Boundary value problems in elastostatics with singular data
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Lithuanian Mathematical Journal
Boundary value problems in elastostatics with singular data Giulio Starita and Alfonsina Tartaglione Dipartimento di Matematica e Fisica, Università degli Studi della Campania “Luigi Vanvitelli”, viale A. Lincoln 5, 81100 Caserta, Italy (e-mail: [email protected]; [email protected]) Received June 7, 2019; revised December 20, 2019
Abstract. We consider the main boundary value problems of linear elastostatics with nonregular data. We prove existence and uniqueness results for bounded and exterior domains of R3 of class C k (k 2). In the case of isotropic body, we prove the results for domains of class C 1,α (α ∈ (0, 1]) and of class C 1 in the case of the displacement problem. MSC: 74B05, 35Q74, 45B05 Keywords: linear elastostatics, layer potentials, boundary value problems, existence and uniqueness theorems, singular data
1 Introduction The boundary-value problems of elastostatics for regular domains and data are today a well-defined part of the variational theory for elliptic systems. For instance, let Ω be a bounded domain of R3 , and let {S1 , S2 } be complementary subsurfaces of ∂Ω . If Ω is of class C k (k 2) and ˆ ∈ W k−1/q, q (S1 ), u
ˆ ∈ W k−1−1/q, q (S2 ), s
q ∈ (1, +∞), then it is well known that the classical mixed problem div C [∇u] = 0 in Ω,
(1.1)
on S1 ,
(1.2)
ˆ u=u
on S2
ˆ τ u + s(u) = s
(τ 0),
(1.3)
has a unique solution u ∈ W k,q (Ω), provided that the elasticity tensor C is regular and satisfies natural ˆ is in equilibrium for S1 = ∅ and τ = 0; see, for example, [11, Chap. VI] definiteness assumptions and s and [1, 3, 6, 12]. Here s(u) = C[∇u]n, with n unit normal on ∂Ω , is the traction field on the boundary. For S2 = ∅ (resp. S1 = ∅ and τ = 0) we have the Dirichlet (or displacement) problem (resp., the Neumann, or traction, problem) [8]. For S1 = ∅ and τ > 0, we have the Robin problem. Clearly, even in view of possible applications, it is quite natural to detect whether the existence and uniqueness for (1.1)–(1.3) still hold under weaker regularity assumptions on the boundary data, for instance, in the c 2020 Springer Science+Business Media, LLC 0363-1672/20/6002-0001
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presence of concentrated loads. From the existence of a regular solution of a boundary value problem with datum in a space H it follows, by transposition, the existence of a so-called very weak solution (see, e.g., [1,12]), which is defined by a suitable integral equation and corresponds to data in the dual space H . In this approach the main problem concerns the sense to give to the attainability of the boundary value (see [1, Chap. 6]). An alternative approach to the existence of a solution to the boundary value problem of elastostatics, confined to homogeneous bodies but undoubtedly more sharp and strictly connected to the structure of system (1.1), is based on the classical theory of layer integral equations [11, Chap. VI]. (From a historical point of view, this approach has been the first one treating i
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