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By Goong Chen; Jianxin Zhou

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2). 2) are Lebesgue integrals (as are those throughout the rest of the book), but for all practical purposes in this book they may be regarded as Riemann integrals. We define the Sobolev space W k,p (Ω) = f : Ω → C or R | f k,p <∞ . 3) W k,p (Ω) becomes a normed linear space with the usual sum of functions and scalar multiplication. 2) if, as we indicated earlier, the derivatives are taken in the sense of distributions and the integrals are the Lebesgue integrals. ) For later use, we also define | f |k,p = ⎧ ⎪ ⎪ ⎪ ⎨ 1/p ∑ |α |=k Ω ⎪ ⎪ ⎪ ⎩ ess.

3) N where α ∈ (Z+ ) is a given multi-index. , a distribution. 3) amounts to no more than integration by parts. 3) says that we can still shift all the “burden of differentiability” from T to φ and define a new distribution. 1. In R, the Heaviside function 0, x H(x) = 1, x > 0, is in L1loc , so it defines a distribution H, φ = ∞ −∞ 0, H(x)φ (x) dx = ∞ 0 φ (x) dx ∀ φ ∈ D. H is differentiable everywhere with pointwise derivative 0, except at the point x = 0, where it is nondifferentiable in the classical sense.

20)? 24 Boundary Element Methods with Applications to Nonlinear Problems We can now reexamine the definition of H k (Ω), k ∈ N, for Ω a proper subdomain of RN . t. 3). 22) have the advantage that k does not have to be a nonnegative integer; it can be any nonnegative real number. 22) as our definitions for the Sobolev space W k,2 (Ω) = H k (Ω) on a domain Ω, for k 0. Note that if Ω is a “pathological” domain such as those depicted in Fig. 3). So we must be rather careful about the way in which the Sobolev spaces are defined.

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