Basics and Highlights in Fundamental Physics by Italy) International School of Subnuclear Physics 1999

By Italy) International School of Subnuclear Physics 1999 (Erice, A. Zichichi, Antonino Zichichi

In August and September 1999, a gaggle of sixty eight physicists from forty eight laboratories in 17 nations met in Erice, Italy, to take part within the thirty seventh process the overseas university of Subnuclear Physics. This quantity constitutes the court cases of that assembly. It makes a speciality of the elemental harmony of primary physics at either the theoretical and the experimental point.

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Un converges weakly to v and hence v = u because of the uniqueness of weak limit. Now define fi (λ) = |λi |q for any 1 ≤ q < p. 17) q = K φ(x)|ui | dx as n tends to ∞. 3. 3 Embedding Theorems In this section we shall establish a compactness embedding theorem which is a generalization of Murat Lemma [Mu] or just an interpolation inequality (cf. [Tr]). 3. 1 Let Ω ⊂ RN be bounded and open, and f ∈ W −1,p (Ω) for 1 < p < ∞, supp f ⊂⊂ Ω. Let u be the solution of equation −∆u = f, in Ω, u = 0, on ∂Ω.

For any η > 0, there exists a function ψ η ∈ D(Ω) such that ψ η − ψ¯η W01,s (Ω) ≤ η. 12) 24 CHAPTER 2. THEORY OF COMPENSATED COMPACTNESS Then for any ϕ ∈ W01,q (Ω), there holds (ψ η − ψ¯η )ϕ W01,p (Ω) (ψ η − ψ¯η )ϕ ≤ CN + N Lp (Ω) + ∂(ψη −ψ¯η ) ϕ Lp (Ω) ∂xk k=1 N ∂ϕ (ψ η − ψ¯η ) ∂x k k=1 Lp (Ω) . From the H¨ older inequality, there hold (ψ η − ψ¯η )ϕ ∂ϕ (ψ η − ψ¯η ) ∂xk Lp (Ω) ≤ ψ η − ψ¯η ≤ ψ η − ψ¯η Lp (Ω) Ls (Ω) Ls (Ω) ϕ ϕ Lq (Ω) , W01,q (Ω) , k = 1, · · · , N and ∂(ψ η − ψ¯η ) ϕ ∂xk Lp (Ω) ≤ ψ η − ψ¯η ϕ W01,s (Ω) Lq (Ω) , k = 1, · · · , N.

2). 1 (1) Let the initial data (u0 (x), v0 (x)) be bounded measurable. 2) exists and is uniformly bounded with respect to the viscosity parameter ε. 6) holds, then there exists a subsequence of r ε = (uε )2 + (v ε )2 (still labelled r ε ) which converges pointwisely to a function l(x, t). 2). 1. 2 Since (uε , v ε ) is uniformly bounded with respect to ε, its weak-star limit (u, v) always exists. However, the strong limit l(x, t) of (uε )2 + (v ε )2 needs not equal u(x, t)2 + v(x, t)2 . 2) without any more condition such as given in part (3).

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