Sobolev Constants in Disconnected Domains

Author(s):  
Catherine Bandle
1996 ◽  
Vol 48 (2) ◽  
pp. 225-243
Author(s):  
Alexandru Aleman ◽  
Stefan Richter ◽  
William T. Ross

AbstractFor a bounded region G ⊂ ℂ and a compact set K ⊂ G, with area measure zero, we will characterize the invariant subspaces ℳ (under ƒ → zƒ) of the Bergman space (G \ K), 1 ≤ p < ∞, which contain (G) and with dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K. When G \ K is connected, we will see that dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K and thus in this case we will have a complete description of the invariant subspaces lying between (G) and (G \ K). When p = ∞, we will remark on the structure of the weak-star closed z-invariant subspaces between H∞(G) and H∞(G \ K). When G \ K is not connected, we will show that in general the invariant subspaces between (G) and (G \ K) are fantastically complicated. As an application of these results, we will remark on the complexity of the invariant subspaces (under ƒ → ζƒ) of certain Besov spaces on K. In particular, we shall see that in the harmonic Dirichlet space , there are invariant subspaces ℱ such that the dimension of ζℱ in ℱ is infinite.


2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Roberta Musina ◽  
Alexander I. Nazarov

Abstract We consider different fractional Neumann Laplacians of order {s\in(0,1)} on domains {\Omega\subset\mathbb{R}^{n}} , namely, the restricted Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{R}}}} , the semirestricted Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sr}}}} and the spectral Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sp}}}} . In particular, we are interested in the attainability of Sobolev constants for these operators when Ω is a half-space.


2004 ◽  
Vol 14 (4) ◽  
pp. 1741-1765 ◽  
Author(s):  
Mark Jerrum ◽  
Jung-Bae Son ◽  
Prasad Tetali ◽  
Eric Vigoda

Author(s):  
Rafael Ortega ◽  
Meirong Zhang

In this paper we will show that the optimal bounds for certain static and dynamic bifurcation values of periodic solutions of some superlinear differential equations can be expressed explicitly using Sobolev constants.


2011 ◽  
Vol 36 (4) ◽  
pp. 607-616 ◽  
Author(s):  
Michiel van den Berg

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