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Higher order variational inequalities with non-standard growth conditions in dimension two: plates with obstacles. (English) Zbl 1014.49031
The authors consider the problem \[ \text{Minimize}\quad J(w):= \int_\Omega f(\nabla^2w) dx\quad\text{subject to}\quad w \geq \Psi \text{ on } \Omega\qquad\qquad\tag{1} \] among real-valued functions \(w\) with zero trace. Here \(\Omega\) is a bounded, star-shaped Lipschitz domain in \(\mathbf R^2\), and \(f\) is a smooth convex function of class \(C^2\), with range \([0,\infty ]\), and which satisfies a sub-quadratic growth condition together with technical conditions on \(f\), its gradient \(Df\), and its Hessian \(D^2 f\). A typical example would be the function \(f(s) = |s|\;\ln (1 + |s|)\). The obstacle function \(\Psi\) is assumed to be of class \(W^3_2 (\Omega)\), where \(W^m_p\) denotes the usual Sobolev space, with \(\left. \Psi \right|_{\partial \Omega} < 0\) and \(\max_{\overline{\Omega}} \Psi > 0\). The problem is of physical interest, in that \(J\) is the functional corresponding to the strain energy of a clamped plate occupying the region \(\Omega\), and the properties of \(f\) allow for modelling of plates having a range of inelastic constitutive laws. The main result is the following: given \(f\) that satisfies the growth and other technical conditions, the obstacle problem (1) admits a unique solution \(u\) that is of class \(W^2_{p,\text{loc}}(\Omega)\) for any finite \(p\). In particular, \(u \) belongs to the Hölder space \(C^{1,\alpha}\) for any \(\alpha < 1\). The key to the proof lies in overcoming the difficulty that there is no analogue to the density property of \(C^\infty_0(\Omega)\) in \(Z := \{ v \in W^2_p (\Omega)\;|\;v \geq \psi \}\) (see [M. Fuchs, G. Li and O. Martio, Ann. Acad. Sci. Fenn., Math. 23, No. 2, 549-558 (1998; Zbl 0910.49002)]). This difficulty is overcome by adopting a more elaborate approximation procedure involving \(J\) and \(\Psi\). In particular, first the problem is regularized by introducing a sequence \(\Psi^\varepsilon \in W^3_2(\Omega)\) of functions that coincide with \(\Psi\) in the neighbourhood of a disc \(D\) compactly embedded in \(\Omega\). The regularized problem uses \(\Psi^\varepsilon\) as the obstacle, and has a unique solution \(u^\varepsilon\) in the space \(K^\varepsilon := \{ v \in \overset{o} W^2_A(\Omega)\;|v \geq \Psi^\varepsilon \text{a.e.} \}\), where \(\overset{o} W^2_A\) is the Orlicz-Sobolev space for the \(N\)-function \(A\). Also, the quadratic regularization \(J_\delta\) of \(J\) is introduced, and this latter problem is considered in the space \({K^\varepsilon}^\prime (\Omega)\), which differs from \(K^\varepsilon\) in the replacement of \(\overset{o} W^2_A\) by \(\overset{o} W^2_2\). Technical properties of the regularized problem are established, for example the weak convergence \(u_\delta^\varepsilon \rightharpoonup u^\varepsilon\) as \(\delta \rightarrow 0\). A further key result is that of the density of \({K^\varepsilon}^\prime\) in \(K^\varepsilon\). Finally, the main result is proved with the aid of convergence properties of the regularized problem and its solution, and the use of the Sobolev embedding theorem.

MSC:
49N60 Regularity of solutions in optimal control
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
74K20 Plates
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