Martingale estimation functions for Bessel processes

In this paper we derive martingale estimating functions for the dimensionality parameter of a Bessel process based on the eigenfunctions of the diffusion operator. Since a Bessel process is non-ergodic and the theory of martingale estimating functions is developed for ergodic diffusions, we use the space-time transformation of the Bessel process and formulate our results for a modified Bessel process. We deduce consistency, asymptotic normality and discuss optimality. It turns out that the martingale estimating function based of the first eigenfunction of the modified Bessel process coincides with the linear martingale estimating function for the Cox Ingersoll Ross process. Furthermore, our results may also be applied to estimating the multiplicity parameter of a one-dimensional Dunkl process and some related polynomial processes.

Multiplicity of positive solutions for (p,q)-Laplace equations with two parameters

We study the zero Dirichlet problem for the equation [Formula: see text] in a bounded domain [Formula: see text], with [Formula: see text]. We investigate the relation between two critical curves on the [Formula: see text]-plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighborhoods of the point [Formula: see text], where [Formula: see text] is the first eigenfunction of the [Formula: see text]-Laplacian, we show the existence of two and, which is rather unexpected, three distinct positive solutions, depending on a relation between the exponents [Formula: see text] and [Formula: see text].

On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains

The $p$-Laplacian has attracted more and more attention in data analysis disciplines in the past decade. However, there is still a knowledge gap about its behavior, which limits its practical application. In this paper, we are interested in its iterative behavior in domains contained in two-dimensional Euclidean space. Given a connected set $\varOmega _0 \subset \mathbb{R}^2$, define a sequence of sets $(\varOmega _n)_{n=0}^{\infty }$ where $\varOmega _{n+1}$ is the subset of $\varOmega _n$ where the first eigenfunction of the (properly normalized) Neumann $p$-Laplacian $ -\varDelta ^{(p)} \phi = \lambda _1 |\phi |^{p-2} \phi $ is positive (or negative). For $p=1$, this is also referred to as the ratio cut of the domain. We conjecture that these sets converge to the set of rectangles with eccentricity bounded by 2 in the Gromov–Hausdorff distance as long as they have a certain distance to the boundary $\partial \varOmega _0$. We establish some aspects of this conjecture for $p=1$ where we prove that (1) the 1-Laplacian spectral cut of domains sufficiently close to rectangles is a circular arc that is closer to flat than the original domain (leading eventually to quadrilaterals) and (2) quadrilaterals close to a rectangle of aspect ratio $2$ stay close to quadrilaterals and move closer to rectangles in a suitable metric. We also discuss some numerical aspects and pose many open questions.

On the viscosity solutions of eigenvalue problems for a class of nonlinear elliptic equations

We consider viscosity solutions of a class of nonlinear degenerate elliptic equations, involving a parameter, on bounded domains. These arise in the study of eigenvalue problems. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many instances, show the existence of the first eigenvalue and an associated positive first eigenfunction.

Properties of the first eigenvalue with sign-changing weight of the discrete p-Laplacian and applications

By establishing some results around the first eigenvalue λ1(m) for the following problem: -Δ(φp(Δu(k - 1)))= λm(k)φp(u(k)); k∈ [1; n]; u(0) = 0 = u(n + 1); where m ∈ M([1; n]) = {m : [1; n] → R /∃ k∈ [1; n]; m(k) > 0} ; as the constant sign of the first eigenfunction with λ1(m); the simplicity of λ1(m); the strict monotonicity property with respect the weight and sign change of any eigenfunction with ( λ > λ1(m)); we prove the existence and non-existence of solutions of the problem (1.1).

On the resonant Lane–Emden problem for the p-Laplacian

We study the positive solutions of the Lane–Emden problem -Δpu = λp|u|q-2u in Ω, u = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded and smooth domain, N ≥ 2, λp is the first eigenvalue of the p-Laplacian operator Δp, p > 1, and q is close to p. We prove that any family of positive solutions of this problem converges in [Formula: see text] to the function θpep when q → p, where ep is the positive and L∞-normalized first eigenfunction of the p-Laplacian and [Formula: see text]. A consequence of this result is that the best constant of the immersion [Formula: see text] is differentiable at q = p. Previous results on the asymptotic behavior (as q → p) of the positive solutions of the nonresonant Lane–Emden problem (i.e. with λp replaced by a positive λ ≠ λp) are also generalized to the space [Formula: see text] and to arbitrary families of these solutions. Moreover, if uλ,q denotes a solution of the nonresonant problem for an arbitrarily fixed λ > 0, we show how to obtain the first eigenpair of the p-Laplacian as the limit in [Formula: see text], when q → p, of a suitable scaling of the pair (λ, uλ,q). For computational purposes the advantage of this approach is that λ does not need to be close to λp. Finally, an explicit estimate involving L∞- and L1-norms of uλ,q is also derived using set level techniques. It is applied to any ground state family {vq} in order to produce an explicit upper bound for ‖vq‖∞ which is valid for q ∈ [1, p + ϵ] where [Formula: see text].