Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Approximate Estimates for a Differential Operator in a Weighted Hilbert Space

Approximate Estimates for a Differential Operator in a Weighted Hilbert Space We consider the self-adjoint operator L (the Friedrichs extension) associated with the closable form am[u,f]=∫Ω(∑|α|=mρ2(x)DαuDαf¯+vm2(x)uf¯)dx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${a_m}[u,f] = \int_\Omega {\left( {\sum\nolimits_{\left| \alpha \right| = m} {{\rho ^2}(x){D^\alpha }u\overline {{D^\alpha }f} + v_m^2(x)u\bar f} } \right)dx}$$\end{document} in the space L2,ω where Ω ⊂ ℝn is a domain, f,u∈C0∞(Ω)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f,u \in C_0^\infty (\Omega )$$\end{document}, and m ∈ ℕ. Here vs(x) = ρ(x)h-s(x), s > 0, and the positive functions ρ(·) and h(·) satisfy some special conditions. The space L2,ω is a weighted Hilbert space with a locally integrable weight ω(x) positive almost everywhere in Ω. For s > 0 and 1 < p < ∞, we introduce the weighted space of potentials Hps(ρ,vs)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_p^s(\rho ,{v_s})$$\end{document}. For s = m ∈ ℕ and p = 2, the space H2m(ρ,vm)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_2^m(\rho ,{v_m})$$\end{document} is the weighted Sobolev space Wm=W2m(ρ,vm)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_2^m(\rho ,{v_m})$$\end{document} with the equivalent norm ∥f;W2m(ρ,vm)∥=am[f,f]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left\| {f;W_2^m(\rho ,{v_m})} \right\| = \sqrt {{a_m}[f,f]}$$\end{document}. Descriptions are given of the interpolation spaces Hs obtained by the complex and real interpolation methods from the pair {Hps0(ρ,vs0),Hps1(ρ,vs1)}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left\{ {H_p^{{s_0}}(\rho ,{v_{{s_0}}}), H_p^{{s_1}}(\rho ,{v_{{s_1}}})} \right\}$$\end{document}, 0 < s0 < s1. Estimates are derived for the linear widths and Kolmogorov widths of the compact sets F=BHs⋂L−1(BL2,ω)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal{F} = B{H_s}\bigcap {{L^{ - 1}}(B{L_{2,\omega }})}$$\end{document}, s > 0 (where BX is the unit ball in a space X). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Approximate Estimates for a Differential Operator in a Weighted Hilbert Space

Download PDF
9 pages

Loading next page...
 
/lp/springer-journals/approximate-estimates-for-a-differential-operator-in-a-weighted-LM37wX4Ian

References (7)

Publisher
Springer Journals
Copyright
Copyright © Pleiades Publishing, Ltd. 2019
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266119120061
Publisher site
See Article on Publisher Site

Abstract

We consider the self-adjoint operator L (the Friedrichs extension) associated with the closable form am[u,f]=∫Ω(∑|α|=mρ2(x)DαuDαf¯+vm2(x)uf¯)dx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${a_m}[u,f] = \int_\Omega {\left( {\sum\nolimits_{\left| \alpha \right| = m} {{\rho ^2}(x){D^\alpha }u\overline {{D^\alpha }f} + v_m^2(x)u\bar f} } \right)dx}$$\end{document} in the space L2,ω where Ω ⊂ ℝn is a domain, f,u∈C0∞(Ω)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f,u \in C_0^\infty (\Omega )$$\end{document}, and m ∈ ℕ. Here vs(x) = ρ(x)h-s(x), s > 0, and the positive functions ρ(·) and h(·) satisfy some special conditions. The space L2,ω is a weighted Hilbert space with a locally integrable weight ω(x) positive almost everywhere in Ω. For s > 0 and 1 < p < ∞, we introduce the weighted space of potentials Hps(ρ,vs)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_p^s(\rho ,{v_s})$$\end{document}. For s = m ∈ ℕ and p = 2, the space H2m(ρ,vm)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_2^m(\rho ,{v_m})$$\end{document} is the weighted Sobolev space Wm=W2m(ρ,vm)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_2^m(\rho ,{v_m})$$\end{document} with the equivalent norm ∥f;W2m(ρ,vm)∥=am[f,f]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left\| {f;W_2^m(\rho ,{v_m})} \right\| = \sqrt {{a_m}[f,f]}$$\end{document}. Descriptions are given of the interpolation spaces Hs obtained by the complex and real interpolation methods from the pair {Hps0(ρ,vs0),Hps1(ρ,vs1)}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left\{ {H_p^{{s_0}}(\rho ,{v_{{s_0}}}), H_p^{{s_1}}(\rho ,{v_{{s_1}}})} \right\}$$\end{document}, 0 < s0 < s1. Estimates are derived for the linear widths and Kolmogorov widths of the compact sets F=BHs⋂L−1(BL2,ω)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal{F} = B{H_s}\bigcap {{L^{ - 1}}(B{L_{2,\omega }})}$$\end{document}, s > 0 (where BX is the unit ball in a space X).

Journal

Differential Equations – Springer Journals

Published: Dec 4, 2019

There are no references for this article.