Mathematical Principles of Theoretical Physics
3.2. ANALYSIS ON RIEMANNIAN MANIFOLDS 127 Remark 3.8.WhenMis non-compact, the conclusions (3.2.11) and (3.2.12) are also valid o ...
128 CHAPTER 3. MATHEMATICAL FOUNDATIONS for 0<R<1 andβp>1. It follows from (3.2.16) that ∫ BR |∇u|pdx<∞⇔k=n+ (α− 1 ) ...
3.2. ANALYSIS ON RIEMANNIAN MANIFOLDS 129 3) the Laplace operator:DkDk=div·∇, 4) the Laplace-Beltrami operator:∆=dδ+δd, 5) the w ...
130 CHAPTER 3. MATHEMATICAL FOUNDATIONS Letu∈W^1 ,p(T∗M),u= (u 1 ,···,un). Then, divu=Dkuk=gklDl(uk) =Dl(glkuk) =(by( 3. 2. 24 ) ...
3.2. ANALYSIS ON RIEMANNIAN MANIFOLDS 131 and its gradient (i.e. covariant derivatives) read as (3.2.29) ∇u={Dkui}, Dkui= ∂ui ∂x ...
132 CHAPTER 3. MATHEMATICAL FOUNDATIONS which can be generalized to tensor fields on Riemann manifolds. LetMbe a Riemannian mani ...
3.2. ANALYSIS ON RIEMANNIAN MANIFOLDS 133 which is the formula (3.2.36). The Gauss formula (3.2.36) can be generalized to more g ...
134 CHAPTER 3. MATHEMATICAL FOUNDATIONS Theorem 3.14.Let the metric gijbe W^2 ,∞, and f,g be L^2. If (3.2.41) ∫ M [(f,∇φ)−(g,φ)] ...
3.3 Orthogonal Decomposition for Tensor Fields Remark 3.16.IfM=R^1 ×M ̃with metric (3.2.43) and∂M ̃=/0, then the problem (3.2.45 ...
136 CHAPTER 3. MATHEMATICAL FOUNDATIONS 2.Leray decomposition.LetΩ∈Rnbe a domain, andu∈L^2 (TΩ)be ann-dimensional vector field. ...
3.3. ORTHOGONAL DECOMPOSITION FOR TENSOR FIELDS 137 1) The tensor field u can be orthogonally decomposed into (3.3.9) u=∇Aφ+v wi ...
138 CHAPTER 3. MATHEMATICAL FOUNDATIONS Proof of Theorem3.17.We proceed in several steps as follows. STEP1. PROOF OFASSERTION(1) ...
3.3. ORTHOGONAL DECOMPOSITION FOR TENSOR FIELDS 139 where HDk={u∈Hk(E)|divAu= 0 }, Gk={u∈Hk(E)|u=∇Aψ}. Define an operator∆ ̃:HD^ ...
140 CHAPTER 3. MATHEMATICAL FOUNDATIONS 3.3.3 Uniqueness of orthogonal decompositions In this subsection we only consider the ca ...
3.3. ORTHOGONAL DECOMPOSITION FOR TENSOR FIELDS 141 Theorem 3.20.Let u∈L^2 (T 20 M)be symmetric, i.e. uij=uji, and the first Bet ...
142 CHAPTER 3. MATHEMATICAL FOUNDATIONS is closed. Hence it follows that there is aφ∈H^1 (M)such that φk= ∂ φ ∂xk for 1≤k≤n. Ass ...
3.3. ORTHOGONAL DECOMPOSITION FOR TENSOR FIELDS 143 3.3.4 Orthogonal decomposition on manifolds with boundary In the above subse ...
144 CHAPTER 3. MATHEMATICAL FOUNDATIONS with the metric (3.3.43) (gμ ν) = ( −1 0 0 G ) . HereM ̃is a closed Riemannian manifold, ...
3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 145 whereδFis the derivative operator ofF. Given a variational problem, it is importan ...
146 CHAPTER 3. MATHEMATICAL FOUNDATIONS On the other hand, by (3.4.5) and 〈δF(u),v〉= ∫ Rn δF(u)vdx, we deduce that ∫ Rn δF(u)vdx ...
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