From (4.4.7) we immediately obtain that the integrand in (4.4.6) has the form. (4.4.8). 3. 3 sin- ( y ]T aij(v)sij(v) - ^ (ni
http://dx.doi.org/10.1090/surv/085/04
CHAPTER 4
Other boundary value problems for the Lame system The present chapter is concerned with the spectral properties of operator pencils generated by some boundary value problems for the Lame system in a threedimensional cone /C. In Section 4.1 we consider a mixed boundary value problem with the following three types of boundary conditions: (i) U = 0, (ii) *7n = 0 a n d < r n , r ( * 7 ) = 0 , (iii) UT = 0 and an,n(u) = °We use the notation: £/ = (C/i, C/2, E/3) is the displacement vector, n = (ni,ri2,n3) is the exterior normal to 9/C\{0}, Un = U • n is the normal component of the vector J7, UT = U — Unn is the tangential component of the vector U on the boundary, a (U) — {&i,j{U)} is the stress tensor connected with the strain tensor {£^(u)} = {1-(dXjui + dXiuj)} by the Hooke law
(/i is the shear modulus, v is the Poisson ratio, v < 1/2, and Sij denotes the Kronecker symbol), &n,n(U) is the normal component of the vector crn(U) — cr(U) n, i. e., 3
&n,r{U) is the tangential component of the vector an(U). We modify the method used in Chapter 3. As in the case of the Dirichlet problem, we show that the spectrum of the corresponding operator pencil in a certain strip centered about the line Re A = —1/2 consists only of real eigenvalues and that the eigenvectors corresponding to eigenvalues in the interior of this strip do not have generalized eigenvectors. This result combined with Theorem 1.4.4 implies that the solution U has the following asymptotics in a neighborhood of a conic vertex: J
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4. O T H E R B O U N D A R Y VALUE P R O B L E M S F O R T H E L A M E S Y S T E M
108
Here (3 is an arbitrary positive real number less than
«§-*)'+a-*)'r-;, Cj are constants, Xj are the real eigenvalues of the above mentioned pencil in the interval (—1/2,/?), and u^ are the corresponding eigenvectors. The feature of the above asymptotic representation is that the exponents Xj are real and that it does not contain logarithmic terms. At the end of Section 4.1 we derive a variational principle for the eigenvalues A i , . . . , Xj. Sections 4.2-4.4 are dedicated to the Neumann boundary conditions for the Lame system. First we analyze the plane problem in the spirit of Section 3.1. For the three-dimensional case the method used previously does not work and we apply a different approach. We start with the situation when the cone K is given by the inequality X3 > (j){xi,X2)) where 0 is a smooth positively homogeneous of degree 1 function on R 2 \{0}. Then we pass to a three-dimensional anisotropic medium with a crack which has the form of a plane angle. We prove that in both cases the operator pencil generated by the Neumann problem has only the eigenvalues 0 and —1 in the strip — 1 < Re A < 0 and that both eigenvalues have geometric and algebraic multiplicity 3. These assertions imply the Holder continuity of the solution to the Neumann problem for the Lame system in a neighborhood of the vertex of a polyhedral angle as well as the Holder continuity of the displacement field in an anisotropic medium with a polygonal crack. 4.1. A mixed boundary value problem for the Lame system 4.1.1. Formulation of the problem. Let /C be the cone {x = (#1,0:2, #3) € R 3 : x/\x\ G SI}, where Q is a domain on the unit sphere with Lipschitz boundary dfl = 7y1 U • • • U 7jy and 7 1 , . . . , 7 ^ are pairwise disjoint open arcs. Then dK = Fi U • • • U TN , where Tk = {x : x/\x\ G 7/J. The homogeneous Lame system AC/ + ( l - 2 i / ) - 1 V V - C / = 0 can be written in the form
(4.1.1)
E ^ ^ = ° 3= 1
fori
= 1,2,3,
J
where {)U
