Asymptotics of eigenvalues of A regular boundary-value problem

We study a boundary-value problem x ⁽ⁿ⁾ + Fx = λx, U _h(x) = 0, h = 1,..., n, where functions x are given on the interval [0, 1], a linear continuous operator F acts from a Hölder space H ^y into a Sobolev space W ₁ ^n+s , U _h are linear continuous functional defined in the space $H^{k_h } $ , and k _h ≤ n + s - 1 are nonnegative integers. We introduce a concept of k-regular-boundary conditions U _h(x)=0, h = 1, ..., n and deduce the following asymptotic formula for eigenvalues of the boundary-value problem with boundary conditions of the indicated type: $\lambda _v = \left( {i2\pi v + c_ \pm + O(|v|^\kappa )} \right)^n $ , v = ± N, ± N ± 1,..., which is true for upper and lower sets of signs and the constants κ≥0 and c _± depend on boundary conditions.     

Optimal Estimates for the Gradient of Harmonic Functions in the Unit Disk

Let U be the unit disk, p ≥ 1 and let h ^p (U) be the Hardy space of complex harmonic functions. We find the sharp constants C _p and the sharp functions C _p = C _p (z) in the inequality $$|Dw (z)|\leq {C_p}(1-|z|^2)^{-1-1/p} \|w\|_{h^p(\mathbf U)}, w\in h^p(\mathbf U), z\in \mathbf U,$$ in terms of Gaussian hypergeometric and Euler functions. This generalizes some results of Colonna related to the Bloch constant of harmonic mappings of the unit disk into itself and improves some classical inequalities by Macintyre and Rogosinski.     

C∞ functions in infinite dimension and linear partial differential difference equations with constant coefficients

This work is closed to [2] where a dense linear subspace \(\mathbb{E}\) (E) of the space ?(E) of the Silva C ^∞ functions on E is defined; the dual of \(\mathbb{E}\) (E) is described via the Fourier transform by a Paley-Wiener-Schwartz theorem which is formulated exactly in the same way as in the finite dimensional case. Here we prove existence and approximation result for solutions of linear partial differential difference equations in \(\mathbb{E}\) (E) with constant coefficients. We also obtain a Hahn-Banach type extension theorem for some C^∞ functions defined on a closed subspace of a DFN space, which is analogous to a Boland’s result in the holomorphic case [1].     

Topological Aspects of Differential Chains

In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U) $ , and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus??the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or? $\mathcal{D}'(U)$ . For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U?M, but not generally in the strong dual topology of? $\mathcal{B}'(U)$ .     

The Schwarz-Pick lemma and Julia lemma for real planar harmonic mappings

The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic mapping U of D into the open interval I = (?1, 1), $$\frac{{\Lambda _U (z)}} {{\cos \tfrac{{U(z)\pi }} {2}}} \leqslant \frac{4} {\pi }\frac{1} {{1 - \left| z \right|^2 }}$$ holds for z ∈ D, where Λ _U (z) is the maximum dilation of U at z. The inequality is sharp for any z ∈ D and any value of U(z), and the equality occurs for some point in D if and only if $U(z) = \tfrac{4} {\pi }\operatorname{Re} \{ \arctan \phi (z)\}$ , z ∈ D, with a Möbius transformation φ of D onto itself.     

Lions-type compactness and Rubik actions on the Heisenberg group

In this paper we prove a Lions-type compactness embedding result for symmetric unbounded domains of the Heisenberg group. The natural group action on the Heisenberg group ${\mathbb{H}^n=\mathbb{C}^n \times \mathbb{R}}$ is provided by the unitary group U(n) × {1} and its appropriate subgroups, which will be used to construct subspaces with specific symmetry and compactness properties in the Folland-Stein’s horizontal Sobolev space ${HW_0^{1,2}(\mathbb{H}^n)}$ . As an application, we study the multiplicity of solutions for a singular subelliptic problem by exploiting a technique of solving the Rubik-cube applied to subgroups of U(n) × {1}. In our approach we employ concentration compactness, group-theoretical arguments, and variational methods.     

Eigenvalue problem for finite difference equations with p-Laplacian

In this paper, we consider a discrete four-point boundary value problem $$\triangle\bigl(\phi_p\bigl(\triangle u(k-1)\bigr)\bigr)+ \lambda e(k)f\bigl(u(k)\bigr)=0,\quad k\in N(1,T),$$ subject to boundary conditions $$\triangle u(0)-\alpha u(l_{1})=0,\qquad\triangle u(T)+\beta u(l_{2})=0,$$ by a simple application of a fixed point theorem. If e(k),f(u(k)) are nonnegative, the solutions of the above problem may not be nonnegative, this is the main difficulty for us to study positive solution of this problem. In this paper, we give restrictive conditions ??l ₁??1, ??(T+1?l ₂)??1 to guarantee the solutions of this problem are nonnegative, if it has, under the conditions e(k),f(u(k)) are nonnegative. We first construct a new operator equation which is equivalent to the problem and provide sufficient conditions for the nonexistence and existence of at least one or two positive solutions. In doing so, the usual restrictions $f_{0}=\lim_{u\rightarrow 0^{+}}\frac{f(u)}{\phi_{p}(u)}$ and $f_{\infty}=\lim_{u\rightarrow\infty}\frac{f(u)}{\phi_{p}(u)}$ exist are removed.     

The submartingale problem for a class of degenerate elliptic operators

We consider the degenerate elliptic operator acting on ${C^2_b}$ functions on [0,∞) ^d : $$\mathcal{L}f(x)=\sum_{i=1}^d a_i(x) x_i^{\alpha_i} \frac{\partial^2 f}{\partial x_i^2} (x) +\sum_{i=1}^d b_i(x) \frac{\partial f}{\partial x_i}(x), $$ where the a _i are continuous functions that are bounded above and below by positive constants, the b _i are bounded and measurable, and the ${\alpha_i\in (0,1)}$ . We impose Neumann boundary conditions on the boundary of [0,∞) ^d . There will not be uniqueness for the submartingale problem corresponding to ${\mathcal{L}}$ . If we consider, however, only those solutions to the submartingale problem for which the process spends 0 time on the boundary, then existence and uniqueness for the submartingale problem for ${\mathcal{L}}$ holds within this class. Our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations $$ {\rm d}X_t^i=\sqrt{2a_i(X_t)} (X_t^i)^{\alpha_i/2}{\rm d}W^i_t + b_i(X_t) {\rm d}t + {\rm d}L_t^{X^i},\quad X^i_t \geq 0, $$ where ${W_t^i}$ are independent Brownian motions and ${L^{X_i}_t}$ is a local time at 0 for X ⁱ .     

Homoclinic Solutions for a Class of Second Order Hamiltonian Systems

In this paper we consider the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system $${\ddot q}-L(t)q+\nabla W(t,q)=0, \quad\quad\quad\quad\quad\quad\quad (\rm HS)$$ where ${L\in C({\mathbb R},{\mathbb R}^{n^2})}$ is a symmetric and positive definite matrix for all ${t\in {\mathbb R}}$ , W(t, q)?=?a(t)U(q) with ${a\in C({\mathbb R},{\mathbb R}^+)}$ and ${U\in C^1({\mathbb R}^n,{\mathbb R})}$ . The novelty of this paper is that, assuming L is bounded from below in the sense that there is a constant M?>?0 such that (L(t)q, q)?≥ M |q|² for all ${(t,q)\in {\mathbb R}\times {\mathbb R}^n}$ , we establish one new compact embedding theorem. Subsequently, supposing that U satisfies the global Ambrosetti–Rabinowitz condition, we obtain a new criterion to guarantee that (HS) has one nontrivial homoclinic solution using the Mountain Pass Theorem, moreover, if U is even, then (HS) has infinitely many distinct homoclinic solutions. Recent results from the literature are generalized and significantly improved.     

Well‐posedness and approximation of solutions of linear divergence‐form elliptic problems on exterior regions

This paper describes well‐posedness, spectral representations, and approximations of solutions of uniformly elliptic, second‐order, divergence form elliptic boundary value problems on exterior regions U in when N ≥ 3. Inhomogeneous Dirichlet, Neumann, and Robin boundary conditions are treated. These problems are first shown to be well‐posed in the space E¹(U) of finite‐energy functions on U using variational methods. Spectral representations of these solutions involving Steklov eigenfunctions and solutions subject to zero Dirichlet boundary conditions are described. Some approximation results for the A‐harmonic components are obtained. Positivity and comparison results for these solutions are given. Copyright © 2014 John Wiley & Sons, Ltd.     

Harmonic divisors and rationality of zeros of Jacobi polynomials

Let $P_{n}^{ ( \alpha,\beta ) } ( x ) $ be the Jacobi polynomial of degree n with parameters α,β. The main result of the paper states the following: If b≠1,3 and c are non-zero relatively prime natural numbers then $P_{n}^{ ( k+ ( d-3 ) /2,k+ ( d-3 ) /2 ) } ( \sqrt{b/c} ) \neq0$ for all natural numbers d,n and $k\in\mathbb{N}_{0}$ . Moreover, under the above assumption, the polynomial $Q ( x ) = \frac{b}{c} ( x_{1}^{2}+\cdots+x_{d-1}^{2} ) + ( \frac{b}{c}-1 ) x_{d}^{2}$ is not a harmonic divisor, and the Dirichlet problem for the cone {Q(x)<0} has polynomial harmonic solutions for polynomial data functions.     

Hyperbolic Laplace Operator and the Weinstein Equation in {\mathbb{R}^3}

We study the Weinstein equation $$\Delta u - \frac{k}{{x}_{2}} \frac{\partial}{\partial{x}_{2}} + \frac{l}{x^{2}_{2}}u = 0$$ , on the upper half space ${\mathbb{R}^3_{+} = \{ (x_{0}, x_{1}, x_{2}) \in \mathbb{R}^{3} | x_2 > 0\}}$ in case ${4l \leq (k + 1)^{2}}$ . If l =  0 then the operator ${x^{2k}_{2} (\Delta - \frac{k}{x_{2}} \frac{\partial}{\partial{x}_{2}})}$ is the Laplace- Beltrami operator of the Riemannian metric ${ds^2 = x^{-2k}_{2} (\sum^{2}_{i = 0} dx^{2}_{i})}$ . The general case ${\mathbb{R}^{n}_{+}}$ has been studied earlier by the authors, but the results are improved in case ${\mathbb{R}^3_{+}}$ . If k =  1 then the Riemannian metric is the hyperbolic distance of Poincaré upper half-space. The Weinstein equation is connected to the axially symmetric potentials. We compute solutions of the Weinstein equation depending only on the hyperbolic distance and x ₂. The solutions of the Weinstein equation form a socalled Brelot harmonic space and therefore it is known that they satisfy the mean value properties with respect to the harmonic measure. However, without using the theory of Brelot harmonic spaces, we present the explicit mean value properties which give a formula for a harmonic measure evaluated in the center point of the hyperbolic ball. Earlier these results were proved only for k =  1 and l =  0 or k =  1 and l =  1. We also compute the fundamental solutions. The main tools are the hyperbolic metric and its invariance properties. In the consecutive papers, these results are applied to find explicit kernels for k-hypermonogenic functions that are higher dimensional generalizations of complex holomorphic functions.     

Some remarks on the existence of optimal controls for quasilinear systems

Let a quasilinear control system having the state space \(\bar X \subseteq R^n \) be governed by the vector differential equation $$\dot x = G(u(t))x,$$ wherex(0) =x ₀ andU is the family of all bounded measurable functions from [0,T] intoU, a compact and convex subset ofR ^m.LetG:U ?R be a bounded measurable nonlinear function, such thatG(U) is compact and convex.G ^?1 can be convex onG(U) or concave. The main results of the paper establish the existence of a controlu ∈U which minimizes the cost functional $$I(u) = \int_0^T {L(u(t))x(t)dt,} $$ whereL(·) is convex. A practical example of application for chemical reactions is worked out in detail.     

On Titchmarsh–Weyl Functions and Eigenfunction Expansions of First-Order Symmetric Systems

We study general (not necessarily Hamiltonian) first-order symmetric system J y′(t)?B(t)y(t) = Δ(t) f(t) on an interval ${\mathcal{I}=[a,b) }$ with the regular endpoint a. It is assumed that the deficiency indices n _±(T _min) of the minimal relation T _min associated with this system in ${L^2_\Delta(\mathcal{I})}$ satisfy ${n_-(T_{\rm min})\leq n_+(T_{\rm min})}$ . We are interested in boundary conditions playing the role similar to that of separated self-adjoint boundary conditions for Hamiltonian systems. Instead we define λ-depending boundary conditions with Nevanlinna type spectral parameter τ = τ(λ) at the singular endpoint b. With this boundary value problem we associate the matrix m-function m(·) of the size ${N_\Sigma = {\rm dim} {\rm ker} (iJ+I)}$ . Its role is similar to that of the Titchmarsh–Weyl coefficient for the Hamiltonian system. In turn, it allows one to define the Fourier transform ${V: L^2_\Delta(\mathcal{I}) \to L^2(\Sigma)}$ where Σ (·) is a spectral matrix function of m(·). If V is an isometry, then the (exit space) self-adjoint extension ${\tilde{T}}$ of T _min induced by the boundary problem is unitarily equivalent to the multiplication operator in L ²(Σ). Hence the multiplicity of spectrum of ${\tilde{T}}$ does not exceed N _Σ. We also parameterize a set of spectral functions Σ(·) by means of the set of boundary parameters τ. Similar parameterizations for various classes of boundary value problems have earlier been obtained by Kac and Krein, Fulton, Hinton and Shaw, and others.     

Semi-linear Wave Equations with Effective Damping

The authors study the Cauchy problem for the semi-linear damped wave equation $$u_{tt} - \Delta u + b\left( t \right)u_t = f\left( u \right), u\left( {0,x} \right) = u_0 \left( x \right), u_t \left( {0,x} \right) = u_1 \left( x \right)$$ in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for |f(u)| ≈ |u| ^p in the supercritical case of $p > \tfrac{2} {n}$ and $p \leqslant \tfrac{n} {{n - 2}}$ for n ≥ 3 is proved.     

Maximum principle of optimal periodic control for functional differential systems

This paper discusses the optimal periodic control problem to minimize the cost function $$J(u) = \int_0^1 {g(t,x(t),u(t))dt} $$ subject to the functional differential system $$dx(t)/dt = f(t,x_t ,u(t)),x_1 = x_0 $$ andu(·) εU _ad. The maximum principle as a necessary condition of optimal control is proved under the assumption that Eq. (4) and its adjoint equation (5) both have no nontrivial periodic solution with period of 1. In this paper, the control domainU is an arbitrary set inR ^m.     

Об аппроксимации свермками и баэисах иэ сдвигов функции

For 2π-periodic functions f ∈ L ^p ( $ \mathbb{T} $ ), 1 ≤ p < ∞, σ ∈ V ( $ \mathbb{T} $ ) and g ∈ L( $ \mathbb{T} $ ), we consider the convolutions $$ (f*d\sigma )_T (x) = \int_0^{2\pi } {f(x - t)d\sigma (t), } (f*g)_T (x) = \int_0^{2\pi } {f(x - t)g(t)dt.} $$ For fixed functions σ ∈ V ( $ \mathbb{T} $ ) and g ∈ L( $ \mathbb{T} $ ), necessary and sufficient conditions are obtained for the density of the ranges of these operators in L ^p . Similar result is proved for the dyadic convolution $$ (f*g)_2 (x) = \int_0^1 {f(x \oplus t)g(t)dt,} $$ where ⊕ is the operation of dyadic addition on [0, 1). Moreover, it is proved that in the spaces L ^p ( $ \mathbb{T} $ ), 1 ≤ p ∞, and C( $ \mathbb{T} $ ) there exist no bases of shifts of a function. Similar results are obtained for the spaces L ^p [0, 1]*, 1 ≤ p < ∞, and C[0, 1]* relative to dyadic shifts, where [0, 1]* is the modified segment [0, 1]. It is also proved that in the space L(?₊) there exists no basis of dyadic shifts of a function.     

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

Boundary-value problems for a nonlinear hyperbolic equation with variable coefficients and the Lévy Laplacian

For a nonlinear hyperbolic equation with variable coefficients and the infinite-dimensional Lévy Laplacian Δ _L , $$\beta \left( {\sqrt 2 \left\| x \right\|_H \frac{{\partial U(t,x)}} {{\partial t}}} \right)\frac{{\partial ^2 U(t,x)}} {{\partial t^2 }} + \alpha (U(t,x))\left[ {\frac{{\partial U(t,x)}} {{\partial t}}} \right]^2 = \Delta _L U(t,x),$$ we present algorithms for the solution of the boundary-value problem U(0, x) = u ₀, U(t, 0) = u ₁ and the exterior boundary-value problem U(0, x) = v ₀, \(\left. {U(t,x)} \right|_{\Gamma = v_1 }\) , \(\lim _{\left\| x \right\|_{H \to \infty } } \left. {U(t,x) = v_2 } \right|\) for the class of Shilov functions depending on the parameter t.     

On solvability of derivative dependent doubly singular boundary value problems

In this paper we establish existence of solutions of singular boundary value problem ?(p(x)y ^′(x))^′=q(x)f(x,y,py′) for 0<x≤b and $\lim_{x\rightarrow0^{+}}p(x)y^{\prime}(x)=0$ , α ₁ y(b)+β ₁ p(b)y ^′(b)=γ ₁ with p(0)=0 and q(x) is allowed to have integrable discontinuity at x=0. So the problem may be doubly singular. Here we consider $\lim_{x\rightarrow0^{+}}\frac{q(x)}{p'(x)}\neq0$ therefore $\lim_{x\rightarrow0^{+}}p(x)y'(x)=0$ does not imply y′(0)=0 unless $\lim_{x\rightarrow0^{+}}f(x,y(x),p(x)y'(x))=0$ .