On the improved interior regularity of the solution of a fourth order elliptic problem modelling the displacement of a linearly elastic shallow shell lying subject to an obstacle

Abstract

In this paper we show that the solution of an obstacle problem for linearly elastic shallow shells enjoys higher differentiability properties in the interior of the domain where it is defined.

Keywords

Shallow shell obstacle problems elliptic variational inequalities improved regularity fourth order problems

1. Introduction

In this paper we study the regularity of the solution of an obstacle problem for shallow shells, which was obtained as a result of a rigorous asymptotic analysis by Léger and Miara in the papers [16,17]. Since the solution of this fourth order problem is uniquely determined and takes the form of a Kirchhoff–Love field (cf., e.g., [4]), the problem in object is formulated as a set of variational equations for the tangential components of the solution, and a set of variational inequalities posed over a non-empty, closed and convex subset of the Sobolev space $H_{0}^{2}$ for the transverse component of the solution. It is also worth mentioning the following recent literature about obstacle problems [8,9,11,12,18,22].

The augmentation of regularity for fourth order variational inequalities was first addressed by Frehse in the early Seventies [14,15]. In the late Seventies and early Eighties, Caffarelli and his collaborators published the two papers [1,2], where they proved that the solution of an obstacle problem for the biharmonic operator (cf., e.g., Section 6.7 of [7]) could not be too regular.

To our best knowledge, there is no record in the literature treating the augmentation of regularity of fourth order variational inequalities whose solution is a vector field. The purpose of this paper is to extend the result obtained by Frehse in the pioneering work [14] to the case where the fourth order obstacle problem devised by Léger and Miara is considered. The generalisation of the result obtained by Frehse to the vectorial case has been inspired by the recent paper [21], where such a higher regularity is entailed in the contrivance of a suitable finite element method for approximating the solution of the model devised by Léger and Miara in the aforementioned literature. The fact that the variational equations and variational inequalities appearing in such a model are coupled, makes the analysis substantially more complicated than in the scalar case.

This paper is divided into four sections (including this one). In Section 2 we present some background and notation. In Section 3 we prove an augmentation of regularity result for the tangential components of the solution of the problem under consideration, following somehow the ideas in the celebrated work of Nirenberg [20]. In particular, the proof concerning the augmentation of regularity for the tangential components of the solution will be conducted using an alternative approach from the one originally proposed by Nirenberg. The novelty in our proof consists in considering the boundary conditions associated with the problem under consideration, which are not considered in the literature (cf., the remarks preceding the proof of Theorem 1 in Section 6.3 of [13]). As we will see, such a feature will be fundamental in order to exploit the ellipticity of the bilinear form under consideration through a specific inequality of Korn’s type with boundary conditions. Finally, in Section 4, we generalise the improved regularity result in [14] for the transverse component of the solution of the problem under consideration.

2. Background and notation

For an overview about the classical notions of differential geometry used in this paper see, e.g., [5] or [6] while, for an overview about the classical notions of functional analysis used in this paper see, e.g., [7]. We let Latin indices, except h, take their values in the set ${1, 2, 3}$ and we let Greek indices, except ρ, ν and ε, take their values in the set ${1, 2}$ and, unless differently indicated, we use the Einstein summation convention with respect to repeated indices in conjunction with this rule. The notations $δ_{i j}$ , $δ^{i j}$ , $δ_{j}^{i}$ and $δ_{α β}$ designate the Kronecker symbol. Given an open subset Ω of $R^{n}$ , where $n ⩾ 1$ , we denote the usual Lebesgue and Sobolev spaces by $L^{2} (Ω)$ , $H^{1} (Ω)$ , $H_{0}^{1} (Ω)$ , $H^{2} (Ω)$ , or $H_{0}^{2} (Ω)$ . We denote by $D (Ω)$ the space of functions which are infinitely differentiable in Ω and whose support is contained in Ω. The Euclidean norm of any point $x \in Ω$ is denoted by $| x |$ . The special notation $‖ \cdot ‖_{m, Ω}$ , where $m ⩾ 1$ is an integer, denotes the norm of the space $H^{m} (Ω)$ . If $m = 0$ , then: $\begin{matrix} ‖ \cdot ‖_{0, Ω} : = ‖ \cdot ‖_{L^{2} (Ω)} . \end{matrix}$

The special notation $| \cdot |_{m, Ω}$ , where $m ⩾ 1$ is an integer, denotes the standard semi-norm of the space $H^{m} (Ω)$ . Spaces of vector-valued functions are denoted in boldface.

The boundary Γ of an open subset Ω in $R^{n}$ is said to be Lipschitz-continuous if the following conditions are satisfied (cf., e.g., Section 1.18 of [7]): Given an integer $s ⩾ 1$ , there exist constants $α_{1} > 0$ and $L > 0$ , and a finite number of local coordinate systems, with coordinates $ϕ_{r}^{'} = (ϕ_{1}^{r}, \dots, ϕ_{n - 1}^{r}) \in R^{n - 1}$ and $ϕ_{r} = ϕ_{n}^{r}$ , and corresponding functions $\begin{matrix} {\tilde{θ}}_{r} : {\tilde{ω}}_{r} : = {ϕ_{r} \in R^{n - 1}; | ϕ_{r} | < α_{1}} \to R, 1 ⩽ r ⩽ s, \end{matrix}$ such that $\begin{matrix} Γ = ⋃_{r = 1}^{s} {(ϕ_{r}^{'}, ϕ_{r}); ϕ_{r}^{'} \in {\tilde{ω}}_{r} and ϕ_{r} = {\tilde{θ}}_{r} (ϕ_{r}^{'})}, \end{matrix}$ and $\begin{matrix} | {\tilde{θ}}_{r} (ϕ_{r}^{'}) - {\tilde{θ}}_{r} (υ_{r}^{'}) | ⩽ L | ϕ_{r}^{'} - υ_{r}^{'} |, for all ϕ_{r}^{'}, υ_{r}^{'} \in {\tilde{ω}}_{r}, and all 1 ⩽ r ⩽ s . \end{matrix}$

We observe that the second last formula takes into account overlapping local charts, while the last set of inequalities express the Lipschitz continuity of the mappings ${\tilde{θ}}_{r}$ .

An open set Ω is said to be locally on the same side of its boundary Γ if, in addition, there exists a constant $α_{2} > 0$ such that $\begin{array}{l} {(ϕ_{r}^{'}, ϕ_{r}); ϕ_{r}^{'} \in {\tilde{ω}}_{r} and {\tilde{θ}}_{r} (ϕ_{r}^{'}) < ϕ_{r} < {\tilde{θ}}_{r} (ϕ_{r}^{'}) + α_{2}} \subset Ω, for all 1 ⩽ r ⩽ s, \\ {(ϕ_{r}^{'}, ϕ_{r}); ϕ_{r}^{'} \in {\tilde{ω}}_{r} and {\tilde{θ}}_{r} (ϕ_{r}^{'}) - α_{2} < ϕ_{r} < {\tilde{θ}}_{r} (ϕ_{r}^{'})} \subset R^{n} ∖ \overline{Ω}, for all 1 ⩽ r ⩽ s . \end{array}$

Let $ω \subset R^{2}$ be a domain, namely a non-empty bounded open connected subset of $R^{2}$ with Lipschitz continuous boundary $γ : = \partial ω$ and such that ω is all on the same side of γ.

Let $ω_{1} \subset ω$ ; the special notation $ω_{1} \subset \subset ω$ means that $\overline{ω_{1}} \subset ω$ and $dist (γ, \partial ω_{1}) > 0$ . We denote by $H_{loc}^{m} (ω)$ the space of measurable functions that are of class $H^{m} (ω_{1})$ , for all $ω_{1} \subset \subset ω$ . Let $y = (y_{α})$ denote a generic point in the set $\overline{ω}$ and let $\partial_{α} : = \partial / \partial y_{α}$ and $\partial_{α β} : = \partial^{2} / \partial y_{α} \partial y_{β}$ .

In what follows $ν$ denotes the outer unit normal vector field to the boundary γ and $\partial_{ν}$ denotes the outer unit normal derivative operator along γ.

As a model of the three-dimensional “physical” space $R^{3}$ , we take a real three-dimensional affine Euclidean space, i.e., a set in which a point $O \in R^{3}$ has been chosen as the origin and with which a real three-dimensional Euclidean space, denoted $E^{3}$ , is associated. We equip $E^{3}$ with an orthonormal basis consisting of three vectors $e^{i}$ , with components $e_{j}^{i} = δ_{j}^{i}$ . The Euclidean inner product of two elements $a$ and $b$ of $E^{3}$ is denoted by $a \cdot b$ .

The definition of $R^{3}$ as an affine Euclidean space means that with any point $x \in R^{3}$ is associated an uniquely determined vector $O x \in E^{3}$ . The origin $O \in R^{3}$ and the orthonormal vectors $e^{i} \in E^{3}$ together constitute a Cartesian frame in $R^{3}$ and the three components $x_{i}$ of the vector $O x$ over the basis formed by $e^{i}$ are called the Cartesian coordinates of $x \in R^{3}$ , or the Cartesian components of $O x \in E^{3}$ . Once a Cartesian frame has been chosen, any point $x \in R^{3}$ may be thus identified with the vector $O x = x_{i} e^{i} \in E^{3}$ . As a result, a set in $R^{3}$ can be identified with a “physical” body in the Euclidean space $E^{3}$ .

For each $ε > 0$ , we define the sets $\begin{matrix} Ω^{ε} : = ω \times] - ε, ε [, Γ_{0}^{ε} : = γ \times [- ε, ε], Γ_{\pm}^{ε} : = ω \times {\pm ε}, \end{matrix}$ and we let $x^{ε} = (x_{i}^{ε})$ denote a generic point in the set $\overline{Ω^{ε}}$ . We then have that $\begin{matrix} x_{α}^{ε} = y_{α} and - ε ⩽ x_{3}^{ε} ⩽ ε . \end{matrix}$

Let us define the sets $\begin{matrix} Ω : = ω \times] - 1, 1 [, Γ_{0} : = γ \times [- 1, 1], Γ_{\pm} : = ω \times {\pm 1}, \end{matrix}$ and let us denote by $x = (x_{i})$ a generic point in the set $\overline{Ω}$ . We observe that it is possible to define a bijection $π^{ε} : \overline{Ω} \to \overline{Ω^{ε}}$ as follows $\begin{matrix} π^{ε} x = x^{ε}, for all x \in \overline{Ω}, \end{matrix}$ which means that $x_{α}^{ε} = x_{α} = y_{α}$ and $x_{3}^{ε} = ε x_{3}$ . Consequently, we have that $\partial_{α}^{ε} = \partial_{α}$ and $\partial_{3}^{ε} = ε^{- 1} \partial_{3}$ .

Define the surface ${\hat{ω}}^{ε}$ by $\begin{matrix} {\hat{ω}}^{ε} : = {(y, θ^{ε} (y)); y \in \overline{ω}}, \end{matrix}$ and define its corresponding outer unit normal vector field $a_{3}^{ε}$ by $\begin{matrix} a_{3}^{ε} : = \frac{1}{\sqrt{α^{ε}}} (- \partial_{1} θ^{ε}, - \partial_{2} θ^{ε}, 1), with α^{ε} : = {| \partial_{1} θ^{ε} |}^{2} + {| \partial_{2} θ^{ε} |}^{2} + 1 . \end{matrix}$

The reference configuration ${\hat{Ω}}^{ε}$ in Cartesian coordinates of the three-dimensional body under consideration is thus obtained from ${\hat{ω}}^{ε}$ via the following expression: $\begin{matrix} {\hat{Ω}}^{ε} = {{\hat{x}}^{ε} = ({\hat{x}}_{i}^{ε}) \in E^{3}; {\hat{x}}^{ε} = (y, θ^{ε} (y)) + x_{3}^{ε} a_{3}^{ε} (y) for all x^{ε} = (y, x_{3}^{ε}) \in Ω^{ε}} . \end{matrix}$

The latter expression thus defines a mapping $Θ^{ε} : \overline{Ω^{ε}} \to {{\hat{Ω}}^{ε}}^{-} \subset E^{3}$ . By resorting to the assumed smoothness of the mapping $θ^{ε}$ , it is possible to show (cf., e.g., [5]) that the mapping $Θ^{ε}$ is, for ε sufficiently small, a $C^{1}$ -diffeomorphism. We let ${\hat{x}}^{ε} = ({\hat{x}}_{i}^{ε})$ denote a generic point of the set ${{\hat{Ω}}^{ε}}^{-}$ , and we let ${\hat{\partial}}_{i}^{ε} : = \partial / \partial {\hat{x}}_{i}^{ε}$ .

Furthermore, we assume that the three-dimensional body in object is made of a isotropic homogeneous linearly elastic material. Its elastic behaviour is thus described in terms of its two Lamé coefficients $λ^{ε} ⩾ 0$ and $μ^{ε} > 0$ . As it is customarily (cf., e.g., [4] and [5]) assumed for many models in linearised elasticity, we hypothesise that there exist constants $λ ⩾ 0$ and $μ > 0$ , both independent of ε, such that $λ^{ε} \equiv λ$ and $μ^{ε} \equiv μ$ .

For such small parameter $ε > 0$ (the parameter ε is “small” compared to the dimensions of ω), the linearly elastic body whose reference configuration is ${{\hat{Ω}}^{ε}}^{-}$ is thus called a linearly elastic shell with thickness $2 ε$ and middle surface ${\hat{ω}}^{ε} = Θ^{ε} (\overline{ω} \times {0})$ .

Besides, by virtue of the form of the surface ${\hat{ω}}^{ε}$ , we note that, up to an additive constant, the mapping $θ^{ε} : \overline{ω} \to R$ , measuring the deviation of the middle surface of the reference configuration of the shell from a plane, should be of the same order as the thinness of the shell, i.e., of order $O (ε)$ – big “Oh” in Landau notation. In view of these geometrical properties, we say that the linearly elastic shell under consideration is shallow.

Referring to [10] (see also Section 3.1 of [4]), we recall the rigorous criterion for defining a linearly elastic shallow shell (from now on shallow shell). A linearly elastic shell is a shallow shell if and only if there exists a function $θ \in C^{3} (\overline{ω})$ , independent of ε, such that $\begin{matrix} (1) & θ^{ε} (y) = ε θ (y), for all y \in \overline{ω} . \end{matrix}$

We assume that the reference configuration of the shallow shell under consideration is subject to applied body forces, denoted by ${\hat{f}}^{ε}$ , whose density per unit volume is defined by means of their components ${\hat{f}}_{i}^{ε} \in L^{2} ({\hat{Ω}}^{ε})$ , and to applied surface forces on the upper surface of the reference configuration, denoted by ${\hat{g}}^{+, ε}$ , whose density per unit area is defined by means of their components ${\hat{g}}_{i}^{+, ε} \in L^{2} ({\hat{Γ}}_{+}^{ε})$ . Since we do not take into account any friction, applied surface forces associated with the “lower” face of the shallow shell reference configuration are not to be taken into account.

We write the Cartesian components of a vector $O {\hat{x}}_{-}^{ε}$ corresponding to a point ${\hat{x}}_{-}^{ε} \in {\hat{Γ}}_{-}^{ε}$ as follows: $\begin{matrix} O {\hat{x}}_{-}^{ε} = (\begin{matrix} y_{1} + ε \frac{\partial_{1} θ^{ε}}{\sqrt{α^{ε}}} \\ y_{2} + ε \frac{\partial_{2} θ^{ε}}{\sqrt{α^{ε}}} \\ θ^{ε} (y) - ε \frac{1}{\sqrt{α^{ε}}} \end{matrix}) = (\begin{matrix} x_{1}^{ε} + ε \frac{\partial_{1} θ^{ε}}{\sqrt{α^{ε}}} \\ x_{2}^{ε} + ε \frac{\partial_{2} θ^{ε}}{\sqrt{α^{ε}}} \\ x_{3}^{ε} - ε \frac{1}{\sqrt{α^{ε}}} \end{matrix}), with y = (y_{α}) \in ω . \end{matrix}$

We can thus define the following mappings over the “lower” face of the reference configuration ${\hat{Γ}}_{-}^{ε}$ : $\begin{matrix} {\hat{θ}}^{ε} = θ^{ε} \circ {(Θ^{ε})}^{- 1} and {\hat{α}}^{ε} = α^{ε} \circ {(Θ^{ε})}^{- 1} . \end{matrix}$

The unilateral contact feature is expressed in terms of the displacement ${\hat{u}}^{ε}$ the shallow shell is subject to. Such a displacement ${\hat{u}}^{ε}$ is firstly required to satisfy the following non-penetrability condition on the “lower” face ${\hat{Γ}}_{-}^{ε}$ of the reference configuration: $\begin{matrix} (O {\hat{x}}_{-}^{ε} + {\hat{u}}^{ε} ({\hat{x}}_{-}^{ε})) \cdot e^{3} ⩾ - ε, for all {\hat{x}}_{-}^{ε} \in {\hat{Γ}}_{-}^{ε} . \end{matrix}$

Since $e^{3} = (0, 0, 1)$ , the latter equivalently reads: $\begin{matrix} ({\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{α}}^{ε}}}) ({\hat{x}}_{-}^{ε}) + {\hat{u}}_{3}^{ε} ({\hat{x}}_{-}^{ε}) ⩾ - ε, for all {\hat{x}}_{-}^{ε} \in {\hat{Γ}}_{-}^{ε} . \end{matrix}$

In view of the non-penetrability condition, it is licit to assume that $\begin{matrix} (2) & θ^{ε} > 0 in \overline{ω}, \end{matrix}$ which, by virtue of the criterion defining a shallow shell, is equivalent to saying that $θ > 0$ in $\overline{ω}$ . The latter inequality (2) will play a fundamental role for defining the soundness of the two-dimensional limit model obtained as a result of the rigorous asymptotic analysis carried out in [16], which we introduce next.

We denote by $\hat{A}$ the fourth order three-dimensional elasticity tensor associated with the reference configuration in Cartesian coordinates. Such a tensor components take the following classical form (cf., e.g., [3] for the definition and additional properties) $\begin{matrix} {\hat{A}}^{i j k ℓ} : = λ δ^{i j} δ^{k ℓ} + μ (δ^{i k} δ^{j ℓ} + δ^{i ℓ} δ^{j k}) . \end{matrix}$

The linearised strain tensor associated with the displacement ${\hat{u}}^{ε}$ is defined by $\begin{matrix} {\hat{e}}^{ε} ({\hat{u}}^{ε}) = ({\hat{e}}_{i j}^{ε} ({\hat{u}}^{ε})) : = \frac{1}{2} (\nabla {\hat{u}}^{ε} + {[\nabla {\hat{u}}^{ε}]}^{T}) . \end{matrix}$

The second order linearised stress tensor associated with the reference configuration (cf., e.g., [3]) is denoted by ${\hat{σ}}^{ε}$ and is defined by $\begin{matrix} {\hat{σ}}^{ε} : = \hat{A} {\hat{e}}^{ε} . \end{matrix}$

The uni-laterality is described by adding the two following constraints, associated with the absence of friction, to the non-penetrability condition. The first constraint is that the no tensile forces, but only compressive forces, are exerted on the “lower” face of the shell by the obstacle. In formulas, we have $\begin{matrix} - ({\hat{σ}}^{ε} ({\hat{x}}_{-}^{ε}) \cdot a_{3}^{ε}) \cdot e^{3} \equiv {\hat{σ}}_{3}^{ε} ({\hat{x}}_{-}^{ε}) ⩾ 0, for all {\hat{x}}_{-}^{ε} \in {\hat{Γ}}_{-}^{ε} . \end{matrix}$

The second constraint is that either a point on the “lower” face is not in contact with the obstacle – so that the non-penetrability condition is strictly satisfied and the reaction of the obstacle at this point vanishes – or a point on the “lower” face is in contact with the obstacle and the non-penetrability condition is equal to zero, although the obstacle reaction is non-zero. In formulas, we have $\begin{matrix} {\hat{σ}}_{3}^{ε} ({\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{α}}^{ε}}} + {\hat{u}}_{3}^{ε} + ε) = 0, on {\hat{Γ}}_{-}^{ε} . \end{matrix}$

The prototypical physical example of the obstacle problem we are considering in this paper is the one of an inflated membrane placed close to a wall.

We are thus in a position to write, at least formally, the three-dimensional (in the sense that it is posed over a non-zero volume subset of the three-dimensional Euclidean space $E^{3}$ ) boundary value problem describing the equilibrium of a shallow shell subject to a planar horizontal obstacle at the level $- ε$ .

Problem $P ({\hat{Ω}}^{ε})$ .
Find ${\hat{u}}^{ε}$ satisfying: $\begin{array}{l} - div {\hat{σ}}^{ε} = {\hat{f}}^{ε}, in {\hat{Ω}}^{ε}, \\ {\hat{u}}^{ε} = 0, on {\hat{Γ}}_{0}^{ε}, \\ {\hat{σ}}^{ε} \cdot a_{3}^{ε} = {\hat{g}}^{+, ε}, on {\hat{Γ}}_{+}^{ε}, \\ {\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{α}}^{ε}}} + {\hat{u}}_{3}^{ε} + ε ⩾ 0, on {\hat{Γ}}_{-}^{ε}, \\ {\hat{σ}}_{3}^{ε} ⩾ 0, on {\hat{Γ}}_{-}^{ε}, \\ {\hat{σ}}_{3}^{ε} ({\hat{θ}}^{ε} - \frac{ε}{\sqrt{{\hat{α}}^{ε}}} + {\hat{u}}_{3}^{ε} + ε) = 0, on {\hat{Γ}}_{-}^{ε} . \end{array}$ ■

The natural functional framework for the equilibrium Problem $P ({\hat{Ω}}^{ε})$ is the non-empty, closed, and convex cone: $\begin{matrix} {\hat{K}}^{ε} ({\hat{Ω}}^{ε}) : = {{\hat{v}}^{ε} = ({\hat{v}}_{i}^{ε}) \in H^{1} ({\hat{Ω}}^{ε}); {\hat{v}}^{ε} = 0 on Γ_{0}^{ε} and {\hat{v}}_{3}^{ε} ⩾ - {\hat{θ}}^{ε} - ε + \frac{ε}{\sqrt{{\hat{α}}^{ε}}} on {\hat{Γ}}_{-}^{ε}} . \end{matrix}$

As a result, the boundary value Problem $P ({\hat{Ω}}^{ε})$ can be rigorously expressed in terms of a set of variational inequalities posed over the cone ${\hat{K}}^{ε} ({\hat{Ω}}^{ε})$ .
Problem $P ({\hat{Ω}}^{ε})$ .
Find ${\hat{u}}^{ε} \in {\hat{K}}^{ε} ({\hat{Ω}}^{ε})$ satisfying the following variational inequalities: $\begin{matrix} \int_{{\hat{Ω}}^{ε}} {\hat{A}}^{i j k ℓ} {\hat{e}}_{k ℓ}^{ε} ({\hat{u}}^{ε}) {\hat{e}}_{i j}^{ε} ({\hat{v}}^{ε} - {\hat{u}}^{ε}) d {\hat{x}}^{ε} ⩾ \int_{{\hat{Ω}}^{ε}} {\hat{f}}^{ε} \cdot ({\hat{v}}^{ε} - {\hat{u}}^{ε}) d {\hat{x}}^{ε} + \int_{{\hat{Γ}}_{+}^{ε}} {\hat{g}}^{+, ε} \cdot ({\hat{v}}^{ε} - {\hat{u}}^{ε}) d {\hat{Γ}}^{ε}, \end{matrix}$ for all ${\hat{v}}^{ε} \in {\hat{K}}^{ε} ({\hat{Ω}}^{ε})$ . ■

The function space over which the two-dimensional problem constituting the object of interest of this paper shall be posed is the following: $\begin{matrix} V (ω) : = {η = (η_{i}) \in H^{1} (ω) \times H^{1} (ω) \times H^{2} (ω); η_{i} = \partial_{ν} η_{3} = 0 on γ} . \end{matrix}$

We equip the space $V (ω)$ with the norm $‖ \cdot ‖_{V (ω)}$ defined as follows: $\begin{matrix} ‖ ξ ‖_{V (ω)} : = ‖ ξ_{1} ‖_{1, ω} + ‖ ξ_{2} ‖_{1, ω} + ‖ ξ_{3} ‖_{2, ω}, for all ξ \in V (ω) . \end{matrix}$

The corresponding semi-norm $| \cdot |_{V (ω)}$ is defined by: $\begin{matrix} | η |_{V (ω)} : = | η_{1} |_{1, ω} + | η_{2} |_{1, ω} + | η_{3} |_{2, ω}, for all η \in V (ω) . \end{matrix}$

We define the space associated with the tangential components by: $\begin{matrix} V_{H} (ω) : = {η_{H} = (η_{α}) \in H^{1} (ω) \times H^{1} (ω); η_{α} = 0 on γ} . \end{matrix}$

Observe that $\begin{matrix} V (ω) = V_{H} (ω) \times H_{0}^{2} (ω) . \end{matrix}$

More specifically, the transverse component of the displacement field belongs to the following non-empty, closed, and convex set of the space $H_{0}^{2} (ω)$ (see [16]): $\begin{matrix} (3) & K_{3} (ω) : = {η_{3} \in H_{0}^{2} (ω); θ + η_{3} ⩾ 0 almost everywhere in ω} . \end{matrix}$

By virtue of the Rellich–Kondrachov theorem (cf., e.g., Theorem 6.6-3 of [7]), the compact embedding $H^{2} (ω) ↪ ↪ C^{0} (\overline{ω})$ holds (the symbol “ $↪ ↪$ ” denotes a compact embedding and the space $C^{0} (\overline{ω})$ is equipped with the sup-norm). Hence, by virtue of the fact that $θ \in C^{3} (\overline{ω})$ , the set $K_{3} (ω)$ defined in (3) also takes the following form: $\begin{matrix} K_{3} (ω) = {η_{3} \in H_{0}^{2} (ω); θ + η_{3} ⩾ 0 in \overline{ω}} . \end{matrix}$

Moreover, by virtue of the assumption (2), we have that the function $η_{3} = 0$ belongs to $K_{3} (ω)$ . Define $f^{ε} \in L^{2} (Ω^{ε})$ by $f^{ε} : = {\hat{f}}^{ε} \circ Θ^{ε}$ , and define $g^{+, ε} \in L^{2} (Γ_{+}^{ε})$ by $g^{+, ε} : = {\hat{g}}^{+, ε} \circ Θ^{ε}$ . The vectors $f^{ε}$ and $g^{+, ε}$ , respectively corresponding to the applied forces ${\hat{f}}^{ε}$ and ${\hat{g}}^{+, ε}$ the reference configuration ${{\hat{Ω}}^{ε}}^{-}$ is subject to, enter the two-dimensional de-scaled model we shall introduce next.

We also assume that there exist functions $f_{i} \in L^{2} (Ω)$ and $g_{i}^{+} \in L^{2} (Γ_{+})$ independent of ε such that the following standard assumptions on the data hold (cf., e.g., [4]): $\begin{array}{l} f_{α}^{ε} (x^{ε}) = ε^{2} f_{α} (x) at each x = (x_{i}) \in Ω, \\ f_{3}^{ε} (x^{ε}) = ε^{3} f_{3} (x) at each x = (x_{i}) \in Ω, \\ g_{α}^{+, ε} (x^{ε}) = ε^{3} g_{α}^{+} (x) at each x = (x_{i}) \in Γ_{+}, \\ g_{3}^{+, ε} (x^{ε}) = ε^{4} g_{3}^{+} (x) at each x = (x_{i}) \in Γ_{+} . \end{array}$

We are now ready to state the scaled two-dimensional limit problem, obtained as a result of the rigorous asymptotic analysis carried out in [16] and [17], with Problem $P ({\hat{Ω}}^{ε})$ as a point of departure.
Problem $P (ω)$ .
Find $ζ = (ζ_{H}, ζ_{3}) \in V_{H} (ω) \times K_{3} (ω)$ satisfying the following variational inequalities: $\begin{matrix} - \int_{ω} m_{α β} (ζ_{3}) \partial_{α β} (η_{3} - ζ_{3}) d y + \int_{ω} n_{α β}^{θ} (ζ) (\partial_{α} θ) \partial_{β} (η_{3} - ζ_{3}) d y \\ ⩾ \int_{ω} p_{3} (η_{3} - ζ_{3}) d y - \int_{ω} s_{α} \partial_{α} (η_{3} - ζ_{3}) d y \\ for all η_{3} \in K_{3} (ω), \end{matrix}$ and the following variational equations: $\begin{matrix} \int_{ω} n_{α β}^{θ} (ζ) \partial_{β} η_{α} d y = \int_{ω} p_{α} η_{α} d y for all η_{H} = (η_{α}) \in V_{H} (ω), \end{matrix}$ where $\begin{array}{l} \{\begin{matrix} m_{α β} (ζ_{3}) : = - \frac{4 λ μ}{3 (λ + 2 μ)} Δ ζ_{3} δ_{α β} - \frac{4}{3} μ \partial_{α β} ζ_{3}, \\ n_{α β}^{θ} (ζ) : = \frac{4 λ μ}{λ + 2 μ} e_{σ σ}^{θ} (ζ) δ_{α β} + 4 μ e_{α β}^{θ} (ζ), \\ e_{α β}^{θ} (ζ) : = \frac{1}{2} (\partial_{α} ζ_{β} + \partial_{β} ζ_{α}) + \frac{1}{2} (\partial_{α} θ \partial_{β} ζ_{3} + \partial_{β} θ \partial_{α} ζ_{3}), \\ p_{i} : = \int_{- 1}^{1} f_{i} d x_{3} + g_{i}^{+}, \\ s_{α} : = \int_{- 1}^{1} x_{3} f_{α} d x_{3} + g_{α}^{+} . \end{matrix} \end{array}$ ■

The two-dimensional obstacle problem obtained in [16] and [17] is modelled by a set of variational equations and a set of variational inequalities and, besides, its solution is a Kirchhoff–Love field (see, for instance, Section 3.4 of [4]). As a result, we can “separate” the transverse component of the displacement vector field from the tangential components of the displacement vector field. Clearly, only the transverse component of the displacement is subject to the geometrical constraint associated with the obstacle.

Likewise, since $θ^{ε} \in C^{3} (\overline{ω})$ , we define the non-empty, closed, and convex set $K_{3}^{ε} (ω)$ by: $\begin{matrix} K_{3}^{ε} (ω) : = {η_{3} \in H_{0}^{2} (ω); θ^{ε} + η_{3} ⩾ 0 in \overline{ω}} . \end{matrix}$

The next step consists in de-scaling Problem $P (ω)$ . More specifically, the solution $ζ = (ζ_{H}, ζ_{3})$ of Problem $P (ω)$ is de-scaled as follows (cf. [4]): $\begin{array}{l} ζ_{H}^{ε} = ε^{2} ζ_{H} in ω, \\ ζ_{3}^{ε} = ε ζ_{3} in ω . \end{array}$

Thanks to (1), if $ζ_{3} \in K_{3} (ω)$ , then $ζ_{3}^{ε} \in K_{3}^{ε} (ω)$ . The following two-dimensional de-scaled problem can be thus derived, and constitutes the point of departure of our analysis concerning the augmentation of regularity.
Problem $P^{ε} (ω)$ .
$P^{ε} (ω)$ Find $ζ^{ε} = (ζ_{H}^{ε}, ζ_{3}^{ε}) \in V_{H} (ω) \times K_{3}^{ε} (ω)$ satisfying the following variational inequalities: $\begin{matrix} (4) & \begin{matrix} - \int_{ω} m_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (η_{3} - ζ_{3}^{ε}) d y + \int_{ω} n_{α β}^{θ, ε} (ζ^{ε}) (\partial_{α} θ^{ε}) \partial_{β} (η_{3} - ζ_{3}^{ε}) d y \\ ⩾ \int_{ω} p_{3}^{ε} (η_{3} - ζ_{3}^{ε}) d y - \int_{ω} s_{α}^{ε} \partial_{α} (η_{3} - ζ_{3}^{ε}) d y \\ for all η_{3} \in K_{3}^{ε} (ω), \end{matrix} \end{matrix}$ and the following variational equations: $\begin{matrix} (5) & \int_{ω} n_{α β}^{θ, ε} (ζ^{ε}) \partial_{β} η_{α} d y = \int_{ω} p_{α}^{ε} η_{α} d y for all η_{H} = (η_{α}) \in V_{H} (ω), \end{matrix}$ where $\begin{array}{l} \{\begin{matrix} m_{α β}^{ε} (ζ_{3}^{ε}) : = - ε^{3} {\frac{4 λ μ}{3 (λ + 2 μ)} Δ ζ_{3}^{ε} δ_{α β} + \frac{4}{3} μ \partial_{α β} ζ_{3}^{ε}}, \\ n_{α β}^{θ, ε} (ζ^{ε}) : = ε {\frac{4 λ μ}{λ + 2 μ} e_{σ σ}^{θ, ε} (ζ^{ε}) δ_{α β} + 4 μ e_{α β}^{θ, ε} (ζ^{ε})}, \\ e_{α β}^{θ, ε} (ζ^{ε}) : = \frac{1}{2} (\partial_{α} ζ_{β}^{ε} + \partial_{β} ζ_{α}^{ε}) + \frac{1}{2} (\partial_{α} θ^{ε} \partial_{β} ζ_{3}^{ε} + \partial_{β} θ^{ε} \partial_{α} ζ_{3}^{ε}), \\ p_{i}^{ε} : = \int_{- ε}^{ε} f_{i}^{ε} d x_{3}^{ε} + g_{i}^{+, ε}, \\ s_{α}^{ε} : = \int_{- ε}^{ε} x_{3}^{ε} f_{α}^{ε} d x_{3}^{ε} + ε g_{α}^{+, ε} . \end{matrix} \end{array}$ ■

Clearly, (4) and (5) can be combined into a single system of variational equations, whose left hand side is associated with the symmetric bilinear form $b (\cdot, \cdot)$ given by (cf. Sections 3.5, 3.6 and 3.7 of [4]) $\begin{matrix} b (ζ^{ε}, η) & = - \int_{ω} m_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} η_{3} d y + \int_{ω} n_{α β}^{θ, ε} (ζ^{ε}) (\partial_{α} θ^{ε}) \partial_{β} η_{3} d y \\ + \int_{ω} n_{α β}^{θ, ε} (ζ^{ε}) \partial_{β} η_{α} d y . \end{matrix}$

A straightforward computation shows that $\begin{matrix} b (η, η) & : = \int_{ω} \frac{4 λ μ}{λ + 2 μ} {\frac{ε^{3}}{3} {(Δ η_{3})}^{2} + ε {(e_{σ σ}^{θ, ε} (η))}^{2}} d y \\ + 4 μ {\frac{ε^{3}}{3} \sum_{α, β} ‖ \partial_{α β} η_{3} ‖_{0, ω}^{2} + ε \sum_{α, β} {‖ e_{α β}^{θ, ε} (η) ‖}_{0, ω}^{2}}, \end{matrix}$ for all $η \in V (ω)$ .

Likewise, we associate the sum of the right hand sides of (4) and (5) with a linear and continuous form ℓ defined as follows: $\begin{matrix} ℓ (η) : = \int_{ω} p_{i}^{ε} η_{i} d y - \int_{ω} s_{α}^{ε} \partial_{α} η_{3} d y for all η \in V (ω) . \end{matrix}$

The energy functional associated with the variational formulation in Problem $P^{ε} (ω)$ takes the following form: $\begin{matrix} J^{ε} (η) = \frac{1}{2} b (η, η) - ℓ (η) for all η \in V_{H} (ω) \times K_{3}^{ε} (ω) . \end{matrix}$

As a result, Problem $P^{ε} (ω)$ is equivalent to finding $ζ^{ε} = (ζ_{H}^{ε}, ζ_{3}^{ε}) \in V_{H} (ω) \times K_{3}^{ε} (ω)$ such that $\begin{matrix} J^{ε} (ζ^{ε}) = min {J^{ε} (η); η \in V_{H} (ω) \times K_{3}^{ε} (ω)} . \end{matrix}$

The bilinear form $b (\cdot, \cdot)$ is continuous, i.e., there exists a constant $M > 0$ such that: $\begin{matrix} b (ξ, η) ⩽ M ‖ ξ ‖_{V (ω)} ‖ η ‖_{V (ω)} for all ξ, η \in V (ω) . \end{matrix}$ By Theorem 3.6-1 of [4], such a bilinear form $b (\cdot, \cdot)$ is $V (ω)$ -elliptic, i.e., there exists a constant $α > 0$ such that $\begin{matrix} b (η, η) ⩾ α ‖ η ‖_{V (ω)}^{2} for all η \in V (ω) . \end{matrix}$

As a result, Problem $P^{ε} (ω)$ admits a unique solution $ζ^{ε} = (ζ_{H}^{ε}, ζ_{3}^{ε})$ which belongs to $V_{H} (ω) \times K_{3}^{ε} (ω)$ and satisfies: $\begin{matrix} b (ζ^{ε}, η - ζ^{ε}) ⩾ ℓ (η - ζ^{ε}) for all η = (η_{H}, η_{3}) \in V_{H} (ω) \times K_{3}^{ε} (ω), \end{matrix}$ or, equivalently, there exists a unique $ζ^{ε} = (ζ_{H}^{ε}, ζ_{3}^{ε}) \in V_{H} (ω) \times K_{3}^{ε} (ω)$ such that $\begin{matrix} J^{ε} (ζ^{ε}) = min {J^{ε} (η); η \in V_{H} (ω) \times K_{3}^{ε} (ω)} . \end{matrix}$

In what follows, we show that the solution $ζ^{ε}$ of Problem $P^{ε} (ω)$ enjoys higher regularity properties in the interior of the domain ω.
3. Improved interior regularity of the tangential components of the displacement field

Our first objective is to show that the tangential components of the solution $ζ^{ε}$ of Problem $P^{ε} (ω)$ , i.e., the components $ζ_{α}^{ε}$ of the vector field $ζ_{H}^{ε}$ , are of class $H_{loc}^{2} (ω) \cap H_{0}^{1} (ω)$ .

In order to establish such a property, we generalise a method developed by Nirenberg in the landmark paper [20] (see also [13]) by resorting to the ideas of [14].

Denote by $D_{ρ h}$ the first order finite difference quotient of either a function or a vector field in the canonical direction $e_{ρ}$ of $R^{2}$ and with increment size h sufficiently small. The first order finite difference quotient of a function ξ in the canonical direction $e_{ρ}$ of $R^{2}$ and with increment size h is defined by: $\begin{matrix} D_{ρ h} ξ (y) : = \frac{ξ (y + h e_{ρ}) - ξ (y)}{h}, \end{matrix}$ for all (or, possibly, a.a.) $y \in ω$ such that $(y + h e_{ρ}) \in ω$ .

The first order finite difference quotient of a vector field $ξ = (ξ_{α})$ in the canonical direction $e_{ρ}$ of $R^{2}$ and with increment size h is defined by $\begin{matrix} D_{ρ h} ξ (y) : = \frac{ξ (y + h e_{ρ}) - ξ (y)}{h}, \end{matrix}$ or, equivalently, $\begin{matrix} D_{ρ h} ξ (y) = (D_{ρ h} ξ_{α} (y)) . \end{matrix}$

We define the second order finite difference quotient of a function ξ in the canonical direction $e_{ρ}$ of $R^{2}$ and with increment size h by $\begin{matrix} δ_{ρ h} ξ (y) : = \frac{ξ (y + h e_{ρ}) - 2 ξ (y) + ξ (y - h e_{ρ})}{h^{2}}, \end{matrix}$ for all $y \in \overline{ω}$ such that $(y \pm h e_{ρ}) \in \overline{ω}$ .

The second order finite difference quotient of a vector field $ξ = (ξ_{α})$ in the canonical direction $e_{ρ}$ of $R^{2}$ and with increment size h is defined by $\begin{matrix} δ_{ρ h} ξ (y) : = (\frac{ξ_{α} (y + h e_{ρ}) - 2 ξ_{α} (y) + ξ_{α} (y - h e_{ρ})}{h^{2}}), \end{matrix}$ for all $y \in \overline{ω}$ such that $(y \pm h e_{ρ}) \in \overline{ω}$ .

Note in passing that the second order finite difference quotient of a function ξ can be expressed in terms of the first order finite difference quotient via the following identity: $\begin{matrix} (6) & δ_{ρ h} ξ = D_{- ρ h} D_{ρ h} ξ . \end{matrix}$

Similarly, the second order finite difference quotient of a vector field $ξ$ can be expressed in terms of the first order finite difference quotient via the following identity: $\begin{matrix} (7) & δ_{ρ h} ξ = D_{- ρ h} D_{ρ h} ξ . \end{matrix}$

Theorem 1.
Each of the tangential components $ζ_{α}^{ε}$ of the solution of Problem $P^{ε} (ω)$ is of class $H_{loc}^{2} (ω) \cap H_{0}^{1} (ω)$ .
Proof.
Let $ω_{0} \subset \subset ω$ and let $ω_{1} \subset ω$ be such that $\begin{matrix} ω_{1} \subset \subset ω_{0} \subset \subset ω . \end{matrix}$

Let $φ \in D (ω_{0})$ be such that $0 ⩽ φ ⩽ 1$ , $supp φ \subset \subset ω_{0}$ , and $φ \equiv 1$ in $ω_{1}$ and assume, without loss of generality, that $ω_{0}$ is a domain; otherwise we carry out the procedure on a polygon Q such that $supp φ \subset \subset Q \subset \subset ω_{0}$ , instead of the set $ω_{0}$ .

Let us consider the variational equations (5), which we re-state for clarity here below: $\begin{matrix} \int_{ω} n_{α β}^{θ, ε} (ζ^{ε}) \partial_{β} η_{α} d y = \int_{ω} p_{α}^{ε} η_{α} d y for all η_{H} = (η_{α}) \in V_{H} (ω) . \end{matrix}$

Let us observe that the tensor $(n_{α β}^{θ, ε})$ is symmetric, i.e., $\begin{matrix} (8) & n_{α β}^{θ, ε} (ξ) = n_{β α}^{θ, ε} (ξ) for all ξ = (ξ_{H}, ξ_{3}) \in V_{H} (ω) \times H_{0}^{2} (ω) . \end{matrix}$

In what follows, the components of the linearised strain tensor are denoted $e_{α β}$ and are defined by: $\begin{matrix} e_{α β} (η_{H}) : = \frac{1}{2} (\partial_{α} η_{β} + \partial_{β} η_{α}) for all η_{H} \in V_{H} (ω) . \end{matrix}$

Hence, the following identity holds $\begin{array}{l} n_{α β}^{θ, ε} (ξ) & = ε {\frac{4 λ μ}{λ + 2 μ} e_{σ σ} (ξ_{H}) δ_{α β} + 4 μ e_{α β} (ξ_{H}) \\ + \frac{2 λ μ}{λ + 2 μ} \partial_{σ} θ^{ε} \partial_{σ} ξ_{3} δ_{α β} + 2 μ (\partial_{α} θ^{ε} \partial_{β} ξ_{3} + \partial_{β} θ^{ε} \partial_{α} ξ_{3})}, \end{array}$ for all $ξ = (ξ_{H}, ξ_{3}) \in V_{H} (ω) \times H_{0}^{2} (ω)$ .

As a result of the symmetry (8) and the previous identity, the variational equations (5) take the following form $\begin{matrix} (9) & \begin{matrix} ε \int_{ω} {\frac{4 λ μ}{λ + 2 μ} e_{σ σ} (ζ_{H}^{ε}) δ_{α β} + 4 μ e_{α β} (ζ_{H}^{ε})} e_{α β} (η_{H}) d y \\ = \int_{ω} p_{α}^{ε} η_{α} d y - ε \frac{2 λ μ}{λ + 2 μ} \int_{ω} \partial_{σ} θ^{ε} \partial_{σ} ζ_{3}^{ε} δ_{α β} e_{α β} (η_{H}) d y \\ - 2 μ ε \int_{ω} (\partial_{α} θ^{ε} \partial_{β} ζ_{3}^{ε} + \partial_{β} θ^{ε} \partial_{α} ζ_{3}^{ε}) e_{α β} (η_{H}) d y, \end{matrix} \end{matrix}$ for all $η_{H} = (η_{α}) \in V_{H} (ω)$ .

Given two functions $f_{1}, f_{2} \in L^{2} (ω_{0})$ , we say that $f_{1} ≲_{H} f_{2}$ if there exists a constant $C > 0$ independent of h and eventually dependent on $‖ ζ_{H}^{ε} ‖_{1, ω}$ , φ and its derivatives such that: $\begin{matrix} \int_{ω_{0}} f_{1} d y ⩽ \int_{ω_{0}} f_{2} d y + C (1 + {‖ D_{ρ h} (φ ζ_{H}^{ε}) ‖}_{1, ω_{0}}) . \end{matrix}$

The relation $≲_{H}$ is clearly reflexive and transitive. Specialise the test function by taking $η_{H} \in V_{H} (ω)$ as the extension by zero outside $ω_{0}$ of the vector field $(- φ^{2} δ_{ρ h} ζ_{H}^{ε})$ . Thanks to the results contained in [19] and the fact that $supp φ \subset \subset ω_{0}$ , we infer that such a function satisfies all the regularity requirements. Testing such a function in the variational equations (9) gives: $\begin{matrix} (10) & \begin{matrix} - ε \int_{ω_{0}} {\frac{4 λ μ}{λ + 2 μ} e_{σ σ} (ζ_{H}^{ε}) δ_{α β} + 4 μ e_{α β} (ζ_{H}^{ε})} e_{α β} (φ^{2} δ_{ρ h} ζ_{H}^{ε}) d y \\ = - \int_{ω_{0}} p_{α}^{ε} (φ^{2} δ_{ρ h} ζ_{α}^{ε}) d y \\ + ε \frac{2 λ μ}{λ + 2 μ} \int_{ω_{0}} \partial_{σ} θ^{ε} \partial_{σ} ζ_{3}^{ε} δ_{α β} e_{α β} (φ^{2} δ_{ρ h} ζ_{H}^{ε}) d y \\ + 2 μ ε \int_{ω_{0}} (\partial_{α} θ^{ε} \partial_{β} ζ_{3}^{ε} + \partial_{β} θ^{ε} \partial_{α} ζ_{3}^{ε}) e_{α β} (φ^{2} δ_{ρ h} ζ_{H}^{ε}) d y . \end{matrix} \end{matrix}$

To begin with, we estimate the right hand side of (10). Since $θ^{ε} \in C^{3} (\overline{ω})$ , $supp φ \subset \subset ω_{0}$ , and $ζ_{3}^{ε} \in H_{0}^{2} (ω)$ , an application of Green’s formula, Cauchy–Schwarz inequality and Theorem 5.3(i) in Section 5.8.2 of [13] gives: $\begin{array}{l} ε \frac{2 λ μ}{λ + 2 μ} \int_{ω_{0}} \partial_{σ} θ^{ε} \partial_{σ} ζ_{3}^{ε} δ_{α β} e_{α β} (φ^{2} δ_{ρ h} ζ_{H}^{ε}) d y \\ + 2 μ ε \int_{ω_{0}} (\partial_{α} θ^{ε} \partial_{β} ζ_{3}^{ε} + \partial_{β} θ^{ε} \partial_{α} ζ_{3}^{ε}) e_{α β} (φ^{2} δ_{ρ h} ζ_{H}^{ε}) d y \\ = - ε \frac{λ μ}{λ + 2 μ} \int_{ω_{0}} \partial_{β} (\partial_{σ} θ^{ε} \partial_{σ} ζ_{3}^{ε} δ_{α β}) φ^{2} δ_{ρ h} ζ_{α}^{ε} d y \\ - μ ε \int_{ω_{0}} \partial_{β} (\partial_{α} θ^{ε} \partial_{β} ζ_{3}^{ε} + \partial_{β} θ^{ε} \partial_{α} ζ_{3}^{ε}) φ^{2} δ_{ρ h} ζ_{α}^{ε} d y \\ - ε \frac{λ μ}{λ + 2 μ} \int_{ω_{0}} \partial_{α} (\partial_{σ} θ^{ε} \partial_{σ} ζ_{3}^{ε} δ_{α β}) φ^{2} δ_{ρ h} ζ_{β}^{ε} d y \\ - μ ε \int_{ω_{0}} \partial_{α} (\partial_{α} θ^{ε} \partial_{β} ζ_{3}^{ε} + \partial_{β} θ^{ε} \partial_{α} ζ_{3}^{ε}) φ^{2} δ_{ρ h} ζ_{β}^{ε} d y \\ ⩽ C {‖ D_{ρ h} (φ ζ_{H}^{ε}) ‖}_{1, ω_{0}} . \end{array}$ In conclusion, we have that the right hand side of (10) is $≲_{H} 0$ .

Let us now proceed to the study of the left hand side of (10). Observe that, since $\partial_{α} (φ δ_{ρ h} (φ ζ_{H}^{ε})) ≲_{H} \partial_{α} (φ^{2} δ_{ρ h} ζ_{H}^{ε})$ , and since $\partial_{α} (D_{ρ h} (φ ζ_{H}^{ε})) = D_{ρ h} (\partial_{α} (φ ζ_{H}^{ε}))$ (cf., e.g., [14]), the following chain of relations holds: $\begin{array}{l} 4 μ e_{α β} (D_{ρ h} (φ ζ_{H}^{ε})) e_{α β} (D_{ρ h} (φ ζ_{H}^{ε})) \\ ≲_{H} - \frac{4 λ μ}{λ + 2 μ} e_{σ σ} (ζ_{H}^{ε}) e_{α α} (φ δ_{ρ h} (φ ζ_{H}^{ε})) - 4 μ e_{α β} (ζ_{H}^{ε}) e_{α β} (φ δ_{ρ h} (φ ζ_{H}^{ε})) \\ ≲_{H} - \frac{4 λ μ}{λ + 2 μ} e_{σ σ} (ζ_{H}^{ε}) e_{α α} (φ^{2} δ_{ρ h} ζ_{H}^{ε}) - 4 μ e_{α β} (ζ_{H}^{ε}) e_{α β} (φ^{2} δ_{ρ h} ζ_{H}^{ε}) . \end{array}$

To summarise, we have: $\begin{matrix} 4 μ e_{α β} (D_{ρ h} (φ ζ_{H}^{ε})) e_{α β} (D_{ρ h} (φ ζ_{H}^{ε})) ≲_{H} 0 . \end{matrix}$

An application of Korn’s inequality with boundary conditions (cf., e.g., Theorem 6.15-4 of [7]) gives: $\begin{matrix} (11) & 4 μ ε \int_{ω_{0}} e_{α β} (D_{ρ h} (φ ζ_{H}^{ε})) e_{α β} (D_{ρ h} (φ ζ_{H}^{ε})) d y ⩾ C {‖ D_{ρ h} (φ ζ_{H}^{ε}) ‖}_{0, ω_{0}}^{2} . \end{matrix}$

A combination of (11) and the fact that the right hand side of (10) is $≲_{H} 0$ gives $\begin{matrix} {‖ D_{ρ h} (φ ζ_{H}^{ε}) ‖}_{1, ω_{0}}^{2} ⩽ C (1 + {‖ D_{ρ h} (φ ζ_{H}^{ε}) ‖}_{1, ω_{0}}), \end{matrix}$ which in turn implies that $\begin{matrix} {‖ D_{ρ h} ζ_{H}^{ε} ‖}_{1, ω_{1}} is uniformly bounded with respect to h . \end{matrix}$

An application of Theorem 3 of Section 5.8.2 of [13] shows that each $ζ_{α}^{ε}$ is also in $H_{loc}^{2} (ω)$ , as it was to be proved. □

4. Improved interior regularity of the transverse component of the displacement field

Our second objective is to show that the transverse component of the solution, i.e., the function $ζ_{3}^{ε}$ , is of class $H_{loc}^{3} (ω) \cap H_{0}^{2} (ω)$ . The arising difficulty is owing to the fact that $ζ_{3}^{ε}$ is required to satisfy a constraint in $\overline{ω}$ . The improved regularity for such a transverse component is obtained by generalising the method developed by Frehse in the seminal paper [14].

We first extend the results contained in two preliminary lemmas to the problem under consideration. Let $ω_{0} \subset ω$ and $ω_{1} \subset ω$ be such that $\begin{matrix} ω_{1} \subset \subset ω_{0} \subset \subset ω . \end{matrix}$

Following the ideas of [14], we assume that there exists a non-zero real number κ and a canonical direction $e_{ρ}$ of $R^{2}$ such that the non-negative quantity $\begin{matrix} d : = inf {| x - y |; x \in \partial ω_{0}, y \in \partial ω_{1} and x - y = κ e_{ρ}} \end{matrix}$ is strictly greater than zero. Let $φ_{1} \in D (ω)$ be such that $\begin{matrix} supp φ_{1} \subset \subset ω_{1} and 0 ⩽ φ_{1} ⩽ 1 . \end{matrix}$

Before proving the main result of this section, we establish two abstract preparatory lemmas.

Lemma 1.

Assume that $ψ \in C^{0} (\overline{ω})$ is concave on $ω_{0}$ . Let $η \in H^{2} (ω)$ be such that $η + ψ ⩾ 0$ in $\overline{ω}$ . Then, for each $0 < h < d$ and all $0 < ϱ < h^{2} / 2$ , the function $\begin{matrix} η_{ϱ} : = η + ϱ φ_{1} δ_{ρ h} η, \end{matrix}$ is such that $η_{ϱ} \in H^{2} (ω_{1})$ and $η_{ϱ} + ψ ⩾ 0$ in $\overline{ω_{1}}$ .

Proof.

The regularity immediately follows by the fact that $η \in H^{2} (ω)$ and $φ_{1}$ is smooth (cf., e.g., [19]). For all $y \in \overline{ω}$ such that $(y \pm h e_{ρ}) \in \overline{ω}$ we have that $\begin{array}{l} η_{ϱ} (y) & = (η + ϱ φ_{1} δ_{ρ h} η) (y) \\ = η (y) + ϱ φ_{1} (y) \frac{η (y + h e_{ρ}) - 2 η (y) + η (y - h e_{ρ})}{h^{2}} \\ = \frac{ϱ}{h^{2}} φ_{1} (y) η (y + h e_{ρ}) + (1 - \frac{2 ϱ}{h^{2}} φ_{1} (y)) η (y) + \frac{ϱ}{h^{2}} φ_{1} (y) η (y - h e_{ρ}) . \end{array}$

By virtue of the properties of $φ_{1}$ and ϱ, the functions $c_{1} (y) = c_{- 1} (y) : = ϱ h^{- 2} φ_{1} (y)$ and $c_{0} (y) : = 1 - 2 ϱ h^{- 2} φ_{1} (y)$ are non-negative in $\overline{ω}$ . Combining these properties with fact that $η + ψ ⩾ 0$ in $\overline{ω}$ gives: $\begin{array}{l} η_{ϱ} (y) & ⩾ - c_{1} ψ (y + h e_{ρ}) - c_{0} ψ (y) - c_{- 1} ψ (y - h e_{ρ}) \\ = - (ψ (y) + ϱ φ_{1} (y) \frac{ψ (y + h e_{ρ}) - 2 ψ (y) + ψ (y - h e_{ρ})}{h^{2}}), \end{array}$ for all $y \in \overline{ω_{1}}$ .

The assumed concavity of ψ in $ω_{0}$ gives $\begin{matrix} δ_{ρ h} ψ = \frac{ψ (y + h e_{ρ}) - 2 ψ (y) + ψ (y - h e_{ρ})}{h^{2}} < 0, \end{matrix}$ for all $0 < h < d$ and all $y \in \overline{ω_{1}}$ . As a result, we derive $η_{ϱ} + ψ ⩾ 0$ in $\overline{ω_{1}}$ , as it was to be proved. □

For treating the case where the concavity assumption fails, we need the following lemma.

Lemma 2.

Let the function ψ be of class $C^{2} (\overline{ω})$ and such that $ψ > 0$ in $\overline{ω}$ . Then, for every $y_{0} = (y_{0, 1}, y_{0, 2}) \in ω$ , there exists a neighbourhood $U$ of $y_{0}$ and numbers $a \in R$ , $a_{0}, r > 0$ such that the function $(- ψ + a) g$ is convex in $U$ , where: $\begin{matrix} g (y) : = 1 - \frac{1}{2} \prod_{ρ = 1}^{2} exp (r y_{ρ} - r y_{0, ρ}), for all y = (y_{1}, y_{2}) \in \overline{ω} . \end{matrix}$

Moreover, it results $g (y) ⩾ a_{0}$ , for all $y \in U$ .

Proof.

Fix $y_{0} \in ω$ . Owing to the fact that $ψ \in C^{2} (\overline{ω})$ and $ψ > 0$ in $\overline{ω}$ , we can find numbers $a \in R$ and $τ > 0$ , and a neighbourhood $U_{0}$ of $y_{0}$ such that $\begin{matrix} - ψ (y) + a ⩽ - τ < 0 for all y \in U_{0} . \end{matrix}$

For all $y \in \overline{ω}$ , define the function: $\begin{matrix} Π (y) : = \prod_{ρ = 1}^{2} exp (r y_{ρ} - r y_{0, ρ}) . \end{matrix}$

For a given $α \in {1, 2}$ , we have: $\begin{array}{l} \frac{1}{Π} \partial_{α α} [(- ψ + a) g] & = \frac{1}{Π} \partial_{α} (- g \partial_{α} ψ + (- ψ + a) \partial_{α} g) \\ = \frac{1}{Π} \partial_{α} (- (1 - \frac{Π}{2}) \partial_{α} ψ - \frac{r}{2} (- ψ + a) Π) \\ = \frac{1}{Π} [- (1 - \frac{Π}{2}) \partial_{α α} ψ + r Π \partial_{α} ψ - \frac{r^{2}}{2} Π (- ψ + a)] \\ = - (\frac{1}{Π} - \frac{1}{2}) \partial_{α α} ψ + r \partial_{α} ψ - \frac{r^{2}}{2} (- ψ + a) . \end{array}$

Thanks to the global uniform boundedness of the first and second derivative of ψ in $\overline{ω}$ , and the properties of the numbers a and τ, we derive that there exists a positive number M such that: $\begin{matrix} \frac{1}{Π} \partial_{α α} ((- ψ + a) g) ⩾ - M | \frac{1}{Π} - \frac{1}{2} | - r M + \frac{r^{2}}{2} τ . \end{matrix}$

We obtain that, for r sufficiently large, we have that $\begin{matrix} p_{1} (r) : = \frac{τ}{2} r^{2} - M r - M > 0 . \end{matrix}$

Let us choose a neighbourhood $U$ of $y_{0}$ such that $U \subset \subset U_{0}$ and $\begin{matrix} | y_{ρ} - y_{0, ρ} | ⩽ \frac{1}{2 r} log \frac{3}{2}, for all y \in U and all ρ \in {1, 2} . \end{matrix}$

Then, we have $\begin{matrix} 0 < Π (y) ⩽ \frac{3}{2}, for all y \in U, \end{matrix}$ and $\begin{matrix} | \frac{1}{Π (y)} - \frac{1}{2} | ⩽ 1, for all y \in U . \end{matrix}$

As a result, we have $\begin{matrix} \frac{1}{Π} \partial_{α α} ((- ψ + a) g) ⩾ - M | \frac{1}{Π} - \frac{1}{2} | - M r + \frac{τ}{2} r^{2} ⩾ p_{1} (r) ⩾ 0, \end{matrix}$ for all $y \in U$ , and also $g (y) ⩾ 1 - Π / 2 ⩾ 1 / 4$ , for all $y \in U$ . Letting $a_{0} : = 1 / 4$ completes the proof. □

We are now ready to prove the main result of this section.

Theorem 2.

The transverse component $ζ_{3}^{ε}$ of the solution of Problem $P^{ε} (ω)$ is of class $H_{loc}^{3} (ω) \cap H_{0}^{2} (ω)$ .

Proof.

Let $y_{0} \in ω$ and let $U$ be the neighbourhood of $y_{0}$ constructed in Lemma 2 in correspondence of $θ^{ε}$ . It results that the function $\begin{matrix} \tilde{θ^{ε}} : = - (- θ^{ε} + a) g, \end{matrix}$ is concave in $U$ . Let $φ \in D (U)$ such that $supp φ \subset \subset U_{1} \subset \subset U$ , for some neighbourhood $U_{1}$ of $y_{0}$ , such that $0 ⩽ φ ⩽ 1$ , and such that $φ \equiv 1$ in a compact strict subset of its support. Without loss of generality, we can assume that $U_{1}$ is a domain; otherwise, we take a polygon Q such that $supp φ \subset \subset Q \subset \subset U_{1}$ instead of $U_{1}$ . Since, by Lemma 2, we have $g ⩾ a_{0} > 0$ in $U$ and since $ζ_{3}^{ε} \in K_{3}^{ε} (ω)$ , it results $\begin{matrix} - \tilde{θ^{ε}} ⩽ (ζ_{3}^{ε} + a) g in \overline{ω} . \end{matrix}$

Thanks to the concavity of $\tilde{θ^{ε}}$ in $U$ , we are in a position to apply Lemma 1 and obtain $\begin{matrix} (12) & - \tilde{θ^{ε}} ⩽ (ζ_{3}^{ε} + a) g + ϱ φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g] in U_{1}, \end{matrix}$ for sufficiently small h and ϱ. Dividing (12) by g and then subtracting a from each member of (12) gives $\begin{matrix} - θ^{ε} ⩽ ζ_{3}^{ε} + ϱ g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g] = : {\tilde{ζ}}_{3, ϱ}^{ε} in U_{1} . \end{matrix}$

Since we have that ${\tilde{ζ}}_{3, ϱ}^{ε} = ζ_{3}^{ε}$ in $U_{1} ∖ supp φ$ , we are in a position to extend ${\tilde{ζ}}_{3, ϱ}^{ε}$ by $ζ_{3}^{ε}$ outside $U_{1}$ ; we denote this extension by $\tilde{{\tilde{ζ}}_{3, ϱ}^{ε}}$ . Since ${\tilde{ζ}}_{3, ϱ}^{ε} \in H^{2} (U_{1})$ and satisfies the geometrical constraint, we can infer that $\tilde{{\tilde{ζ}}_{3, ϱ}^{ε}} \in K_{3}^{ε} (ω)$ .

Specialising $η_{3} = \tilde{{\tilde{ζ}}_{3, ϱ}^{ε}}$ in (4) gives: $\begin{matrix} (13) & \begin{matrix} \int_{U_{1}} m_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) d y \\ - \int_{U_{1}} n_{α β}^{θ, ε} (ζ^{ε}) (\partial_{α} θ^{ε}) \partial_{β} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) d y \\ ⩽ - \int_{U_{1}} p_{3}^{ε} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) d y \\ + \int_{U_{1}} s_{α}^{ε} \partial_{α} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) d y . \end{matrix} \end{matrix}$

Let us define the translation operator E in the canonical direction $e_{ρ}$ of $R^{2}$ and with increment size h for a smooth enough function $v : U_{1} \to R$ by $\begin{array}{l} E_{ρ h} v (y) : = v (y + h e_{ρ}), \\ E_{- ρ h} v (y) : = v (y - h e_{ρ}) . \end{array}$

Moreover, the following identities can be easily checked out: $\begin{array}{l} (14) & D_{ρ h} (v w) = (E_{ρ h} w) (D_{ρ h} v) + v D_{ρ h} w, \\ (15) & D_{- ρ h} (v w) = (E_{- ρ h} w) (D_{- ρ h} v) + v D_{- ρ h} w, \\ (16) & δ_{ρ h} (v w) = w δ_{ρ h} v + (D_{ρ h} w) (D_{ρ h} v) + (D_{- ρ h} w) (D_{- ρ h} v) + v δ_{ρ h} w . \end{array}$

Given two functions $f_{1}, f_{2} \in L^{2} (U_{1})$ , we say that $f_{1} ≲_{3} f_{2}$ if there exists a constant $C > 0$ independent of h and eventually dependent on g, $g^{- 1}$ , $‖ ζ_{3}^{ε} ‖_{2, ω}$ , φ and its derivatives such that: $\begin{matrix} \int_{U_{1}} f_{1} d y ⩽ \int_{U_{1}} f_{2} d y + C (1 + {‖ D_{ρ h} (φ ζ_{3}^{ε}) ‖}_{2, U_{1}}) . \end{matrix}$

By Lemma 2 we know that the derivatives of g and $g^{- 1}$ are uniformly bounded in $U$ . As a result, we have that (13) implies: $\begin{matrix} (17) & \begin{matrix} m_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) \\ ≲_{3} n_{α β}^{θ, ε} (ζ^{ε}) (\partial_{α} θ^{ε}) \partial_{β} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) \\ - p_{3}^{ε} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) \\ + s_{α}^{ε} \partial_{α} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) . \end{matrix} \end{matrix}$

An application of (16) and Hölder’s inequality gives that there exists a constant $C > 0$ such that: $\begin{matrix} (18) & \int_{U_{1}} p_{3}^{ε} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) d y ⩽ C {‖ p_{3}^{ε} ‖}_{0, ω} {‖ D_{- ρ h} ζ_{3}^{ε} ‖}_{1, U_{1}} . \end{matrix}$

An application of (16) and Hölder’s inequality, (6), and the inequality (cf., formula (4.11) of [14]) $\begin{matrix} (19) & {‖ \partial_{α} (δ_{ρ h} (ζ_{3}^{ε} φ)) ‖}_{0, U_{1}} ⩽ C {‖ D_{ρ h} (ζ_{3}^{ε} φ) ‖}_{2, U_{1}}, \end{matrix}$ gives that there exists a constant $C > 0$ such that: $\begin{matrix} (20) & \begin{matrix} \int_{U_{1}} s_{α}^{ε} \partial_{α} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) d y \\ ⩽ C {‖ s_{α}^{ε} ‖}_{0, ω} {‖ ζ_{3}^{ε} ‖}_{2, ω} + | \int_{U_{1}} s_{α}^{ε} g^{- 1} φ \partial_{α} (δ_{ρ h} [(ζ_{3}^{ε} + a) g]) d y | \\ ⩽ C {‖ s_{α}^{ε} ‖}_{0, ω} ({‖ ζ_{3}^{ε} ‖}_{2, ω} + {‖ D_{ρ h} ζ_{3}^{ε} ‖}_{2, U_{1}}) . \end{matrix} \end{matrix}$

Likewise, we infer that $\begin{matrix} (21) & \int_{U_{1}} n_{α β}^{θ, ε} (ζ^{ε}) (\partial_{α} θ^{ε}) \partial_{β} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) d y ⩽ C {‖ D_{ρ h} ζ_{3}^{ε} ‖}_{2, U_{1}}, \end{matrix}$ for some $C > 0$ .

By virtue of (18)–(21), we deduce that the right hand side of (17) satisfies: $\begin{matrix} (22) & \begin{matrix} n_{α β}^{θ, ε} (ζ^{ε}) (\partial_{α} θ^{ε}) \partial_{β} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) \\ - p_{3}^{ε} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) \\ + s_{α}^{ε} \partial_{α} (g^{- 1} φ^{2} δ_{ρ h} [(ζ_{3}^{ε} + a) g]) ≲_{3} 0 . \end{matrix} \end{matrix}$

Using (16) in the left hand side of (17) gives $\begin{array}{l} m_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (g^{- 1} φ^{2} [g δ_{ρ h} ζ_{3}^{ε} + (D_{ρ h} g) (D_{ρ h} ζ_{3}^{ε}) + (D_{- ρ h} g) (D_{- ρ h} ζ_{3}^{ε})]) \\ ≲_{3} m_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (g^{- 1} φ^{2} [g δ_{ρ h} ζ_{3}^{ε} + (D_{ρ h} g) (D_{ρ h} ζ_{3}^{ε}) + (D_{- ρ h} g) (D_{- ρ h} ζ_{3}^{ε}) + ζ_{3}^{ε} δ_{ρ h} g]) \\ = m_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (φ^{2} [δ_{ρ h} ζ_{3}^{ε} + g^{- 1} (D_{ρ h} g) (D_{ρ h} ζ_{3}^{ε}) + g^{- 1} (D_{- ρ h} g) (D_{- ρ h} ζ_{3}^{ε}) + g^{- 1} ζ_{3}^{ε} δ_{ρ h} g]) \\ ≲_{3} m_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (φ^{2} g^{- 1} δ_{ρ h} [(ζ_{3}^{ε} + a) g]), \end{array}$ where the latter term corresponds to the left hand side of (17). Let us now use (14)–(16) for computing $δ_{ρ h} (ζ_{3}^{ε} φ)$ . Define $A : = - φ (D_{ρ h} φ) (D_{ρ h} ζ_{3}^{ε}) - φ (D_{- ρ h} φ) (D_{- ρ h} ζ_{3}^{ε})$ and observe that (16) gives $\begin{matrix} φ δ_{ρ h} (ζ_{3}^{ε} φ) + A = φ [φ δ_{ρ h} ζ_{3}^{ε} + ζ_{3}^{ε} δ_{ρ h} φ] . \end{matrix}$

We can then deduce that $\begin{matrix} (23) & \begin{matrix} m_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (φ δ_{ρ h} (ζ_{3}^{ε} φ) + A) \\ ≲_{3} m_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (φ^{2} (δ_{ρ h} ζ_{3}^{ε})) \\ + \partial_{α β} (g^{- 1} (D_{- ρ h} ζ_{3}^{ε}) (D_{- ρ h} g) + g^{- 1} (D_{ρ h} ζ_{3}^{ε}) (D_{ρ h} g) + ζ_{3}^{ε} δ_{ρ h} g) . \end{matrix} \end{matrix}$

Combining (17), (22), (23), and the fact that the terms appearing in A have order of differentiation less or equal than one, gives: $\begin{matrix} m_{α β}^{ε} (ζ_{3}^{ε}) \partial_{α β} (φ δ_{ρ h} (ζ_{3}^{ε} φ)) ≲_{3} 0 . \end{matrix}$

Exploiting the estimate (19), we can interchange the symbols φ and $\partial_{α β}$ , thus getting: $\begin{matrix} (24) & m_{α β}^{ε} (ζ_{3}^{ε}) φ δ_{ρ h} (\partial_{α β} (ζ_{3}^{ε} φ)) ≲_{3} 0 . \end{matrix}$

Writing the second order finite difference quotient $δ_{ρ h}$ appearing in (24) in the form (6) and recalling the properties of $supp φ$ , we are then in a position to apply the integration-by-parts formula for the first order finite difference quotient $D_{- ρ h}$ (cf., e.g., equation (16) in Section 6.3.1 of [13]) together with the same computations that led to (11), thus getting: $\begin{matrix} - m_{α β}^{ε} (D_{ρ h} (φ ζ_{3}^{ε})) \partial_{α β} (D_{ρ h} (φ ζ_{3}^{ε})) ≲_{3} m_{α β}^{ε} (ζ_{3}^{ε}) φ δ_{ρ h} (\partial_{α β} (ζ_{3}^{ε} φ)) . \end{matrix}$

In conclusion, the latter and (24) give: $\begin{matrix} - m_{α β}^{ε} (D_{ρ h} (φ ζ_{3}^{ε})) \partial_{α β} (D_{ρ h} (φ ζ_{3}^{ε})) ≲_{3} 0 . \end{matrix}$

An application of (19) and Theorem 6.8-4 of [7] gives: $\begin{array}{l} - \int_{U_{1}} m_{α β}^{ε} (D_{ρ h} (φ ζ_{3}^{ε})) \partial_{α β} (D_{ρ h} (φ ζ_{3}^{ε})) d y \\ = \frac{4 λ μ}{3 (λ + 2 μ)} ε^{3} \int_{U_{1}} Δ (D_{ρ h} (ζ_{3}^{ε} φ)) δ_{α β} \partial_{α β} (D_{ρ h} (ζ_{3}^{ε} φ)) d y \\ + \frac{4}{3} μ ε^{3} \int_{U_{1}} \partial_{α β} (D_{ρ h} (ζ_{3}^{ε} φ)) \partial_{α β} (D_{ρ h} (ζ_{3}^{ε} φ)) d y \\ = \frac{4 λ μ}{3 (λ + 2 μ)} ε^{3} \int_{U_{1}} Δ (D_{ρ h} (ζ_{3}^{ε} φ)) Δ (D_{ρ h} (ζ_{3}^{ε} φ)) d y \\ + \frac{4}{3} μ ε^{3} \int_{U_{1}} \partial_{α β} (D_{ρ h} (ζ_{3}^{ε} φ)) \partial_{α β} (D_{ρ h} (ζ_{3}^{ε} φ)) d y \\ ⩾ C {‖ D_{ρ h} (ζ_{3}^{ε} φ) ‖}_{2, U_{1}}^{2} . \end{array}$

In conclusion, we have obtained that there exists a constant $C > 0$ for which: $\begin{matrix} {‖ D_{ρ h} (ζ_{3}^{ε} φ) ‖}_{2, U_{1}}^{2} ⩽ C (1 + {‖ D_{ρ h} (ζ_{3}^{ε} φ) ‖}_{2, U_{1}}) . \end{matrix}$

An application of Theorem 3 of Section 5.8.2 of [13] together with the fact that $φ \equiv 1$ in a compact strict subset of its support shows that $ζ_{3}^{ε} \in H_{loc}^{3} (ω)$ , as it was to be proved. □

As a conclusive remark, we observe that the results proved in Theorems 1 and 2 are independent.

Footnotes

Acknowledgements

The author is greatly indebted to Professor Philippe G. Ciarlet for his encouragement and guidance.

The author is greatly thankful to the anonymous referee for their suggested improvements.

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