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Directional differentiability for elliptic quasi-variational inequalities of obstacle type

Calculus of Variations and Partial Differential Equations volume 58, Article number: 39 (2019) Cite this article

Abstract

The directional differentiability of the solution map of obstacle type quasi-variational inequalities (QVIs) with respect to perturbations on the forcing term is studied. The classical result of Mignot is then extended to the quasi-variational case under assumptions that allow multiple solutions of the QVI. The proof involves selection procedures for the solution set and represents the directional derivative as the limit of a monotonic sequence of directional derivatives associated to specific variational inequalities. Additionally, estimates on the coincidence set and several simplifications under higher regularity are studied. The theory is illustrated by a detailed study of an application to thermoforming comprising of modelling, analysis and some numerical experiments.

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Notes

  1. This exists by Theorem 3.

  2. Under these circumstances, g maps W into W [32, Theorem 1.18].

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  1. Weierstrass Institute, Mohrenstr. 39, 10117, Berlin, Germany

    Amal Alphonse, Michael Hintermüller & Carlos N. Rautenberg

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  1. Amal Alphonse
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This research was carried out in the framework of MATHEON supported by the Einstein Foundation Berlin within the ECMath projects OT1, SE5, CH12 and SE15/SE19 as well as project A-AP24. The authors further acknowledge the support of the DFG through the DFG-SPP 1962: Priority Programme “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” within Projects 10, 11, and 13, through Grant no. HI 1466/7-1 Free Boundary Problems and Level Set Methods, and SFB/TRR154.

A Simplification of the Laplace–Beltrami operator

A Simplification of the Laplace–Beltrami operator

Define \(T(r) := {\hat{T}}(r,{\varPhi }(u)(r))\) so that \(T:[0,1]\rightarrow {\mathbb {R}}\). We want to write \({\varDelta }_{\varGamma } {\hat{T}}\) in terms of \({\varDelta } T\). With \(w:={\varPhi }(u)\), the metric tensor is \(g_{11}=g = 1+ (w')^2\) and its inverse is \(g^{11}={1}\slash (1+(w')^2)=g^{-1}\) and so the Laplace–Beltrami of a function \({\hat{H}}:{\varGamma } \rightarrow {\mathbb {R}}\) can be written as

$$\begin{aligned} {\varDelta }_{\varGamma } {\hat{H}} = \frac{1}{\sqrt{g}}(g^{11}\sqrt{g}H')' \end{aligned}$$

where \(H:[0,L] \rightarrow {\mathbb {R}}\) is defined by \(H(r) = {\hat{H}}(r,w(r))\). Thus

$$\begin{aligned} {\varDelta }_{\varGamma } {\hat{H}} =\frac{H''}{g} - \frac{H'g'}{2g^2}. \end{aligned}$$

Noticing that \(g' = 2w'w''\), the above reads

$$\begin{aligned} {\varDelta }_{\varGamma } {\hat{H}}&=\frac{H''}{1+(w')^2} - \frac{w'w''H'}{(1+(w')^2)^2}. \end{aligned}$$

And now picking \({\hat{H}}={\hat{T}}\), we find

$$\begin{aligned} {\varDelta }_{\varGamma } {\hat{T}}&=\frac{{\varDelta } T}{1+({\varPhi }(u)')^2} - \frac{T'{\varPhi }(u)'{\varDelta } {\varPhi }(u)}{(1+({\varPhi }(u)')^2)^2}. \end{aligned}$$

Then if \(x=(x_1,x_2) = (r, {\varPhi }(r))\) the Eq. (54) becomes

$$\begin{aligned} kT(r) - \frac{{\varDelta } T(r)}{1+({\varPhi }(u)'(r))^2} + \frac{T'(r){\varPhi }(u)'(r){\varDelta } {\varPhi }(u)(r)}{(1+({\varPhi }(u)'(r))^2)^2}&= g({\varPhi }(r)-u(r)). \end{aligned}$$

In order to deduce (50) we have taken \({\varPhi }(u)'\) to be close to zero in the equality above.

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Alphonse, A., Hintermüller, M. & Rautenberg, C.N. Directional differentiability for elliptic quasi-variational inequalities of obstacle type. Calc. Var. 58, 39 (2019). https://doi.org/10.1007/s00526-018-1473-0

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