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Korn’s type inequalities and applications to elasticity. (English) Zbl 0972.35013

International conference in memory of Vito Volterra. Papers of the conference, Rome, Italy, October 8-11, 1990. Rome: Accademia Nazionale dei Lincei, Atti Convegni Lincei. 92, 183-209 (1992).
From the introduction: Korn’s inequalities play an important role in the mathematical theory of elasticity. Using these inequalities, one can prove existence theorems, uniqueness, and stability of solutions of boundary value problems for the system of elasticity in bounded and unbounded domains, estimate approximate solutions, and so on.
In §2 we consider Korn’s and Hardy’s type inequalities for some unbounded domains. In §3 we give applications of these results to study uniqueness and stability of solutions of boundary value problems for elasticity in unbounded domains.
In §4 we solve the problem of the dimension of the kernel for the Dirichlet boundary value problem for the system of elasticity in the class of solutions with the finite energy or the finite Dirichlet integral for any unbounded domain \(\Omega\) on the base of a new concept: \(C_g\)-capacity. Examples show how this dimension depends on the geometry of a domain \(\Omega\).
For the entire collection see [Zbl 0957.00059].

MSC:

35B45 A priori estimates in context of PDEs
74B05 Classical linear elasticity
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
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