By J.W. Slooff and W. Schmidt (Eds.)
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Extra resources for Computational Aerodynamics Based on the Euler Equations
Example text
B) If u ∈ M 1,p (Rn ), then u ∈ W 1,p (Rn ) and u has g ∈ D(u) with g Lp ≤ C(n, p) |∇u| Lp . Remark. In fact W 1,p (Rn ) = M 1,p (Rn ) for all 1 ≤ p < ∞ but the “pointwise” inequality in (a) does not give the information needed. If u ∈ C 1 (Ω), then ∇u = 0 in each open set V ⊂ Ω where u = const. e. in the set where u = const. This does not hold in the space M 1,p (X). Because of this I have not yet seen any serious development for the Potential Theory in the space M 1,p (X). However, for many fractal sets X of Rn the space M 1,p (X) with a natural Hausdorff measure is a natural space to study minimizers of variational integrals and it would be interesting to know if a potential theory similar to the theory based on the Newtonian space can be constructed on M 1,p (X).
April 7, 2008 17:13 World Scientific Review Volume - 9in x 6in 40 O. Martio A curve is rectifiable if l(γ) < ∞. e. γ([a, b]) is not a point, unless otherwise stated. Two important concepts are associated with a rectifiable path γ: the length function Sγ : [a, b] → R and parametrization by arc length. The function Sγ is defined as Sγ (t) = l(γ|[a, t]), a ≤ t ≤ b, and the path γ˜ : [0, l(γ)] → X is the unique 1–lipschitz continuous map such that γ = γ˜ ◦ Sγ . In particular, l(˜ γ |[0, t]) = t, 0 ≤ t ≤ l(γ), and γ˜ is obtained from γ by an increasing change of parameter.
2. 2. Let (X, d) be a metric space with 0 < µ(X) < ∞ and u ∈ M 1,p (X), 1 ≤ p < ∞. 4) X where g ∈ D(u) and uX = − udµ = 1 µ(X) X udµ. X Remarks. (a) Note that |u|dµ < ∞ X by the H¨older inequality since µ(X) < ∞. 5. 2. Since |u(x) − uX | = | −(u(x) − u(y))dµ(y)| X ≤ diam(X) −(g(x) + g(y))dµ(y) = diam(X)(g(x) + gX ), X we obtain by integrating over x |u(x) − uX |p dµ ≤ diam(X)p X (g(x) + gY )p dµ X p p−1 X p p gX dµ g p dµ + = diam(X) 2 where we have used the inequality (a + b) ≤ 2 and p ≥ 1. Now p−1 X p p (a + b ) valid for a, b ≥ 0 p gX dµ = µ(X)( − gdµ)p ≤ µ(X) − g p dµ = X X X g p dµ X April 7, 2008 17:13 World Scientific Review Volume - 9in x 6in 34 O.