By Thomas Craig
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Additional resources for A Treatise on Linear Differential Equations, Volume I: Equations with Uniform Coefficients
4 to weighted Lp Sobolev and weighted H¨older 23 24 2. 8). 9, where the Dirichlet problem for a strongly elliptic differential operator with variable coefficients is studied. 1. Weighted Sobolev spaces in a dihedron In this section, we introduce the weighted Sobolev space Vδl,p (D) in a dihedron D both for positive and negative integer l. For l ≥ 1 we consider also the corresponding trace spaces on the faces Γ+ and Γ− of the dihedron. We prove imbedding theorems and the equivalence of different norms.
Suppose that ηu ∈ W m−1,p (D) , η L(Dx ) u ∈ W l−2m,p (D) , l ≥ m, ◦ and χu ∈W m,p (D) for every χ ∈ C0∞ (D\M ). 13) ζu W l,p (D) ≤c η L(Dx ) u + ηu W l−2m,p (D) W m−1,p (D) with a constant c independent of u. P r o o f. Let σν be an infinitely differentiable functions depending only on x3 such that +∞ ∂xj 3 σν < cj , σν (x3 ) = 0 for x3 ∈ (ν − 1, ν + 1), σν = 1. ν=−∞ Furthermore, let τν = σν−1 + σν + σν+1 . 8 that the norm in W l,p (D) is equivalent to the norm +∞ σν u p W l,p (D) 1/p . 7 that ζσν u ∈ W l,p (D) and ζσν u p W l,p (D) ≤ ≤ c c L(Dx ) (ζσν u) ζσν L(Dx )u W l−2m,p (D) p W l−2m,p (D) + ζσν u + ητν u L1 (D) p W l−1,p (D) .
11) A∗α Dxα , |α|=2m where A∗α is the adjoint matrix of Aα . It is evident, that the differential operators L(Dx ) and L+ (Dx ) are simultaneously strongly elliptic. 12) ∂ k−1 u = gk± on Γ± , k = 1, . . , m, ∂nk−1 is also uniquely solvable in V0m,2 (D) for arbitrary f ∈ V0−m,2 (D) and gk ∈ m−k+1/2,2 V0 (Γ± ) . 4. An a priori estimate for the solution. 8 for boundary value problems in a cone. 7). 7. Let Ω be a bounded domain in RN with smooth (of class C ∞ ) boundary ∂Ω. Furthermore, let L(x, Dx ) be a strongly elliptic differential operator of order 2m with infinitely differentiable coefficients on Ω.
A Treatise on Linear Differential Equations, Volume I: Equations with Uniform Coefficients by Thomas Craig