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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.

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