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Lecture notes on differential equations of mathematical by Sweers G.

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By Sweers G.

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U weakly The sequentially weakly lower semi-continuity goes as follows. If un in W01,2 (Ω), then Z ³ ´ 2 2 1 1 E(un ) − E(u) = 2 |∇un | − 2 |∇u| − f (un − u) dx Ω Z ³ ´ 2 1 |∇u − ∇u| + (∇u − ∇u) · ∇u − f (u − u) dx = n n n 2 ZΩ ((∇un − ∇u) · ∇u − f (un − u)) dx → 0. ≥ Ω ³ ´0 ¢ ¡ R Here we also used that v 7→ Ω f vdx ∈ W01,2 (Ω) . Hence the minimizer exists. 46 In this example we have used zero Dirichlet boundary conditions which allowed us to use the well-known function space W01,2 (Ω). For nonzero boundary conditions a way out is to consider W01,p (Ω)+g where g is an arbitrary W 1,p (Ω)function that satisfies the boundary conditions.

6) E i=1 ∞ C0 -function ¯ and with such that χ(x) = 1 for x ∈ Ω As a last step let χ be a 0 support(χ) ⊂ Ω . We define the extension operator Em by Em (u) = χ X u ¯i + ζ +1 u. 4). It remains to show that Em (u) ∈ W0m,p (Ω0 ). Due to the assumption that support(χ) ⊂ Ω0 we find that support(Jε ∗ Em (u)) ⊂ Ω0 for ε small enough where Jε is a mollifier. Hence (Jε ∗ Em (u)) ∈ C0∞ (Ω0 ) and since Jε ∗ Em (u) approaches Em u in W m,p (Ω0 )-norm for ε ↓ 0 one finds Em u ∈ W0m,p (Ω0 ). © ª 0 2 Exercise 27 A square inside a disk: Ω1 = (−1, 1) and Ω1 = x ∈ R2 ; |x| < 2 .

Similar, if ∂ν y, the normal derivative on M is given then also all tangential ∂ y are determined. Etcetera. So it will be suﬃcient to fix all derivatives of ∂ν normal derivatives from order 0 up to order m − 1. 5 The Cauchy ´ problem for the mth -order partial diﬀerential ∂ ∂m equation F x, y, ∂x1 y, . . , ∂xm y = 0 on Rn with respect to the smooth (n − 1)n dimensional manifold M is the following: ´  ³ ∂ ∂m  F x, y, y, . . 10) ∂ν y = φ1  on M,  ..   .    ∂ m−1 ∂ν m−1 y = φm−1 where ν is the normal on M and where the φi are given.