1,721,009 research outputs found
Spectral stability of Dirichlet second order uniformly elliptic operators
No full textWe prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Dirichlet boundary conditions upon domain perturbation. The main results concern estimates for the variation of the eigenvalues via the Hausdorff distance between the domains or the Lebesgue measure of their symmetric difference. Our analysis includes domains with Lipschitz boundaries as well as domains with boundary degenerations of power type
Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators
No full textWe prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation
Spectral Stability of Higher Order Uniformly Elliptic Operators
Full text linkWe prove estimates for the variation of the eigenvalues of uniformly elliptic operators with homogeneous Dirichlet or Neumann boundary conditions upon variation of the open set on which an operator is defined. We consider operators of arbitrary even order and open sets admitting arbitrary strong degeneration. The main estimate is expressed in terms of a natural and easily computable distance between open sets with continuous boundaries. Another estimate is obtained in terms of the lower Hausdorff-Pompeiu deviation of the boundaries, which in general may be much smaller than the usual Hausdorff-Pompeiu distance. Finally, in the case of diffeomorphic open sets, we obtain an estimate even without the assumption of continuity of the boundaries
Spectral stability of the p-Laplacian
No full textWe study the dependence of the eigenvalues of the p-Laplacian upon domain perturbation. We prove Lipschitz-type estimates for the deviation of the eigenvalues following a domain perturbation. Such estimates are expressed in terms of suitable measures of vicinity between open sets, such as the 'atlas distance' and the 'lower Hausdorff-Pompeiu deviation'. In the case of open sets with Holder continuous boundaries, our results improve a result known for the first eigenvalue
Spectral stability of elliptic selfadjoint differential operators with Dirichlet and Neumann boundary conditions
No full textWe present a general spectral stability theorem for nonnegative selfadjoint operators with compact resolvents, which is based on the notion of a transition operator, and some applications to the study of the dependence of the eigenvalues of uniformly elliptic operators upon domain perturbation
On boundedness of the fractional maximal operator from complementary Morrey-type spaces to Morrey-type spaces.
No full textThe problem of boundedness of the fractional maximal operator M-alpha from complementary Morrey-type spaces to Mortey-type spaces is reduced to the problem of boundedness of the dual Hardy operator in weighted L-p-spaces on the cone of non-negative non-increasing functions, which allows obtaining sharp sufficient conditions for boundedness of M-alpha
Spectral stability of nonnegative selfadjoint operators
No full textWe present a general spectral stability theorem which allows to compare the eigenvalues of two non-negative self-adjoint operators with compact resolvents acting on different Hilbert spaces. This theorem is based on the notion of transition operator which is introduced. Examples and applications to domain perturbation problems are discussed
Spectral stability estimates for elliptic operators in domain perturbation problems
No full textWe consider the problem of finding estimates for the variation of the eigenvalues of partial differential operators of elliptic type following a domain perturbation. We survey several results obtained by using two different approaches, one of which based on Hardy-type inequalities
Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators
No full textWe prove sharp stability estimates for the variation of the eigenvalues of non-negative self-adjoint elliptic operators of arbitrary even order upon variation of the open sets on which they are defined. These estimates are expressed in terms of the Lebesgue measure of the symmetric difference of the open sets. Both Dirichlet and Neumann boundary conditions are considered
Spectral stability of nonnegative self-adjoint operators
No full textThe survey is devoted to spectral stability problems for uniformly elliptic differential operators under the variation of the domain and to the accompanying estimates for the difference of the eigenvalues. Two approaches to the problem are discussed in detail. In the first one it is assumed that the domain is transformed by means of a transformation of a certain class, and the spectral stability with respect to this transformation is investigated. The second approach is based on the notion of a transition operator and allows direct comparison of the eigenvalues on domains which are close in that or another sense
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