EQUATIONS OF ELASTIC HYPERSURFACES

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2 SHELLS CONTENTS 1 INTRODUCTION 3 2 OUTLINE OF DIFFERENTIAL GEOMETRY 6 2.1 Differentiation and implicit function theorem . . . . . . . . . . . . . . . . 6 2.2 Vector fields on R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Vector product and integration on hypersurfaces in R n . . . . . . . . . . . 23 2.5 Gauß and Stoke’s formulae for domains in R n . . . . . . . . . . . . . . . . 31 2.6 Differential forms and Stoke’s formula for hypersurfaces . . . . . . . . . . 36 2.7 Covariant derivative, deformation tensor and Weingarten operator . . . . . 46 2.8 Coercive operators and Lax-Milgram lemma . . . . . . . . . . . . . . . . 58 3 CALCULUS OF DIFFERENTIAL OPERATORS ON HYPERSURFACES 65 3.1 Extension of the unit normal vector field to a hypersurface . . . . . . . . . 65 3.2 Calculus of tangential differential operators . . . . . . . . . . . . . . . . . 70 3.3 Differential operators on hypersurfaces in R n . . . . . . . . . . . . . . . . 79 3.4 The derivation of equation of elastic hypersurface . . . . . . . . . . . . . . 84 3.5 The surface Lamé operator and related PDO’s . . . . . . . . . . . . . . . . 90 3.6 Korn’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.7 Killing’s vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4 BOUNDARY VALUE PROBLEMS: GENERAL RESULTS 101 4.1 The Green formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2 Homogeneous linear ordinary differential equations . . . . . . . . . . . . . 106 4.3 Function spaces and convolution operators . . . . . . . . . . . . . . . . . 111 4.4 On pseudodifferential operators . . . . . . . . . . . . . . . . . . . . . . . 118 4.5 Non-homogeneous linear ordinary differential equations . . . . . . . . . . 122 4.6 Solvability of boundary value problems . . . . . . . . . . . . . . . . . . . 128 4.7 Layer potentials and boundary ΨDEs . . . . . . . . . . . . . . . . . . . . 137 5 BOUNDARY VALUE PROBLEMS: EXAMPLES 146 5.1 Boundary value problems for the Laplace-Beltrami equation . . . . . . . . 146 5.2 Layer potentials and boundary integral equations . . . . . . . . . . . . . . 151 5.3 Boundary value problems for the bi-Laplace-Beltrami equation . . . . . . . 158 5.4 Boundary value problems for the Lamé equation . . . . . . . . . . . . . . 162 5.5 Boundary value problems for equations of anisotropic elasticity . . . . . . 171 References 179
1. INTRODUCTION 3 1 INTRODUCTION Partial differential equations (abbreviated as PDEs) on hypersurfaces and corresponding boundary value problems (abbreviated as BVPs) we encounter rather often in applications: see [Ha1, §72] for the heat conduction by surfaces, [Ar1, §10] for the equations of surface flow, [Ci1] , [Ci3]-[Ci6], [Ko2], [Go1] for thin flexural shell problems in elasticity, [AC1] for the vacuum Einstein equations describing gravitational fields, [TZ1] for the Navier-Stokes equations on spherical domains, [TW1] for Stokes equations, [MM1] for minimal surfaces, [AMM1] for diffusion by surfaces, as well as the references therein. Furthermore, such equations arise naturally while studying the asymptotic behavior of solutions to elliptic boundary value problems in the neighborhood of conical points (see a classical reference [Ko1] ). We propose to write partial differential equations on hypersurface in cartesian coordinates of the ambient space instead of more customary local coordinates and Riemannian metric tensor. On the one hand this is improper having in mind generalization to arbitrary manifolds, which require intrinsic tools, independent of the ambient space. But on the other hand equations written in cartesian coordinates are simpler, have constant coefficients and enable rather simplified proof of basic properties, such as symmetry, Korn’s inequality, explicit Green formula etc. based on these properties it is possible to investigate solvability and uniqueness of classical and non-classical boundary value problems for those operators. Moreover, These results are useful in numerical and engineering applications (cf. [AN1], [Be1], [Ce1], [Co1], [DaL1], [BGS1], [Sm1]). Having in mind applications, equations in cartesian coordinates are simpler for approximation and numerical treatment. The approach is not new and was already applied earlier in [MM1] to minimal surfaces. The present investigation is based on the joint paper with D. Mitrea & M. Mitrea [DMM] and provides a (relatively) simple calculus of Boundary value problems (BVP’s) for partial differential equations (PDE’s) on hypersurfaces in R n . Let us consider an example: a surface S be given by a local immersion Θ : ω → S , ω ⊂ R n−1 , (1.1) which means that the derivatives { g k := ∂ k Θ } n−1 are linearly independent (i.e., the Jakobi k=1 matrix ∇ x Θ x has the maximal rank n−1). Let ν(X ) = (ν 1 (X ), . . . , ν j (X )) ⊤ be the outer unit normal vector (the Gauß mapping) to S at X ∈ S (see § 2.4 for details). The derivatives { } n−1 gk . Let { g k } n−1 k=1 k=1 be the contravariant frame in TS -biorthogonal to the covariant frame 〈g j , g k 〉 = δ jk , j, k = 1, . . . , n − 1. The bases { } n−1 g k and { g k} n−1 constitute, respectively, k=1 k=1 the columns and the rows of the Jacobi matrices ∇ x Θ x and ∇ {Θ1 ,...,Θ n−1 }Θ −1 (note, that the surface gradient of the inverse mapping Θ −1 : S → Ω is taken with respect to the tangential vector fields Θ k := Θ(e k ), k = 1, 2, . . . , n − 1). The Gram matrix G S (X ) = [g jk (X )] n−1×n−1 , g jk := 〈g j , g k 〉, is then positive definite, responsible for the Riemann metric on S and is called the covariant metric tensor. Moreover, it has the inverse matrix G −1 S (X ) = [gjk (X )] n−1×n−1 , g jk := 〈g j , g k 〉 (cf. (2.94)- (2.100)), which is called the contravariant metric tensor. Consider the following differential 1-form ω f ∑n−1 ∑n−1 ω f (V ) := ∂ V f = V k ∂ k f for f ∈ C 1 (S ) , V = V k g k ∈ TS . (1.2) k=1 k=1
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BIBLIOGRAPHY 181 [CD1] O.Chkadua, R
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BIBLIOGRAPHY 183 [Hr1] L. Hörmande
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BIBLIOGRAPHY 185 [Si1] I. Simonenko
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EQUATIONS OF ELASTIC HYPERSURFACES

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