Multipoint Problem for Nonisotropic Partial Differential Equations with Constant Coefficients

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable t and periodic conditions with respect to coordinates x ₁,..., x _p for a nonisotropic (concerning differentiation with respect to t and x ₁,..., x _p) partial differential equation with constant complex coefficients. We establish conditions for the existence and uniqueness of a solution of this problem and prove metric theorems on lower bounds for small denominators appearing in the course of the construction of its solution.     

A Multipoint Problem for Pseudodifferential Equations

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable t and periodic conditions with respect to coordinates x ₁,...,x _p for equations unsolved with respect to the leading derivative with respect to t and containing pseudodifferential operators. We establish conditions for the unique solvability of this problem and prove metric assertions related to lower bounds for small denominators appearing in the course of its solution.     

A Problem with Nonlocal Conditions for Partial Differential Equations with Variable Coefficients

We establish conditions for the unique solvability of a problem for partial differential equations with coefficients dependent on variables t and x in a rectangular domain with nonlocal two-point conditions with respect to t and local boundary conditions with respect to x. We prove metric statements related to lower bounds of small denominators appearing in the course of solution of the problem.     

Solvability of a model problem for the Stokes equations in an infinite angle

We consider a model problem for the Stokes equations in the half-plane ? ₊ ² (x₂>0) with different boundary conditions on the semiaxes (x₂=0, x₁<0) and (x₂=0, x₁>0), which plays an important role in the studies of some free boundary problems, such as problem of filling or drying a capillary. The proof of the solvability of the problem in weighted Sobolev and Hölder spaces is presented, and estimates for the solution as well as the asymptotic formula for the solution in the vicinity of the singular point x=0 are obtained. The proof is based on an explicit formula for the solution in terms of its Mellin transform, which makes it possible to obtain the estimates uniform with respect to one of the parameters of the problem (in the problem of filling a capillary it is proportional to the velocity of filling). Bibliography: 9 titles.     

Multipoint problem for hyperbolic equations with variable coefficients

By using the metric approach, we study the problem of classical well-posedness of a problem with multipoint conditions with respect to time in a tube domain for linear hyperbolic equations of order 2n (n ≥ 1) with coefficients depending onx. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of the problem.     

Existence of the non-radially symmetric ground state for p-Laplacian equations involving Choquard type

We consider some p-Laplacian type equations with sum of nonlocal term and subcritical nonlinearities. We prove the existence of the ground states, which are positive. Because of including p=2, these results extend the results of Li, Ma and Zhang [Nonlinear Analysis: Real World Application 45(2019) 1-25]. When p=2, N=3, by a variant variational identity and a constraint set, we can prove the existence of a non-radially symmetric solution. Moreover, this solution u(x₁, x₂, x₃) is radially symmetric with respect to (x₁, x₂) and odd with respect to x₃.     

Weighted isoperimetric inequalities on ℝn and applications to rearrangements

We study isoperimetric inequalities for a certain class of probability measures on ?ⁿ together with applications to integral inequalities for weighted rearrangements. Furthermore, we compare the solution to a linear elliptic problem with the solution to some “rearranged” problem defined in the domain {x: x₁ < α (x₂, …, x_n)} with a suitable function α (x₂, …, x_n). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Directional Derivative Problem for the Telegraph Equation with a Dirac Potential

In the domain Q = [0,∞)×[0,∞) of the variables (x, t), for the telegraph equation with a Dirac potential concentrated at a point (x₀, t₀) ∈ Q, we consider a mixed problem with initial (at t = 0) conditions on the solution and its derivative with respect to t and a condition on the boundary x = 0 which is a linear combination with coefficients depending on t of the solution and its first derivatives with respect to x and t (a directional derivative). We obtain formulas for the classical solution of this problem under certain conditions on the point (x₀, t₀), the coefficient of the Dirac potential, and the conditions of consistency of the initial and boundary data and the right-hand side of the equation at the point (0, 0). We study the behavior of the solution as the direction of the directional derivative in the boundary condition tends to a characteristic of the equation and obtain estimates of the difference between the corresponding solutions.     

On the Stabilization of a Solution of the Cauchy Problem for One Class of Integro-Differential Equations

We consider a solution of the Cauchy problem u(t, x), t > 0, x ∈ R ², for one class of integro-differential equations. These equations have the following specific feature: the matrix of the coefficients of higher derivatives is degenerate for all x. We establish conditions for the existence of the limit lim _t→∞ u(t, x) = v(x) and represent the solution of the Cauchy problem in explicit form in terms of the coefficients of the equation.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1699 – 1706, December, 2004.     

Solvability of fourth‐order Sturm‐Liouville problems with abstract linear functionals in boundary‐transmission conditions

This study deals with the solvability of one nonclassical boundary‐value problem for fourth‐order differential equation on two disjoint intervals I₁=(−1,0)and I₂=(0,1). The boundary conditions contain not only endpoints x=−1and x=1but also a point of interaction x=0, finite number internal points x_jki∈I_j and abstract linear functionals S_k. So, our problem is not a pure differential one. We investigate such important properties as isomorphism, Fredholmness and coerciveness with respect to the spectral parameter. Note that the obtained results are new even in the case of the boundary conditions without internal points x_jki and without abstract linear functionals S_k.     

Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula

We prove a uniform bound for the density, p _t (x), of the solution at time t(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p _t (x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al. ⁽⁵⁾     

Tikhonov regularization of nonlinear III-posed problems in hilbert scales

In this paper we consider nonlinear ill-posed problems F(x) = y ₀, where x and y ₀ are elements of Hilbert spaces X and Y, respectively. We solve these problems by Tikhonov regularization in a Hilbert scale. This means that the regularizing norm is stronger than the norm in X. Smoothness conditions are given that guarantee convergence rates with respect to the data noise in the original norm in X. We also propose a variant of Tikhonov regularization that yields these rates without needing the knowledge of the smoothness conditions. In this variant F is allowed to be known only approximately and X can be approximated by a finite-dimensional subspace. Finally, we illustrate the required conditions for a simple parameter estimation problem for regularization in Sobolev spaces.     

The Camassa–Holm Equation on the Half-Line: a Riemann–Hilbert Approach

We consider the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation u _t −u _txx +2u _x +3uu _x =2u _x u _xx +uu _xxx on the half-line x≥0. In this article, we aim to provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann–Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated to initial and boundary values of the solution. The construction requires more boundary data than those needed for a well-posed IBV problem. Their dependence is expressed in terms of an algebraic relation to be satisfied by the spectral functions. This RH formulation gives us the long-time asymptotics of a solution of the CH-equation. Dedicated to Gennadi Henkin in great admiration.     

On the identification of degenerate indices in the nonlinear complementarity problem with the proximal point algorithm

In this paper we focus on the problem of identifying the index sets P(x):=i|x_i>0, N(x):={i|F_i(x)>0 and C(x):=i|x_i=F_i(x)=0} for a solution x of the monotone nonlinear complementarity problem NCP(F). The correct identification of these sets is important from both theoretical and practical points of view. Such an identification enables us to remove complementarity conditions from the NCP and locally reduce the NCP to a system which can be dealt with more easily. We present a new technique that utilizes a sequence generated by the proximal point algorithm (PPA). Using the superlinear convergence property of PPA, we show that the proposed technique can identify the correct index sets without assuming the nondegeneracy and the local uniqueness of the solution.This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan.Mathematics Subject Classification (2000): 90C33, 65K10     

Algebraic polynomials least deviating from zero in measure on a segment

We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment [–1, 1] with respect to a measure, or, more precisely, with respect to the functional μ(f) = mes{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1}. We also discuss an analogous problem with respect to the integral functionals ∫_–1¹ φ (∣f (x)∣) dx for functions φ that are defined, nonnegative, and nondecreasing on the semiaxis [0, +∞).     

A layer adaptive B‐spline collocation method for singularly perturbed one‐dimensional parabolic problem with a boundary turning point

In this article, we develop a parameter uniform numerical method for a class of singularly perturbed parabolic equations with a multiple boundary turning point on a rectangular domain. The coefficient of the first derivative with respect to x is given by the formula a₀(x, t)x^p, where a₀(x, t) ≥ α > 0 and the parameter p ∈ [1,∞) takes the arbitrary value. For small values of the parameter ε, the solution of this particular class of problem exhibits the parabolic boundary layer in a neighborhood of the boundary x = 0 of the domain. We use the implicit Euler method to discretize the temporal variable on uniform mesh and a B‐spline collocation method defined on piecewise uniform Shishkin mesh to discretize the spatial variable. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular component. These bounds are applied in the convergence analysis of the proposed scheme on Shishkin mesh. The resulting method is boundary layer resolving and has been shown almost second‐order accurate in space and first‐order accurate in time. It is also shown that the proposed method is uniformly convergent with respect to the singular perturbation parameter ε. Some numerical results are given to confirm the predicted theory and comparison of numerical results made with a scheme consisting of a standard upwind finite difference operator on a piecewise uniform Shishkin mesh. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1143–1164, 2011     

Two inverse problems of finding a coefficient in a parabolic equation

For a parabolic equation, we consider inverse problems of reconstructing a coefficient that depends on the space variables alone. The first problem is to find a lower-order coefficient c(x) multiplying u(x, t), and the second problem is to find the coefficient a(x) multiplying Δu. As additional information, the integral of the solution with respect to time with some weight function is given. The coefficients of the equation depend both on time and on the space variables. We obtain sufficient conditions for the existence of generalized solutions of our problems; moreover, for the first problem, we also prove uniqueness and construct an iterative sequence that converges to the desired coefficient almost everywhere in the domain. We present examples of input data of these problems for which the assumptions of our theorems are necessarily true.     

On a nonlinear equation unsolved with respect to the levy laplacian

We propose a method for the solution of the nonlinear equationf(U(x),ΔU(x))=F(x) (Δ _L is an infinite-dimensional Laplacian, Δ _L U(x)=γ, γ≠0) unsolved with respect to the infinite-dimensional Laplacian, and for the solution of the Dirichlet problem for this equation.     

Mixed problem for a semilinear ultraparabolic equation in an unbounded domain

We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation {fx1870-01} in an unbounded domain with respect to the variables x. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1661–1673, December, 2007.     

Study of a special nonlinear problem arising in convex semi-infinite programming

We consider convex problems of semi-infinite programming (SIP) using an approach based on the implicit optimality criterion. This criterion allows one to replace optimality conditions for a feasible solution x ⁰ of the convex SIP problem by such conditions for x ⁰ in some nonlinear programming (NLP) problem denoted by NLP(I(x ⁰)). This nonlinear problem, constructed on the base of special characteristics of the original SIP problem, so-called immobile indices and their immobility orders, has a special structure and a diversity of important properties. We study these properties and use them to obtain efficient explicit optimality conditions for the problem NLP(I(x ⁰)). Application of these conditions, together with the implicit optimality criterion, gives new efficient optimality conditions for convex SIP problems. Special attention is paid to SIP problems whose constraints do not satisfy the Slater condition and to problems with analytic constraint functions for which we obtain optimality conditions in the form of a criterion. Comparison with some known optimality conditions for convex SIP is provided.