By Knut Aas-Jakobsen, Mathis Grenacher (auth.)
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Extra resources for Analysis of Slender Reinforced Concrete Frames
Appl. Math. Sci. 147, SpringerVerlag, New York, 2006.  M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, Bristol, 1998. 860 GILLES AUBERT AND PIERRE KORNPROBST  D. P. , Athena Scientiﬁc, Nashua, NH, 1999.  J. Bourgain, H. Brezis, and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Diﬀerential Equations, J. L. Menaldi, E. Rofman, and A. , IOS Press, 2001, pp. 439–455.  A. Buades, B. Coll, and J. M.
In this article we propose an alternative method based on a recent new characterization of the Sobolev spaces by Bourgain, Brezis, and Mironescu , and further extended by Ponce  in the BV -case. In  the authors showed that the Sobolev seminorm of a function f can be approximated by a sequence of integral operators involving a diﬀerential quotient of f and a suitable sequence of radial molliﬁers: lim n→∞ Ω Ω |u(x) − u(y)|p ρn (|x − y|)dxdy = KN,p |x − y|p |∇u|p dx. Ω ∗ Received by the editors July 10, 2007; accepted for publication (in revised form) October 15, 2008; published electronically February 6, 2009.
Regularity result for un . We have obtained the existence of a minimizer in Lp (Ω). Let us show that the solution is in fact more regular than just Lp . As for W 1,p (Ω), the space W s,p (Ω) can be characterized by a diﬀerential quotient. For 0 < s < 1 and 1 ≤ p < ∞, we deﬁne W s,p (Ω) = u ∈ Lp (Ω); |u(x) − u(y)| ∈ Lp (Ω × Ω) , |x − y|s+N/p endowed with the norm |u|pW s,p (Ω) = |u|p dx + Ω Ω Ω |u(x) − u(y)|p dxdy. , C(n) may be diﬀerent from one line to the next). 10) Ω Ω |uln (x) − uln (y)|p ρn (|x − y|)dxdy ≤ C(n).
Analysis of Slender Reinforced Concrete Frames by Knut Aas-Jakobsen, Mathis Grenacher (auth.)