On the regularity of solutions of boundary-value problem for quasilinear elliptic systems with quadratic nonlinearity

The partial regularity up to the boundary of a domain is established for a solution u ∈ H¹ (Ω) ∩ L_∞ (Ω) to the boundary-value problem for a second-order elliptic system with strong nonlinearity in the case of dimension n≥3. Bibliography: 12 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15, 1995, pp. 47–69.     

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.     

Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain Ω ⊂ R ² with smooth boundary. By the method of the Green formula, the theory of extensions of differential operators, and the theory of L-traces (i.e., traces associated with the differential operation L), we establish necessary and sufficient (for elliptic operators) conditions of the solvability of each of these problems in the space H ^m (Ω), m ≥ 4.     

Approximate solution of third boundary-value problem for second-order elliptic equation in domains of arbitrary shape

The fictitious domain method is applied to construct a difference scheme for solving the third boundary-value problem for a second-order elliptic equation in domains of arbitrary shape. A rate of convergence bound is derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 3–7, 1986.     

An estimate for the convergence rate of difference schemes for second-order elliptic equations in domains of arbitrary shape

Using the dummy-domains method, a difference scheme is constructed for solving the first boundary-value problem for elliptic equations of the second order in domains of arbitrary shape. An estimate for the convergence rate of order O(h^1/2) in the norm of W ₂ ¹ is found. Bibliography:5 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 77, 1993, pp. 12–18     

A generalization of the fundamental theorem of spherical harmonic theory

We consider a boundary-value problem of the first kind for a self-adjoint differential operator with constant coefficients on a domain in ℝⁿ bounded by an ellipsoid; boundary conditions are defined by an arbitrary polynomial of degree N. It is proved that the solution of the problem is again a polynomial of degree ≤N. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

The generalized solution of ill-posed boundary problem

In this paper, we define a kind of new Sobolev spaces, the relative Sobolev spaces Wk,p0(Ω,∑). Then an elliptic partial differential equation of the second order with an ill-posed boundary is discussed. By utilizing the ideal of the generalized inverse of an operator, we introduce the generalized solution of the ill-posed boundary problem. Eventually, the connection between the generalized inverse and the generalized solution is studied. In this way, the non-instability of the minimal normal least square solution of the ill-posed boundary problem is avoided.     

A generalized solution in W2 1(0,1) of a problem concerning the deflection of a beam

A new method is proposed for formulating a boundary-value problem for a fourth-order ordinary differential equation with a solution in W₂ ¹(0, 1). This generalized formulation is based on a system of second-order equations with coefficients in W₂ ^–1 (0, 1). The existence and uniqueness of the indicated solution in this class is proven.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 90–96, 1989.     

Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I

We consider the periodic boundary-value problem u _tt − u _xx = g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). By representing a solution of this problem in the form u(x, t) = u ⁰(x, t) + ũ(x, t), where u ⁰(x, t) is a solution of the corresponding homogeneous problem and ũ(x, t) is the exact solution of the inhomogeneous equation such that ũ(x, t + ω) u x = ũ(x, t), we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the period ω. We show that the relation obtained for a solution includes known results established earlier. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 912–921, July, 2005.     

Regularity results for quasilinear elliptic systems with nonlinear boundary conditions

It is proved that a solution of the boundary-value problem for a second-order quasilinear system with controlled order of nonlinearity is partially smooth all the way to the boundary of a domain. The boundary condition is imposed by means of a second-order nonlinear operator which can be regarded as a generalization of the “directional derivative” to the case of quasilinear systems. Bibliography: 6 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 23–50.     

The accuracy of a discrete spectral problem with mixed derivatives

The accuracy of a difference spectral problem for a second-order elliptic equation with mixed derivatives and constant coefficients is estimated. In so doing, the fact that the eigenfunctions belong to the Sobolyev space W ₂ ² in a rectangle is used. Bibliography: 6 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 59–66.     

On the Sobolev problem in the complete scale of Banach spaces

In a bounded domainG with boundary ∂G that consists of components of different dimensions, we consider an elliptic boundary-value problem in complete scales of Banach spaces. The orders of boundary expressions are arbitrary; they are pseudodifferential along ∂G. We prove the theorem on a complete set of isomorphisms and generalize its application. The results obtained are true for elliptic Sobolev problems with a parameter and parabolic Sobolev problems as well as for systems with the Douglis-Nirenberg structure. Chernigov Pedagogical Institute, Chernigov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1181–1192, September, 1999.     

Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method

In the case of the Dirichlet problem for a singularly perturbed ordinary differential reaction-diffusion equation, a new approach is used to the construction of finite difference schemes such that their solutions and their normalized first- and second-order derivatives converge in the maximum norm uniformly with respect to a perturbation parameter ɛ ∈(0, 1]; the normalized derivatives are ɛ-uniformly bounded. The key idea of this approach to the construction of ɛ-uniformly convergent finite difference schemes is the use of uniform grids for solving grid subproblems for the regular and singular components of the grid solution. Based on the asymptotic construction technique, a scheme of the solution decomposition method is constructed such that its solution and its normalized first- and second-order derivatives converge ɛ-uniformly at the rate of O(N ⁻²ln² N), where N + 1 is the number of points in the uniform grids. Using the Richardson technique, an improved scheme of the solution decomposition method is constructed such that its solution and its normalized first and second derivatives converge ɛ-uniformly in the maximum norm at the same rate of O(N ⁻⁴ln⁴ N).     

Regularity of the free boundary in a neighborhood of a contact point in the obstacle problem

The obstacle problem for an arbitrary linear elliptic equation is considered. The regularity of the boundary of the noncoincident set is studied in a neighborhood of a contact points of the boundary of a domain. The C1-regularity of the boundary and the continuity up to contact points of the second-order derivatives of the solution are proved. Bibliography: 3 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 101–104.     

Nonnegative solutions of an elliptic equation with a singular potential

We study the behavior of nonnegative solutions of the Dirichlet problem for a linear elliptic equation with a singular potential in the ball B = B(0,R) ⊂ R ⁿ (n ≥ 3), R ≤ 1. We find an exact condition on the potential ensuring the existence or absence of a nonnegative solution of that problem.     

Convergence of the method of lines for the first periodic boundary-value problem for a second-order nonlinear parabolic equation

For solving the first generalized periodic boundary-value problem in the case of a second-order quasilinear parabolic equation of form with periodic condition and boundary conditions there is examined a longitudinal variant of the method of lines, reducing the solving of problem (1)–(3) to the solving of a two-point problem for a system ofN ^-1 first-order ordinary differential equations of form with the two-point conditions An error estimate is established. The convergence of the solutions of problem (4)–(5) to the generalized solution of problem (1)–(3) is established for two methods of choosing the functions. Convergence with orderh ² is guaranteed under the assumption of square-integrability of the third derivative of the solution of problem (1)–(3).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 268–276, 1979.     

On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators

We investigate the problem of the well-posedness of a boundary-value problem for a system of pseudodifferential equations of arbitrary order with nonlocal conditions. The equation and boundary conditions contain pseudodifferential operators whose symbols are defined and continuous in a certain domain H ⊂ ℝ _σ ^m . A criterion for the existence and uniqueness of solutions and for the continuous dependence of the solution on the boundary function is established. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1131 – 1136, August, 2005.     

On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations

We consider the first boundary-value problem for a second-order degenerate elliptic-parabolic equation with, generally speaking, discontinuous coefficients. The matrix of leading coefficients satisfies the parabolic Cordes condition with respect to space variables. We prove that the generalized solution of the problem belongs to the H?lder space {ie831-01} if the right-hand side f belongs to L _p , p > n. Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 723–736, June, 2008.     

Solutions to two dimensional, uniformly elliptic equations, that lie in sobolev spaces and have compact support

A non zero functionν in the plane is constructed, having the following properties:ν hasL ^p (1<p<2) second-order derivatives,ν satisfies a second order, non variational, uniformly elliptic equationLv=0; moreoverν has compact support.ν can be used to show that classical results for two dimensional elliptic equations and systems are sharp.     

The Riemann problem on a finite-sheeted Riemann surface of infinite genus

LetR be the Riemann surface of the functionu(z) specified by the equationu ⁿ=P(z) withn ε ℕ,n ≥ 2, andz ε ℂ, whereP(z) is an entire function with infinitely many simple zeros. OnR, the Riemann boundary-value problem for an arbitrary piecewise smooth contour Γ is considered. Necessary and sufficient conditions for its solvability are obtained, and its explicit solution is constructed. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 25–35, January, 2000.