Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(xⁱ)=c_i for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.     

A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions

Let p∈(1,N), Ω⊂RN a bounded W1,p-extension domain and let μ be an upper d-Ahlfors measure on ∂Ω with d∈(N−p,N). We show in the first part that for every p∈[2N/(N+2),N)∩(1,N), a realization of the p-Laplace operator with (nonlinear) generalized nonlocal Robin boundary conditions generates a (nonlinear) strongly continuous submarkovian semigroup on L²(Ω), and hence, the associated first order Cauchy problem is well posed on Lq(Ω) for every q∈[1,∞). In the second part we investigate existence, uniqueness and regularity of weak solutions to the associated quasi-linear elliptic equation. More precisely, global a priori estimates of weak solutions are obtained.     

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.     

Inequalities for single crystal rod growth by edge-defined film-fed growth (E.F.G.) technique

A boundary value problem in the case of the second order axi-symmetric Young-Laplace differential equation (some of whose solutions describe the static meniscus free surface, i.e. the static liquid bridge free surface between the shaper and the crystal, occurring in single crystal rod growth) is analyzed. The analysis concerns the dependence of the solution of an initial value problem of the equation on a parameter p (the controllable part of the pressure difference Δp across the free surface). Inequalities are established for p which are necessary or sufficient conditions for the existence of a solution which represents a stable and convex free surface of a static meniscus. The analysis is numerically illustrated for the static menisci occurring in the NdYAG laser single crystal rod growth from the melt by edge-defined film-fed growth (E.F.G.) technique. This kind of inequalities can be useful in the experiment planning and technology design.     

A numerical method for nonlinear age-structured population models with finite maximum age

We propose a new numerical method for the approximation of solutions to a non-autonomous form of the classical Gurtin-MacCamy population model with a mortality rate that is the sum of an intrinsic age-dependent rate that becomes unbounded as the age approaches its maximum value, plus a non-local, non-autonomous, bounded rate that depends on some weighted population size. We prove that our new quadrature based method converges to second-order and we show the results of several numerical simulations.     

Elliptic equations with indefinite concave nonlinearities near the origin

In this note existence of infinitely many solutions is proved for an elliptic equation with indefinite concave nonlinearities.     

Blow-up time for solutions to some nonlinear Volterra integral equations

The problem of the estimating of a blow-up time for solutions of Volterra nonlinear integral equation with convolution kernel is studied. New estimates, lower and upper, are found and, moreover, the procedure for the improvement of the lower estimate is presented. Main results are illustrated by examples. The new estimates are also compared with some earlier ones related to a shear band model.     

An approximate solution for nonlinear backward parabolic equations

We consider the backward parabolic equation

     

New invariant sets to nonlinear diffusion equations with x-dependent convection and absorption

In this paper, we introduce a new invariant set

     

A partially linearized sigma point filter for latent state estimation in nonlinear time series models

A new technique for the latent state estimation of a wide class of nonlinear time series models is proposed. In particular, we develop a partially linearized sigma point filter in which random samples of possible state values are generated at the prediction step using an exact moment-matching algorithm and then a linear programming based procedure is used in the update step of the state estimation. The effectiveness of the new filtering procedure is assessed via a simulation example that deals with a highly nonlinear, multivariate time series representing an interest rate process.