Strongly continuous semigroups of operators generated by systems of pseudodifferential operators in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

This paper considers semigroups of operators generated by pseudodifferential operators in weighted L _p -spaces of vector functions on \mathbbRⁿ {\mathbb{R}^n} (or on a compact manifold without boundary). Sufficient conditions for a semigroup to be strongly continuous and analytic are obtained, conditions for it to be completely continuous are found, and the distribution of the eigenvalues of its infinitesimal generator is examined. Also, an integral representation that singles out the principal term of the semigroup as t → 0+ is established.     

Asymptotic analysis of a boundary-value problem with nonlinear multiphase boundary interactions in a perforated domain

We consider a boundary-value problem for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ω _ε that is ε-periodically perforated by small holes. The holes are split into two ε-periodic sets depending on the boundary interaction via their boundary surfaces. Therefore, two different nonlinear boundary conditions σ _ε (u _ε ) + εκ _m (u _ε ) = εg _ε ^(m) , m = 1, 2, are given on the corresponding boundaries of the small holes. The asymptotic analysis of this problem is performed as ε → 0, namely, the convergence theorem for both the solution and the energy integral is proved without using an extension operator, asymptotic approximations for the solution and the energy integral are constructed, and the corresponding approximation error estimates are obtained.     

Boundary Value Problems with Compatible Boundary Conditions

If Y is a subset of the space ℝⁿ × ℝⁿ, we call a pair of continuous functions U, V Y-compatible, if they map the space ℝⁿ into itself and satisfy Ux · Vy ≥ 0, for all (x, y) ∈ Y with x · y ≥ 0. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential n-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer's fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points P _δ is obtained. Then passing to the limits as δ tends to zero the so-obtained accumulation points are solutions of the problem.     

Approximating a Feller Semigroup by Using the Yosida Approximation of the Symbol of its Generator

We show that if the pseudodifferential operator −q(x,D) generates a Feller semigroup (T_t)_t≥0 then the Feller semigroups (T_t^(v))_t≥0 generated by the pseudodifferential operators with symbol will converge strongly to (T_t)_t≥0 as ν →∞.     

On homogenization of the mixed boundary-value problem for the heat equation in a domain whose part contains channels arranged in a periodic way

In this paper, we study the asymptotic behavior of the solutionsu _ε (ε is a small parameter) of boundaryvalue problems for the heat equation in the domain Ω_ε=Ω⁻∪Ω _ε ⁺ ∪γ one part of which (Ω _ε ⁺ ) contains ε-periodically situated channels with diameters of order ε and the other part of which (Ω⁺) is a homogeneous medium; γ=∂Ω _ε ⁺ ∩∂Ω⁺. On the boundary of the channels the Neumann boundary condition is posed, and on ∂Ω_ε∩∂Ω the Dirichlet boundary condition is prescribed. The homogenized problem is the Dirichlet problem in Ω with the transmission condition on γ. The estimates for the difference betweenu _ε and the solution of the homogenized problem are obtained. Bibliography: 14 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 20, pp. 27–47, 1997.     

Asymptotic analysis of a parabolic semilinear problem with nonlinear boundary multiphase interactions in a perforated domain

We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ω_ε that is ε-periodically perforated by small holes of size O\mathcal {O}(ε). The holes are divided into two ε-periodical sets depending on the boundary interaction at their surfaces, and two different nonlinear Robin boundary conditions σ_ε(u _ε) + εκ _m (u _ε) = εg ^(m) _ε, m = 1, 2, are imposed on the boundaries of holes. We study the asymptotics as ε → 0 and establish a convergence theorem without using extension operators. An asymptotic approximation of the solution and the corresponding error estimate are also obtained. Bibliography: 60 titles. Illustrations: 1 figure.     

A singular perturbation free boundary problem for elliptic equations in divergence form

In this paper we study the free boundary problem arising as a limit as ɛ → 0 of the singular perturbation problem , where A = A(x) is Holder continuous, β _ɛ converges to the Dirac delta δ₀. By studying some suitable level sets of u _ɛ, uniform geometric properties are obtained and show to hold for the free boundary of the limit function. A detailed analysis of the free boundary condition is also done. At last, using very recent results of Salsa and Ferrari, we prove that if A and Γ are Lipschitz continuous, the free boundary is a C ^1,γ surface around a.e. point on the free boundary.     

Spectral invariance for algebras of pseudodifferential operators on besov-triebel-lizorkin spaces

The algebra of pseudodifferential operators with symbols inS _1,δ ⁰ , δ<1, is shown to be a spectrally invariant subalgebra of ℒ(b _p,q ^s ) and ℒ(F _p,q ^s ). The spectrum of each of these pseudodifferential operators acting onB _p,q ^s orF _p,q ^s is independent of the choice ofs, p, andq.     

Conservative finite difference scheme for a singularly perturbed elliptic reaction-diffusion equation: Approximation of solutions and derivatives

A boundary value problem for a singularly perturbed elliptic reaction-diffusion equation in a vertical strip is considered. The derivatives are written in divergent form. The derivatives in the differential equation are multiplied by a perturbation parameter ɛ², where ɛ takes arbitrary values in the interval (0, 1]. As ɛ → 0, a boundary layer appears in the solution of this problem. Using the integrointerpolational method and the condensing grid technique, conservative finite difference schemes on flux grids are constructed that converge ɛ-uniformly at a rate of O(N ₁⁻²ln² N ₁ + N ₂⁻²), where N ₁ + 1 and N ₂ + 1 are the number of mesh points on the x ₁-axis and the minimal number of mesh points on a unit interval of the x ₂-axis respectively. The normalized difference derivatives ɛ ^k (∂ ^k /∂x ₁ ^k )u(x) (k = 1, 2), which are ɛ-uniformly bounded and approximate the normalized derivatives in the direction across the boundary layer, and the derivatives along the boundary layer (∂ ^k /∂ x ₂ ^k )u(x) (k = 1, 2) converge ɛ-uniformly at the same rate.     

On the maximal L p -regularity of parabolic mixed-order systems

We study maximal L _p -regularity for a class of pseudodifferential mixed-order systems on a space–time cylinder \mathbbRⁿ ×\mathbbR{\mathbb{R}^n \times \mathbb{R}} or X ×\mathbbR{X \times \mathbb{R}} , where X is a closed smooth manifold. To this end, we construct a calculus of Volterra pseudodifferential operators and characterize the parabolicity of a system by the invertibility of certain associated symbols. A parabolic system is shown to induce isomorphisms between suitable L _p -Sobolev spaces of Bessel potential or Besov type. If the cross section of the space–time cylinder is compact, the inverse of a parabolic system belongs to the calculus again. As applications, we discuss time-dependent Douglis–Nirenberg systems and a linear system arising in the study of the Stefan problem with Gibbs–Thomson correction.     

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

Multidimensional ultrametric pseudodifferential equations

We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and the space D′₀(X) of generalized functions on a multidimensional ultrametric space. We also consider some family of pseudodifferential operators on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of ultrametric characteristics are introduced. We prove an existence theorem and describe all solutions for the Cauchy problem (an analog of the Kovalevskaya theorem).     

The Riesz Basis Property of an Indefinite Sturm-Liouville Problem with a Non-Odd Weight Function

For the Sturm-Liouville eigenvalue problem − f′′ = λrf on [−1, 1] with Dirichlet boundary conditions and with an indefinite weight function r changing its sign at 0 we discuss the question whether the eigenfunctions form a Riesz basis of the Hilbert space L ² _|r|[− 1, 1]. So far a number of sufficient conditions on r for the Riesz basis property are known. However, a sufficient and necessary condition is only known in the special case of an odd weight function r. We shall here give a generalization of this sufficient and necessary condition for certain generally non-odd weight functions satisfying an additional assumption.      

Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones

The boundary value problem for the similar stream function f = f(η;λ) of the Cheng–Minkowycz free convection flow over a vertical plate with a power law temperature distribution T_w(x) = T_∞ + Ax^λ in a porous medium is revisited. It is shown that in the λ-range − 1/2 < λ < 0 , the well known exponentially decaying “first branch” solutions for the velocity and temperature fields are not some isolated solutions as one has believed until now, but limiting cases of families of algebraically decaying multiple solutions. For these multiple solutions well converging analytical series expressions are given. This result yields a bridging to the historical quarreling concerning the feasibility of exponentially and algebraically decaying boundary layers. Owing to a mathematical analogy, our results also hold for the similar boundary layer flows induced by continuous surfaces stretched in viscous fluids with power-law velocities u_w(x)∼ x^λ. (Received: June 7, 2005)     

On a grid-method solution of the Laplace equation in an infinite rectangular cylinder under periodic boundary conditions

We study the Dirichlet problem for the Laplace equation in an infinite rectangular cylinder. Under the assumption that the boundary values are continuous and bounded, we prove the existence and uniqueness of a solution to the Dirichlet problem in the class of bounded functions that are continuous on the closed infinite cylinder. Under an additional assumption that the boundary values are twice continuously differentiable on the faces of the infinite cylinder and are periodic in the direction of its edges, we establish that a periodic solution of the Dirichlet problem has continuous and bounded pure second-order derivatives on the closed infinite cylinder except its edges. We apply the grid method in order to find an approximate periodic solution of this Dirichlet problem. Under the same conditions providing a low smoothness of the exact solution, the convergence rate of the grid solution of the Dirichlet problem in the uniform metric is shown to be on the order of O(h ² ln h ⁻¹), where h is the step of a cubic grid.     

Uniqueness theorem of solutions for stochastic differential equation in the plane

LetM={M _z, z ∈ R ₊ ² } be a continuous square integrable martingale andA={A _z, z ∈ R ₊ ² be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane:dX _z=α(z, X_z)dM_z+β(z, X_z)dA_z, z∈R ₊ ² , X_z=Z_z, z∈∂R ₊ ² , whereR ₊ ² =[0, +∞)×[0,+∞) and ∂R ₊ ² is its boundary,Z is a continuous stochastic process on ∂R ₊ ² . We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate (see [1, Theorem 3.2]). Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in [2]. Supported by the National Science Foundation and the Postdoctoral Science Foundation of China     

Transfer of absolute continuity by a flow generated by a stochastic equation with reflection

Let ϕ_t(x), x ∈ ℝ₊ be a value taken at time t ≥ 0 by a solution of a stochastic equation with normal reflection from a hyperplane starting at initial time from x. We characterize the absolutely continuous (with respect to Lebesgue measure) component and the singular component of a stochastic measure-valued process μ_t = μ ○ ϕ _t ⁻¹ that is the image of a certain absolutely continuous measure μ under random mapping ϕ_t(·). We prove that the restriction of the Hausdorff measure H ^d−1 to the support of the singular component is σ-finite and give sufficient conditions guaranteeing that the singular component is absolutely continuous with respect to H ^d−1. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1663–1673, December, 2006.     

Recovering of Damping Coefficients for a System of Coupled Wave Equations with Neumann Boundary Conditions: Uniqueness and Stability

The authors study the inverse problem of recovering damping coefficients for two coupled hyperbolic PDEs with Neumann boundary conditions by means of an additional measurement of Dirichlet boundary traces of the two solutions on a suitable, explicit subportion Γ1 of the boundary Γ, and over a computable time interval T > 0. Under sharp conditions on Γ0 = ΓnΓ1, T > 0, the uniqueness and stability of the damping coefficients are established. The proof uses critically the Carleman estimate due to Lasiecka et al. in 2000, together with a convenient tactical route “post-Carleman estimates” suggested by Isakov in 2006.