By Goong Chen; Jianxin Zhou
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Extra resources for Boundary element methods with applications to nonlinear problems
Let f be a real- or complex-valued function defined on Ω. Let α = (α1 , α2 , . . , αN ) ∈ G. Chen and J. 1007/978-94-91216-27-5_2, © 2010 Atlantis Press/World Scientific 17 18 Boundary Element Methods with Applications to Nonlinear Problems (Z+ )N be a multi-index of nonnegative integers. 1) to denote a mixed partial derivative of f of order |α | ≡ α1 + α2 + · · · + αN . 1), we actually want to take the derivatives of f in the sense of distributions (see Chapter 3). 1) as the classical derivative: the one about which we have learned in advanced calculus.
We also call such a continuous linear functional a distribution or a generalized function. The space of all distributions will be denoted by D (Rn ), or D . The Dirac delta function δ is now well defined as a distribution because δ , φ ≡ φ (0) is a continuous linear functional on D(RN ). We may add distributions or multiply them by C∞ functions to form new distributions. However, the product of two distributions is not well defined in general. Every L1loc (RN ) function f defines a distribution via f,φ ≡ RN f (x)φ (x) dx, ∀ φ ∈ D.
We want to show that there exists a constant C > 0 such that sup |u(x)| C u x∈RN H s (RN ) ∀u ∈ H s (RN ). 23) is continuous [161, Chapter 9]. Since u = F u (u = F u), if we can show u ∈ L1 (RN ) the proof will be complete. But 1 + |ξ |2 u(ξ ) = s/2 u(ξ ) 1 . 24) The first factor on the right is in L2 (RN ) because u ∈ H s (RN ). The second factor is in L2 (RN ) provided that s > 21 N, because RN dξ < ∞ iff 2s > N. (1 + |ξ |2)s Hence u is the product of two L2 (RN ) functions, and is in L1 (RN ) by the Cauchy–Schwarz inequality.