the Ginzburg–Landau equation by translations and rotations. The specific contribution of our work is to prove that in the other cases, the system does not admit any bounded solutions. We show that this bounded solution problem is related to an eigenvalue problem.

We consider the focussing fractional periodic Korteweg–deVries (fKdV) and fractional periodic non-linear Schrödinger equations (fNLS) equations, with L2 sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each $\lambda > 0$, there is a travelling wave solution to fKdV and fNLS $\phi : \|\phi \|_{L^2[-T,T]}^2=\lambda $, which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in Hα/2[ − T, T] and a natural technical condition. This is done rigorously, without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers.

In this article, we study some Kramers–Fokker–Planck operators with a polynomial potential $V(q)$ of degree greater than two having quadratic limiting behaviour. This work provides an accurate global subelliptic estimate for Kramers–Fokker–Planck operators under some conditions imposed on the potential.

For Laplacians defined by measures on a bounded domain in ℝn, we prove analogues of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Pólya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.

We show that the spectrum of the relativistic mean curvature operator on a bounded domain Ω ⊂ ℝN (N ⩾ 1) having smooth boundary, subject to the homogeneous Dirichlet boundary condition, is exactly the interval (λ1(2), ∞), where λ1(2) stands for the principal frequency of the Laplace operator in Ω.

We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order $\unicode[STIX]{x1D70C}<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M,g)$ and we show that there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}=c^{4}g$ such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric $g$ and do not satisfy the unique continuation principle.

We obtain generalizations of the classical Menchov–Rademacher theorem to the case of continuous orthogonal systems. These results are applied to show the existence of Moller wave operators in Schrödinger evolution.

In this paper we study a general eigenvalue problem for the so called (p, 2)-Laplace operator on a smooth bounded domain Ω ⊂ ℝN under a nonlinear Steklov type boundary condition, namely

\[\left\{ \begin{aligned} -\Delta_pu-\Delta u & =\lambda a(x)u \quad {\rm in}\ \Omega,\\ (|\nabla u|^{p-2}+1)\dfrac{\partial u}{\partial\nu} & =\lambda b(x)u \quad {\rm on}\ \partial\Omega . \end{aligned} \right.\]

Given a smooth compact hypersurface $M$ with boundary $\unicode[STIX]{x1D6F4}=\unicode[STIX]{x2202}M$, we prove the existence of a sequence $M_{j}$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\unicode[STIX]{x1D70E}_{k}(M_{j})$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_{j}$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.