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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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