Improved scales of spaces and elliptic boundary-value problems. I

We study improved scales of functional Hilbert spaces over ?ⁿ and smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The theory of elliptic boundary-value problems in these spaces is developed.     

Improved scales of spaces and elliptic boundary-value problems. II

We study improved scales of functional Hilbert spaces over ?ⁿ and smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The theory of elliptic boundary-value problems in these spaces is developed.     

Elliptic boundary-value problems in complete scales of Nikol'skii-type spaces

We consider an elliptic boundary-value problem on an infinitely smooth manifold with, generally speaking, disconnected boundary. It is established that the operator of this problem is a Fredholm operator when considered in complete scales of functional spaces that depend on the parameterss ,p[1, ] and, for sufficiently large s0, coincide with the classical Nikol'skii spaces on a manifold.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1647–1654, December, 1994.In conclusion, the author expresses his deep gratitude to V. A. Mikhailets and Ya. A. Roitberg for helpful discussions.     

Implicit elliptic boundary-value problems with discontinuous nonlinearities

Let be a bounded domain in ⁿ (n3) having a smooth boundary, let be an essentially bounded real-valued function defined on × ^h, and let be a continuous real-valued function defined on a given subset Y of Y ^h. In this paper, the existence of strong solutions u W ^2,p (, ^h) W _o ^1,p (n/2<p<+) to the implicit elliptic equation (–u)=(x,u), with u=(u₁, u₂, ..., u_h) and u=(u ₁, u ₂, ..., u _h), is established. The abstract framework where the problem is placed is that of set-valued analysis.     

A class of Noetherian overdetermined elliptic boundary-value problems

In this paper one investigates overdetermined elliptic systems with constant coefficients for which one can pose general boundary-value problems without overdetermination in the boundary conditions. One finds the algebraic condition which characterizes the indicated class of systems. It is shown that the general boundary-value problems for such systems are Noetherian in the subspace of vector-functions satisfying some purely differential consistency conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. nA. Steklova AN SSSR, Vol. 47, pp. 138–154, 1974.     

Self-adjoint elliptic boundary-value problems. The polynomial property and formally positive operators

The formally self-adjoint boundary-value problem possesses the polynomial property if the corresponding sesquilinear form degenerates only on a finite-dimensional lineal P of vector-valued polynomials. A problem with the polynomial property is elliptic. The kernel and co-kernel of the corresponding operator are described in terms of P. It is shown that every formally positive operator has the polynomial property. Bibliography: 22 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 167–192.     

Condition of extremum for eigenvalues of elliptic boundary-value problems

In this paper, we consider problems of eigenvalue optimization for elliptic boundary-value problems. The coefficients of the higher derivatives are determined by the internal characteristics of the medium and play the role of control. The necessary conditions of the first and second order for problems of the first eigenvalue maximization are presented. In the case where the maximum is reached on a simple eigenvalue, the second-order condition is formulated as completeness condition for a system of functions in Banach space. If the maximum is reached on a double eigenvalue, the necessary condition is presented in the form of linear dependence for a system of functions. In both cases, the system is comprised of the eigenfunctions of the initial-boundary value problem. As an example, we consider the problem of maximization of the first eigenvalue of a buckling column that lies on an elastic foundation.     

Error of the Bubnov-Galerkin method for elliptic boundary-value problems

In this paper the author, developing one of his own methods and making use of more refined Markov inequalities, obtains estimates for the error of the Bubnov-Galerkin method for the derivatives of the solutions of non-self-adjoint elliptic boundary-value problems. Regarding the order of accuracy, they coincide with the best known estimates of the Ritz method for the solutions of positivedefinite problems.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 111, pp. 137–144, 1981.     

A note on initial-value formulations of elliptic boundary-value problems

Summary An elliptic boundary-value problem may be formulated as an initial-value problem by approximating it as a two-point boundary-value problem. A method proposed in [6] not requiring this approximation is disproved by means of an example.

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Zusammenfassung Ein elliptisches Randwertproblem lässt sich als Anfangswerproblem, formulieren, indem es durch ein Zweipunkt-Randwertproblem approximiert wird. Eine in [6] vorgeschlagene Methode, die auf diese Approximation verzichtet, wird durchein Beispiel widerlegt.

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The author is thankful for support by the AFOSR, through contract No. F4462-69-C-0124 and NASA-UAH Agreement Modification 4.     

Finite-element approximation for singularly perturbed nonlinear elliptic boundary-value problems

Numerical solution of a two-dimensional nonlinear singularly perturbed elliptic partial differential equation ∈ Δu = f(x, u), 0 < x, y < 1, with Dirichlet boundary condition is discussed here. The modified Newton method of third-order convergence is employed to linearize the nonlinear problem in place of the standard Newton method. The finite-element method is used to find the solution of the nonlinear differential equation. Numerical results are provided to demonstrate the usefulness of the method.     

A smoothness preserving fictitious domain method for elliptic boundary-value problems

^** Email: mommer{at}math.uu.nl We introduce a new fictitious domain method for the solutionof second-order elliptic boundary-value problems with Dirichletor Neumann boundary conditions on domains with C² boundary.The main advantage of this method is that it extends the solutionssmoothly, which leads to better performance by achieving higheraccuracy with fewer degrees of freedom. The method is basedon a least-squares interpretation of the fundamental requirementsthat the solution produced by a fictitious domain method shouldsatisfy. Careful choice of discretization techniques, togetherwith a special solution strategy, leads then to smooth solutionsof the resulting underdetermined problem. Numerical experimentsare provided which illustrate the performance and flexibilityof the approach.