Investigation of solvability of the multidimensional two-phase Stefan and the nonstationary filtration Florin problems for second order parabolic equations in weighted Hölder spaces of functions

Multidimensional two-phase Stefan (k=1) and nonstationary filtration Florin (k=0) problems for second order parabolic equations in the case when the free boundary is a graph of a functionx _n =Ψ _k (x′t),x′∈ ^n?1 ,n≥2,t∈(0,T) are studied. A unique solvability theorem in weighted Hölder spaces of functions with time power weight is proved, coercive estimates for solutions are obtained. Bibliography: 30 titles.     

Applications of Bernstein-Durrmeyer operators in estimating the covering number

The paper deals with estimates of the covering number for some Mercer kernel Hilbert space with Bernstein-Durrmeyer operators. We first give estimates of l2-norm of Mercer kernel matrices reproducing by the kernels K(α,β)(x,y):=∞∑k=0 C(α,β)k Qk(α,β)(x)Qk(α,β)(y),where Qα,βk(x) are the Jacobi polynomials of order k on (0, 1), Cα,βk > 0 are real numbers,and from which give the lower and upper bounds of the covering number for some particular reproducing kernel Hilbert space reproduced by K(α,β)(x,y).     

Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach

In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation ut(x,t)=(k(x)uxx(x,t)), with Dirichlet boundary conditions u(0,t)=ψ₀, u(1,t)=ψ₁. Main goal of this study is to investigate the distinguishability of the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f(t):=k(0)ux(0,t) or/and h(t):=k(1)ux(1,t). In addition to these, the values k^′(0) and k^′(1) of the unknown coefficient k(x) at x=0 and x=1, respectively, are also determined via the input data. Furthermore, it is shown that measured output dataf(t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x=0 and x=1.     

Weakly Singular Integral Operators in Weighted L∞–Spaces

We study integral operators on (−1, 1) with kernels k(x, t) which may have weak singularities in (x, t) with x ∈N₁, t ∈N₂, or x=t, where N₁,N₂ are sets of measure zero. It is shown that such operators map weighted L^∞–spaces into certain weighted spaces of smooth functions, where the degree of smoothness is the higher the smoother the kernel k(x, t) as a function in x is. The spaces of smooth function are generalizations of the Ditzian-Totik spaces which are defined in terms of the errors of best weighted uniform approximation by algebraic polynomials.     

The approximate identity kernels of product type for the Walsh system

At present there are only a few approximate identity kernels for the Walsh system, for example, the p^N-truncated Dirichlet kernel D_{p^N − 1}(t) = ∑_{j = 0}^{pN − 1} w_j(t) [6]; the Abel-Poisson kernel λ_γ(t) = ∑_{k = 0}^∞ γ^kw_k(t) [3], and so on. In [6], Zheng has introduced a new kind of approximate identity kernels for the Walsh system—the kernels of product type. In the present paper we discuss the approximation properties of such product type kernels. Estimates of their moments as well as a direct approximation theorem are obtained. Then, to establish an inverse approximation theorem, we need the p-adic derivative of product type kernels and we estimate this derivative in L¹-norm.     

Littlewood–Paley and Spectral Multipliers on Weighted L p Spaces

Let L be a non-negative self-adjoint operator acting on L ²(X), where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e ^?tL whose kernel p _t (x,y) has a Gaussian upper bound but there is no assumption on the regularity in variables x and y. In this article we study weighted L ^p -norm inequalities for spectral multipliers of L. We show that a weighted Hörmander-type spectral multiplier theorem follows from weighted L ^p -norm inequalities for the Lusin and Littlewood–Paley functions, Gaussian heat kernel bounds, and appropriate L ² estimates of the kernels of the spectral multipliers.     

Carleman Estimates for Second-Order Hyperbolic Equations

In the space of variables (x, t) ∈ ? ⁿ⁺¹, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ? ? ⁿ⁺¹ whose projection to the x-space is a compact domain Ω, we consider the question of construction of a stability estimate for a solution to the Cauchy problem with data on the lateral boundary S of D. The well-known method for obtaining such estimates bases on the Carleman estimates with an exponential-type weight function exp(2τ?(x, t)) whose construction faces certain difficulties in case of hyperbolic equations with variable coefficients. We demonstrate that if D is symmetric with respect to the plane t = 0 then we can take ?(x, t) to be the function ?(x, t) = s ²(x, x ⁰) ? pt ², where s(x, x ⁰) is the distance between points x and x ⁰ in the Riemannian metric induced by the differential equation, p is some positive number less than 1, and the fixed point x ⁰ can either belong to the domain Ω or lie beyond it. As for the metric, we suppose that the sectional curvature of the corresponding Riemannian space is bounded above by some number k ₀ ≥ 0. In case of space of nonpositive curvature the parameter p can be taken arbitrarily close to 1; in this case as p → 1 the stability estimates lead to a uniqueness theorem which describes exactly the domain of the solution continuation through S. It turns out that, in case of space of bounded positive curvature, construction of a Carleman estimate is possible only if the product of k ₀ and sup _x∈Ω s ²(x, x ⁰) satisfies some smallness condition.     

On the eigenvalues and eigenfunctions of some integral operators

Some properties of the eigenvalues of the integral operator K_gt defined as K_τf(x) = ∫₀^τK(x ? y) f (y) dy were studied by Vittal Rao (J. Math. Anal. Appl.53 (1976), 554–566), with some assumptions on the kernel K(x). In this paper the eigenfunctions of the operator K_τ are shown to be continuous functions of τ under certain circumstances. Also, the results of Vittal Rao and the continuity of eigenfunctions are shown to hold for a larger class of kernels.     

A product quadrature algorithm by Hermite interpolation

This paper is concerned with numerical integration of ∫¹₋₁f(x)k(x)dx by product integration rules based on Hermite interpolation. Special attention is given to the kernel k(x) = e^iτx, with a view to providing high precision rules for oscillatory integrals. Convergence results and error estimates are obtained in the case where the points of integration are zeros of p_n(W; x) or of (1 − x²)p_n−2(W; x), where p_n(W; x), n = 0, 1, 2…, are the orthonormal polynomials associated with a generalized Jacobi weight W. Further, examples are given that test the performance of the algorithm for oscillatory weight functions.     

Oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments

The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) L_nx(t) + δq(t) f(x[g₁(t)],…, x[g_m(t)]) = 0, where δ = ± 1 and L₀x(t) = x(t), L_kx(t) = a_k(t)(L_{k ? 1}x(t))^., $\text{k = 1, 2,…, n (}{}^{\mathrm{.}}\text{=}\text{d}\text{dt}\text{)}$, is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y^.(t) + c₁(t) f(y[g₁(t)],…, y[g_m(t)]) = 0 and (a_{n ? 1}(t)z^.(t))^. ? c₂(t) f(z[g₁(t)],…, z[g_m(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos.     

Single point blow-up for a semilinear initial value problem

The initial value problem on [?R, R] is considered: u_t(t, x) = u_xx(t, x) + u(t, x)^γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥_∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψ_xx + ψ^γ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then lim_{t → T}u(t, x) and lim_{t → T}u_x(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0.     

On a class of some special sets on thek-skeleton of a convex compact set

In this paper, generalizing the notion of a path we define ak-area to be the setD={g(t):t ∈J} on thek-skeleton of a convex compact setK in a Hilbert space, whereg is a continuous injection map from thek-dimensional convex compact setJ to thek-skeleton ofK. We also define anE ^k-area onK, whereE ^k is ak-dimensional subspace, to be ak-area with the propertyπ(g(t))=t,t ∈π(K), whereπ is the orthogonal projection onE ^k. This definition generalizes the notion of an increasing path on the 1-skeleton ofK. The existence of such sets is studied whenK is a subset of a Euclidean space or of a Hilbert space. Finally some conjectures are quoted for the number of such sets in some special cases.     

Applications of the Fourier—Wiener transform to differential equations on infinite dimensional spaces,II

In this paper, we investigate the existence and regularity of solutions for Cauchy problems associated with the following two equations: (1) u_t(x, t) = (?1)^{k + 1}Δ^ku(x, t), and (2) u_tt(x, t) = (?1)^{k + 1}Δ^ku on infinite dimensional spaces. The nonhomogeneous cases associated with (1) and (2) are also considered.     

On the numerical characterization of the reachability cone for an essentially nonnegative matrix

Given an n×n real matrix A with nonnegative off-diagonal entries, the solution to , x₀=x(0), t?0 is x(t)=e^tAx₀. The problem of identifying the initial points x₀ for which x(t) becomes and remains entrywise nonnegative is considered. It is known that such x₀ are exactly those vectors for which the iterates x^(k)=(I+hA)^kx₀ become and remain nonnegative, where h is a positive, not necessarily small parameter that depends on the diagonal entries of A. In this paper, this characterization of initial points is extended to a numerical test when A is irreducible: if x^(k) becomes and remains positive, then so does x(t); if x(t) fails to become and remain positive, then either x^(k) becomes and remains negative or it always has a negative and a positive entry. Due to round-off errors, the latter case manifests itself numerically by x^(k) converging with a relatively small convergence ratio to a positive or a negative vector. An algorithm implementing this test is provided, along with its numerical analysis and examples. The reducible case is also discussed and a similar test is described.     

A generalised Christoffel-Darboux formula and reproducing kernels in spaces of polymonogenic and polyharmonic functions

Let ?_x be the Dirac operator in R^m+1, let t ∈ N and let K_t(x,y) be the Bergman kernel for the space L ₂M_t of square integrable polymonogenic functions of order t in the unit ball B(1) of R^m+1. Then expressions for K_t(x,y) are derived using a generalised Christoffel-Darboux formula related to the orthogonal basis of generalised Gegenbauer polynomials in the weighted L₂-space L₂(B(1);(1 + x²⁾α), α > ?1.     

Approximate solutions of functional differential equations

The author discusses the best approximate solution of the functional differential equation x′(t) = F(t, x(t), x(h(t))), 0 < t < l satisfying the initial condition x(0) = x₀, where x(t) is an n-dimensional real vector. He shows that, under certain conditions, the above initial value problem has a unique solution y(t) and a unique best approximate solution $\text{p}\text{?}{}_{\mathrm{k}}\text{(t)}$ of degree k (cf. [1]) for a given positive integer k. Furthermore, $\text{sup}{}_{\mathrm{0?t?l}}\text{¦}\text{p}\text{?}{}_{\mathrm{k}}\text{(t) ? y(t)¦ → 0}\text{as}\text{k → ∞}$, where ¦ · ¦ is any norm in Rⁿ.     

On the stability properties of perturbed linear nonstationary systems

Several results are presented that relate the stability properties of a perturbed linear nonstationary system ?(t) = (A(t) + B(t)) x(t) to those of an unperturbed linear system ?(t) = A(t) x(t). Similarly, the stability properties of the discrete system x_{k + 1} = (A_k + B_k) x_k are related to those of x_{k + 1} = A_kx_k.     

On the invariance of wave operators

Let H, K be self-adjoint operators on a Hilbert space. Kato's Invariance Principle (T. Kato, “Perturbation Theory for Linear Operators,” Springer-Verlag, Berlin, 1966) states that under certain conditions W(φ(H), φ(K)) = s-lim_{t → + ∞}e^it(φ)ke^?it(φ)H exists and is independent of the monotone function φ whenever W(H, K) exists. The purpose of this paper is to present a new proof of Kato's result based upon a study of the variation of W(φ(H), φ(K)) with respect to φ. It is shown (Theorem 1) that this variation vanishes, and (Theorem 2) that the invariance principle holds provided that K-H belongs to a large subset of the Hilbert-Schmidt class of compact operators.     

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L²(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L² estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.