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Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces | | Charles L. Fefferman
; Karol W. Hajduk
; James C. Robinson
; | | Date: |
6 Apr 2019 | | Abstract: | We show how to approximate functions defined on smooth bounded domains by
elements of eigenspaces of linear operators (e.g. the Laplacian or the Stokes
operator) in such a way that the approximations are bounded and converge in
both Sobolev and Lebesgue spaces simultaneously. We prove an abstract result
referred to fractional power spaces of positive, self-adjoint, compact-inverse
operators on Hilbert spaces, and then obtain our main result by identifying
explicitly these fractional power spaces for the Dirichlet Laplacian and
Dirichlet Stokes operators. As a simple application we prove that all weak
solutions of the convective Brinkman-Forchheimer equations posed on a bounded
domain in $mathbb{R}^3$ satisfy the energy equality. | | Source: | arXiv, 1904.3337 | | Services: | Forum | Review | PDF | Favorites | |
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