The concepts of boundary relations and the corresponding Weyl families are introduced. Let be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space , let be an auxiliary Hilbert space, let
and let be defined analogously. A unitary relation from the Krein space to the Krein space is called a boundary relation for the adjoint if . The corresponding Weyl family is defined as the family of images of the defect subspaces , , under . Here need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space and the class of unitary relations , it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every -valued maximal dissipative (for ) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.
In this work, we describe and analyze two models that were recently proposed for modeling generalized von Kármán plates and generalized Marguerre–von Kármán shallow shells.
First, we briefly review the “classical” von Kármán and Marguerre–von Kármán equations, their physical meaning, and their mathematical justification. We then consider the more general situation where only a portion of the lateral face of a nonlinearly elastic plate or shallow shell is subjected to boundary conditions of von Kármán type, while the remaining portion is free. Using techniques from formal asymptotic analysis, we obtain in each case a two-dimensional boundary value problem that is analogous to, but is more general than, the classical equations.
In particular, it is remarkable that the boundary conditions for the Airy function can still be determined on the entire boundary of the nonlinearly elastic plate or shallow shell solely from the data.
Following recent joint works, we then reduce these more general equations to a single “cubic” operator equation, which generalizes an equation introduced by Berger and Fife, and whose sole unknown is the vertical displacement of the shell. We next adapt an elegant compactness method due to Lions for establishing the existence of a solution to this operator equation. 相似文献
The convergence rate of a fast-converging second-order accurate iterative method with splitting of boundary conditions constructed by the authors for solving an axisymmetric Dirichlet boundary value problem for the Stokes system in a spherical gap is studied numerically. For R/r exceeding about 30, where r and R are the radii of the inner and outer boundary spheres, it is established that the convergence rate of the method is lower (and considerably lower for large R/r) than the convergence rate of its differential version. For this reason, a really simpler, more slowly converging modification of the original method is constructed on the differential level and a finite-element implementation of this modification is built. Numerical experiments have revealed that this modification has the same convergence rate as its differential counterpart for R/r of up to 5 × 103. When the multigrid method is used to solve the split and auxiliary boundary value problems arising at iterations, the modification is more efficient than the original method starting from R/r ~ 30 and is considerably more efficient for large values of R/r. It is also established that the convergence rates of both methods depend little on the stretching coefficient η of circularly rectangular mesh cells in a range of η that is well sufficient for effective use of the multigrid method for arbitrary values of R/r smaller than ~ 5 × 103. 相似文献
We investigate the Method of Fundamental Solutions (MFS) for the solution of certain elliptic boundary value problems. In
particular, we study the case in which the number of collocation points exceeds the number of singularities, which leads to
an over-determined linear system. In such a case, the resulting linear system is over-determined and the proposed algorithm
chooses the approximate solution for which the error, when restricted to the boundary, minimizes a suitably defined discrete
Sobolev norm. This is equivalent to a weighted least-squares treatment of the resulting over-determined system. We prove convergence
of the method in the case of the Laplace’s equation with Dirichlet boundary data in the disk. We develop an alternative way
of implementing the numerical algorithm, which avoids the inherent ill-conditioning of the MFS matrices. Finally, we present
numerical experiments suggesting that introduction of Sobolev weights improves the approximation.
AMS subject classification (2000) 35E05, 35J25, 65N12, 65N15, 65N35, 65T50 相似文献
The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating
boundary is analyzed. The leading terms of the asymptotic expansions for eigenelements are constructed, and the asymptotics
are substantiated for simple eigenvalues.
The text was submitted by the authors in English. 相似文献
Conservative finite-difference schemes are constructed for the problems of self-action of a femtosecond laser pulse and of second-harmonic generation in a one-dimensional nonlinear photonic crystal with nonreflecting boundary conditions. The invariants of the governing equations are found taking into account these conditions. Nonreflecting conditions substantially improve the efficiency of conservative finite-difference schemes used in the modeling of complex nonlinear effects in photonic crystals, which require much smaller steps in space and time than those used in the case of linear propagation. The numerical experiments performed show that the boundary reflects no more than 0.01% of the transmitted energy, which corresponds to the truncation error in the boundary conditions. The amplitude of the reflected pulse is less than that of the pulse transmitted through the boundary by two (and more) orders of magnitude. The simulation is based on the approach proposed by the authors for the given class of problems. 相似文献
It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary. 相似文献