introduction to quasi-variational inequalities in hilbert spaces...
TRANSCRIPT
![Page 1: Introduction to Quasi-variational Inequalities in Hilbert Spaces …math.gmu.edu/~crautenb/ContentHP/Prints/Lecture1.pdf · 2020-03-04 · Introduction to Quasi-variational Inequalities](https://reader033.vdocuments.us/reader033/viewer/2022050515/5f9ef56218825b2cd92f2eea/html5/thumbnails/1.jpg)
Introduction to Quasi-variational Inequalities in Hilbert Spaces
Motivations, and elliptic problems
C. N. RautenbergDepartment of Mathematical Sciences
George Mason UniversityFairfax, VA, U.S.
September 2019
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Contents
1. Introduction, motivation, and lack of energy formulation
2. The general elliptic QVI problem
3. Historical note on QVIs
4. Existence - The compactness approach
5. Existence - The contraction approach
5.1 Some problems with the QVI literature
2/33 QVIs
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Introduction, motivation, and lack of energyformulation
3/33 QVIs
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Introduction - The Dirichlet principle
Consider the deformation of an elastic string on [0, 1]. Suppose that
displacement is given by y : (0, 1) → R.
endpoints are clamped, i.e., y(0) = y(1) = 0.
the string is loaded by a normal uniform force f : (0, 1) → R.
The displacement satisfies the problem
−λy′′(x) = f (x), in (0, 1)y(s) = 0 for s = 0, 1; (BVP)
y<latexit sha1_base64="mEcz1FLhuG1BpP6c5hi50qAIJ0g=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOl5qRfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJu1b1Lqq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8f6QuNAQ==</latexit>
x<latexit sha1_base64="hL+FaLtOT9luwfLW3Ut08xl3Pcw=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx4hkUcCGzI79MLI7OxmZtZICF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOeHjQA=</latexit>
1<latexit sha1_base64="TtIgPQprnJE4HSS++PuM3etxya8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlptcvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPe+uMuQ==</latexit>
y00 = f = 1<latexit sha1_base64="RTBn99K0RmfN1BmI2PRMWbv1aWM=">AAAB+XicbVDLSsNAFL3xWesr6tLNYJG6sSRV0E2h6MZlBfuANpTJdNIOnUzCzKQQQv/EjQtF3Pon7vwbp20W2npg4HDOudw7x485U9pxvq219Y3Nre3CTnF3b//g0D46bqkokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH9/P/PaESsUi8aTTmHohHgoWMIK1kfq2fdnjJj3AKC2Xa0HN7dslp+LMgVaJm5MS5Gj07a/eICJJSIUmHCvVdZ1YexmWmhFOp8VeomiMyRgPaddQgUOqvGx++RSdG2WAgkiaJzSaq78nMhwqlYa+SYZYj9SyNxP/87qJDm69jIk40VSQxaIg4UhHaFYDGjBJieapIZhIZm5FZIQlJtqUVTQluMtfXiWtasW9qlQfr0v1u7yOApzCGVyACzdQhwdoQBMITOAZXuHNyqwX6936WETXrHzmBP7A+vwBYl6SMQ==</latexit>
λ > 0 depends on elastic material properties.
Problem (BVP) is equivalent to minimizing the potential energy:
minv
λ
2
∫ 1
0|v′(x)|2 dx −
∫ 1
0f (x)v(x) dx;
subject to (s.t.) v(0) = v(1) = 0;
4/33 QVIs
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Introduction - The Dirichlet principle
Consider the deformation of an elastic string on [0, 1]. Suppose that
displacement is given by y : (0, 1) → R.
endpoints are clamped, i.e., y(0) = y(1) = 0.
the string is loaded by a normal uniform force f : (0, 1) → R.
The displacement satisfies the problem
−λy′′(x) = f (x), in (0, 1)y(s) = 0 for s = 0, 1; (BVP)
y<latexit sha1_base64="mEcz1FLhuG1BpP6c5hi50qAIJ0g=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOl5qRfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJu1b1Lqq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8f6QuNAQ==</latexit>
x<latexit sha1_base64="hL+FaLtOT9luwfLW3Ut08xl3Pcw=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx4hkUcCGzI79MLI7OxmZtZICF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOeHjQA=</latexit>
1<latexit sha1_base64="TtIgPQprnJE4HSS++PuM3etxya8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlptcvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPe+uMuQ==</latexit>
y00 = f = 1<latexit sha1_base64="RTBn99K0RmfN1BmI2PRMWbv1aWM=">AAAB+XicbVDLSsNAFL3xWesr6tLNYJG6sSRV0E2h6MZlBfuANpTJdNIOnUzCzKQQQv/EjQtF3Pon7vwbp20W2npg4HDOudw7x485U9pxvq219Y3Nre3CTnF3b//g0D46bqkokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH9/P/PaESsUi8aTTmHohHgoWMIK1kfq2fdnjJj3AKC2Xa0HN7dslp+LMgVaJm5MS5Gj07a/eICJJSIUmHCvVdZ1YexmWmhFOp8VeomiMyRgPaddQgUOqvGx++RSdG2WAgkiaJzSaq78nMhwqlYa+SYZYj9SyNxP/87qJDm69jIk40VSQxaIg4UhHaFYDGjBJieapIZhIZm5FZIQlJtqUVTQluMtfXiWtasW9qlQfr0v1u7yOApzCGVyACzdQhwdoQBMITOAZXuHNyqwX6936WETXrHzmBP7A+vwBYl6SMQ==</latexit>
λ > 0 depends on elastic material properties.
Problem (BVP) is equivalent to minimizing the potential energy:
minv
λ
2
∫ 1
0|v′(x)|2 dx −
∫ 1
0f (x)v(x) dx;
subject to (s.t.) v(0) = v(1) = 0;
4/33 QVIs
![Page 6: Introduction to Quasi-variational Inequalities in Hilbert Spaces …math.gmu.edu/~crautenb/ContentHP/Prints/Lecture1.pdf · 2020-03-04 · Introduction to Quasi-variational Inequalities](https://reader033.vdocuments.us/reader033/viewer/2022050515/5f9ef56218825b2cd92f2eea/html5/thumbnails/6.jpg)
Introduction - The Dirichlet principle
Consider the deformation of an elastic string on [0, 1]. Suppose that
displacement is given by y : (0, 1) → R.
endpoints are clamped, i.e., y(0) = y(1) = 0.
the string is loaded by a normal uniform force f : (0, 1) → R.
The displacement satisfies the problem
−λy′′(x) = f (x), in (0, 1)y(s) = 0 for s = 0, 1; (BVP)
y<latexit sha1_base64="mEcz1FLhuG1BpP6c5hi50qAIJ0g=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOl5qRfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJu1b1Lqq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8f6QuNAQ==</latexit>
x<latexit sha1_base64="hL+FaLtOT9luwfLW3Ut08xl3Pcw=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx4hkUcCGzI79MLI7OxmZtZICF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOeHjQA=</latexit>
1<latexit sha1_base64="TtIgPQprnJE4HSS++PuM3etxya8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlptcvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPe+uMuQ==</latexit>
y00 = f = 1<latexit sha1_base64="RTBn99K0RmfN1BmI2PRMWbv1aWM=">AAAB+XicbVDLSsNAFL3xWesr6tLNYJG6sSRV0E2h6MZlBfuANpTJdNIOnUzCzKQQQv/EjQtF3Pon7vwbp20W2npg4HDOudw7x485U9pxvq219Y3Nre3CTnF3b//g0D46bqkokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH9/P/PaESsUi8aTTmHohHgoWMIK1kfq2fdnjJj3AKC2Xa0HN7dslp+LMgVaJm5MS5Gj07a/eICJJSIUmHCvVdZ1YexmWmhFOp8VeomiMyRgPaddQgUOqvGx++RSdG2WAgkiaJzSaq78nMhwqlYa+SYZYj9SyNxP/87qJDm69jIk40VSQxaIg4UhHaFYDGjBJieapIZhIZm5FZIQlJtqUVTQluMtfXiWtasW9qlQfr0v1u7yOApzCGVyACzdQhwdoQBMITOAZXuHNyqwX6936WETXrHzmBP7A+vwBYl6SMQ==</latexit>
λ > 0 depends on elastic material properties.
Problem (BVP) is equivalent to minimizing the potential energy:
minv
λ
2
∫ 1
0|v′(x)|2 dx −
∫ 1
0f (x)v(x) dx;
subject to (s.t.) v(0) = v(1) = 0;
4/33 QVIs
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Introduction - The Dirichlet principle with obstacle
Suppose in addition that
there is an obstacle ϕ : (0, 1) → R such that y(x) ≤ ϕ(x) a.e. in (0, 1).
The displacement satisfies y ≤ ϕ and
λ
∫ 1
0y′(v′ − y′) dx ≥
∫ 1
0f (v − y) dx,
y(s) = 0 for s = 0, 1;(VI)
For all v smooth s.t. v(0) = v(1) = 0 and v ≤ ϕ.
<latexit sha1_base64="U7EeqmYiKeTo/r/N2HofpG7Xero=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lqQY9FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq777ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmGG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NEwiNqrW3Lq7AFonXkFqUKA9qn4NxzFJBZWGcKz1wHMT42dYGUY4nVeGqaYJJlM8oQNLJRZU+9ni1jm6sMoYhbGyJQ1aqL8nMiy0nonAdgpsIr3q5eJ/3iA14Y2fMZmkhkqyXBSmHJkY5Y+jMVOUGD6zBBPF7K2IRFhhYmw8FRuCt/ryOuk26t5VvfHQrLVuizjKcAbncAkeXEML7qENHSAQwTO8wpsjnBfn3flYtpacYuYU/sD5/AEU1o5D</latexit>
y<latexit sha1_base64="mEcz1FLhuG1BpP6c5hi50qAIJ0g=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOl5qRfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJu1b1Lqq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8f6QuNAQ==</latexit>
x<latexit sha1_base64="hL+FaLtOT9luwfLW3Ut08xl3Pcw=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx4hkUcCGzI79MLI7OxmZtZICF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOeHjQA=</latexit>
1<latexit sha1_base64="TtIgPQprnJE4HSS++PuM3etxya8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlptcvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPe+uMuQ==</latexit>
Problem (VI) is equivalent to minimizing the (constrained) potential energy:
minv
λ
2
∫ 1
0|v′(x)|2 dx −
∫ 1
0f (x)v(x) dx;
subject to (s.t.) v(0) = v(1) = 0;v ≤ ϕ
5/33 QVIs
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Introduction - The Dirichlet principle with obstacle
Suppose in addition that
there is an obstacle ϕ : (0, 1) → R such that y(x) ≤ ϕ(x) a.e. in (0, 1).
The displacement satisfies y ≤ ϕ and
λ
∫ 1
0y′(v′ − y′) dx ≥
∫ 1
0f (v − y) dx,
y(s) = 0 for s = 0, 1;(VI)
For all v smooth s.t. v(0) = v(1) = 0 and v ≤ ϕ.
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Problem (VI) is equivalent to minimizing the (constrained) potential energy:
minv
λ
2
∫ 1
0|v′(x)|2 dx −
∫ 1
0f (x)v(x) dx;
subject to (s.t.) v(0) = v(1) = 0;v ≤ ϕ
5/33 QVIs
![Page 9: Introduction to Quasi-variational Inequalities in Hilbert Spaces …math.gmu.edu/~crautenb/ContentHP/Prints/Lecture1.pdf · 2020-03-04 · Introduction to Quasi-variational Inequalities](https://reader033.vdocuments.us/reader033/viewer/2022050515/5f9ef56218825b2cd92f2eea/html5/thumbnails/9.jpg)
Introduction - The Dirichlet principle with obstacle
Suppose in addition that
there is an obstacle ϕ : (0, 1) → R such that y(x) ≤ ϕ(x) a.e. in (0, 1).
The displacement satisfies y ≤ ϕ and
λ
∫ 1
0y′(v′ − y′) dx ≥
∫ 1
0f (v − y) dx,
y(s) = 0 for s = 0, 1;(VI)
For all v smooth s.t. v(0) = v(1) = 0 and v ≤ ϕ.
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Problem (VI) is equivalent to minimizing the (constrained) potential energy:
minv
λ
2
∫ 1
0|v′(x)|2 dx −
∫ 1
0f (x)v(x) dx;
subject to (s.t.) v(0) = v(1) = 0;v ≤ ϕ
5/33 QVIs
![Page 10: Introduction to Quasi-variational Inequalities in Hilbert Spaces …math.gmu.edu/~crautenb/ContentHP/Prints/Lecture1.pdf · 2020-03-04 · Introduction to Quasi-variational Inequalities](https://reader033.vdocuments.us/reader033/viewer/2022050515/5f9ef56218825b2cd92f2eea/html5/thumbnails/10.jpg)
Introduction - Lack of Dirichlet principle with implicit obstacleSuppose in addition that
there is an implicit obstacle ϕ : (0, 1) → R depending on y, i.e., ϕ = Φ(y).
The displacement(s) satisfies y ≤ Φ(y) and
λ
∫ 1
0y′(v′ − y′) dx ≥
∫ 1
0f (v − y) dx,
y(s) = 0 for s = 0, 1;(QVI)
∀v smooth s.t. v(0) = v(1) = 0 and v ≤ Φ(y).x
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Problem (QVI) is NOT (in general) equivalent to minimizing the (implicitly con-strained) potential energy:
minv
λ
2
∫ 1
0|v′(x)|2 dx −
∫ 1
0f (x)v(x) dx;
subject to (s.t.) v(0) = v(1) = 0;v ≤ Φ(v)
6/33 QVIs
![Page 11: Introduction to Quasi-variational Inequalities in Hilbert Spaces …math.gmu.edu/~crautenb/ContentHP/Prints/Lecture1.pdf · 2020-03-04 · Introduction to Quasi-variational Inequalities](https://reader033.vdocuments.us/reader033/viewer/2022050515/5f9ef56218825b2cd92f2eea/html5/thumbnails/11.jpg)
Introduction - Lack of Dirichlet principle with implicit obstacleSuppose in addition that
there is an implicit obstacle ϕ : (0, 1) → R depending on y, i.e., ϕ = Φ(y).
The displacement(s) satisfies y ≤ Φ(y) and
λ
∫ 1
0y′(v′ − y′) dx ≥
∫ 1
0f (v − y) dx,
y(s) = 0 for s = 0, 1;(QVI)
∀v smooth s.t. v(0) = v(1) = 0 and v ≤ Φ(y).x
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Problem (QVI) is NOT (in general) equivalent to minimizing the (implicitly con-strained) potential energy:
minv
λ
2
∫ 1
0|v′(x)|2 dx −
∫ 1
0f (x)v(x) dx;
subject to (s.t.) v(0) = v(1) = 0;v ≤ Φ(v)
6/33 QVIs
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Introduction - Lack of Dirichlet principle with implicit obstacleSuppose in addition that
there is an implicit obstacle ϕ : (0, 1) → R depending on y, i.e., ϕ = Φ(y).
The displacement(s) satisfies y ≤ Φ(y) and
λ
∫ 1
0y′(v′ − y′) dx ≥
∫ 1
0f (v − y) dx,
y(s) = 0 for s = 0, 1;(QVI)
∀v smooth s.t. v(0) = v(1) = 0 and v ≤ Φ(y).x
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Problem (QVI) is NOT (in general) equivalent to minimizing the (implicitly con-strained) potential energy:
minv
λ
2
∫ 1
0|v′(x)|2 dx −
∫ 1
0f (x)v(x) dx;
subject to (s.t.) v(0) = v(1) = 0;v ≤ Φ(v)
6/33 QVIs
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The general elliptic QVI problem
7/33 QVIs
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The class of elliptic QVIs
Let A : V → V ′ and f ∈ V ′ for some (real) Hilbert space V . Consider
Find y ∈ K(y) : ⟨A(y) − f, v − y⟩ ≥ 0, ∀v ∈ K(y) (QVI)
whereK(w) := z ∈ V : Ψ(Gz) ≤ Φ(w).
V ∈ H10(Ω), H1(Ω), L2(Ω), . . ., and part of a Gelfand triple (V, H, V ′).
A is Lipschitz continuous and strongly monotone, i.e.,
⟨A(u) − A(v), u − v⟩ ≥ c∥u − v∥2V .
G : V → Lp(Ω)d is linear, and Ψ : Rd → R is convex.
Φ(v) : Ω → R is measurable.
Obstacle type
G = id, and Ψ(x) = x.
Gradient type
G = ∇, and Ψ(x) = ∥x∥Rn.
8/33 QVIs
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The class of elliptic QVIs
Let A : V → V ′ and f ∈ V ′ for some (real) Hilbert space V . Consider
Find y ∈ K(y) : ⟨A(y) − f, v − y⟩ ≥ 0, ∀v ∈ K(y) (QVI)
whereK(w) := z ∈ V : Ψ(Gz) ≤ Φ(w).
Many contributors Adly, Alphonse, Aubin, Aussel, Barrett, Bensoussan,Bergounioux, Biroli, Caffarelli, Facchinei, Friedman, Frehse, Fukao, Fukushima,Gwinner, Hanouzet, Harker, Hintermüller, Joly, Kano, Kanzow, Kenmochi, Lions,Mignot, Mordukhovich, Mosco, Murase, Outrata, Pang, Prigozhin, Rodrigues,Santos, Tartar, Yousept, . . . .
A. Bensoussan, J.-L. Lions , In a series of papers concerned with impulse control (1973/1974).
9/33 QVIs
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The class of elliptic QVIs : Prototypicals V, A, and K
The typical setting is given by
(V, H, V ′) = (H10(Ω), L2(Ω), H−1(Ω)).
The map A : H10(Ω) → H−1(Ω) is given
⟨Av, w⟩ =∫
Ω
∑i,j
aij(x) ∂v
∂xj
∂w
∂xi+∑
i
ai(x)∫
Ω
∂v
∂xjw + a0(x)vw
dx,
with usual assumptions over coefficients. Also fractional powers As for s ∈ (0, 1)are suitable.
Φ : H10(Ω) → H1
0(Ω) is
A superposition operator, i.e., Φ(y)(x) = φ(y(x)) for some φ.
A solution operator coming from a PDE, e.g., Φ(y) = (−∆)−1y + ϕ0.
10/33 QVIs
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The class of elliptic QVIs
Let A : V → V ′ and f ∈ V ′ for some (real) Hilbert space V . Consider
Find y ∈ K(y) : ⟨A(y) − f, v − y⟩ ≥ 0, ∀v ∈ K(y) (QVI)
whereK(w) := z ∈ V : Ψ(Gz) ≤ Φ(w).
Main difficulties:
The problem is non-smooth and non-convex.
In general there are multiple solutions. The solution set
Q(f )
might be of any cardinality!
In general, the problem does not arise as first order condition of anoptimization problem.
11/33 QVIs
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The class of elliptic QVIs - Cardinality of Q(f )
Find y ≤ Φ(y) : ⟨Ay − f, v − y⟩ ≥ 0, ∀v ≤ Φ(y). (QVI)
If Φ : V → V is Lipschitz with small Lipschitz constant =⇒ Unique solution.
Let A = −∆, V = H10(Ω), f ∈ R =⇒ Two solutions or one.
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Let Φ(y) = y, then if Ay ≤ f then y is a solution of (QVI) =⇒ Mutiplesolutions - Not isolated solutions.
12/33 QVIs
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The class of elliptic QVIs - Cardinality of Q(f )
Find y ≤ Φ(y) : ⟨Ay − f, v − y⟩ ≥ 0, ∀v ≤ Φ(y). (QVI)
If Φ : V → V is Lipschitz with small Lipschitz constant =⇒ Unique solution.
Let A = −∆, V = H10(Ω), f ∈ R =⇒ Two solutions or one.
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Let Φ(y) = y, then if Ay ≤ f then y is a solution of (QVI) =⇒ Mutiplesolutions - Not isolated solutions.
12/33 QVIs
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The class of elliptic QVIs - Cardinality of Q(f )
Find y ≤ Φ(y) : ⟨Ay − f, v − y⟩ ≥ 0, ∀v ≤ Φ(y). (QVI)
If Φ : V → V is Lipschitz with small Lipschitz constant =⇒ Unique solution.
Let A = −∆, V = H10(Ω), f ∈ R =⇒ Two solutions or one.
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Let Φ(y) = y, then if Ay ≤ f then y is a solution of (QVI) =⇒ Mutiplesolutions - Not isolated solutions.
12/33 QVIs
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Historical note on quasi-variationalinequalities
13/33 QVIs
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A little history on QVIs
For x ∈ RN , σ > 0, w a Wiener process and g : RN → RN Lipschitz andbounded, consider
dz = g(z) dt + σI dw(t) +∑
i
δ(t − θi)ξi
z(0) = x.(Px)
Admissible controls U for (Px) are given by (θi, ξi)∞i=1 where
Time instants θi such that 0 = θ0 ≤ θ1 ≤ θ2 ≤ . . . and limi θi = +∞ State jumps ξi on some compact subset K ⊂ RN ,
z(θ+i ) = z(θ−
i ) + ξi.
A. Bensoussan and J. L. Lions considered the optimal impulse control problem
Given k > 0 and α > 0, consider
minν∈U
J(x, ν) := E
∫ ∞
0e−αtf (zx(t)) dt + k
∑i
e−αθi
, (P)
where ν = (θi, ξi), and zx solves (Px).
14/33 QVIs
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A little history on QVIs
For x ∈ RN , σ > 0, w a Wiener process and g : RN → RN Lipschitz andbounded, consider
dz = g(z) dt + σI dw(t) +∑
i
δ(t − θi)ξi
z(0) = x.(Px)
Admissible controls U for (Px) are given by (θi, ξi)∞i=1 where
Time instants θi such that 0 = θ0 ≤ θ1 ≤ θ2 ≤ . . . and limi θi = +∞ State jumps ξi on some compact subset K ⊂ RN ,
z(θ+i ) = z(θ−
i ) + ξi.
A. Bensoussan and J. L. Lions considered the optimal impulse control problem
Given k > 0 and α > 0, consider
minν∈U
J(x, ν) := E
∫ ∞
0e−αtf (zx(t)) dt + k
∑i
e−αθi
, (P)
where ν = (θi, ξi), and zx solves (Px).
14/33 QVIs
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A little history on QVIs
Given k > 0 and α > 0, consider
minν∈U
J(x, ν) := E
∫ ∞
0e−αtf (zx(t)) dt + k
∑i
e−αθi
, (P)
where ν = (θi, ξi), and zx solves (Px).
dz = g(z) dt + σI dw(t) +∑
i
δ(t − θi)ξi
z(0) = x.(Px)
Define the value function
y∗(x) := infν∈U
J(x, ν∗).
Benoussan-Lions(1974)
Under certain conditions, ∃y solution to: Find y ∈ H1(RN) such y ≤ My and⟨−σ
2∆y −
∑i
gi(x)∂y
∂x+ αy − f, v − y
⟩≥ 0, ∀v ≤ My,
where My(x) := k + infξ∈K y(x + ξ). The value function y∗ is a solution to the QVI above.
There exists a solution to (P) → it can be constructed via a solution y.
![Page 25: Introduction to Quasi-variational Inequalities in Hilbert Spaces …math.gmu.edu/~crautenb/ContentHP/Prints/Lecture1.pdf · 2020-03-04 · Introduction to Quasi-variational Inequalities](https://reader033.vdocuments.us/reader033/viewer/2022050515/5f9ef56218825b2cd92f2eea/html5/thumbnails/25.jpg)
A little history on QVIs
Given k > 0 and α > 0, consider
minν∈U
J(x, ν) := E
∫ ∞
0e−αtf (zx(t)) dt + k
∑i
e−αθi
, (P)
where ν = (θi, ξi), and zx solves (Px).
dz = g(z) dt + σI dw(t) +∑
i
δ(t − θi)ξi
z(0) = x.(Px)
Define the value function
y∗(x) := infν∈U
J(x, ν∗).
Benoussan-Lions(1974)
Under certain conditions, ∃y solution to: Find y ∈ H1(RN) such y ≤ My and⟨−σ
2∆y −
∑i
gi(x)∂y
∂x+ αy − f, v − y
⟩≥ 0, ∀v ≤ My,
where My(x) := k + infξ∈K y(x + ξ). The value function y∗ is a solution to the QVI above.
There exists a solution to (P) → it can be constructed via a solution y.
![Page 26: Introduction to Quasi-variational Inequalities in Hilbert Spaces …math.gmu.edu/~crautenb/ContentHP/Prints/Lecture1.pdf · 2020-03-04 · Introduction to Quasi-variational Inequalities](https://reader033.vdocuments.us/reader033/viewer/2022050515/5f9ef56218825b2cd92f2eea/html5/thumbnails/26.jpg)
A little history on QVIs
Given k > 0 and α > 0, consider
minν∈U
J(x, ν) := E
∫ ∞
0e−αtf (zx(t)) dt + k
∑i
e−αθi
, (P)
where ν = (θi, ξi), and zx solves (Px).
dz = g(z) dt + σI dw(t) +∑
i
δ(t − θi)ξi
z(0) = x.(Px)
Define the value function
y∗(x) := infν∈U
J(x, ν∗).
Benoussan-Lions(1974)
Under certain conditions, ∃y solution to: Find y ∈ H1(RN) such y ≤ My and⟨−σ
2∆y −
∑i
gi(x)∂y
∂x+ αy − f, v − y
⟩≥ 0, ∀v ≤ My,
where My(x) := k + infξ∈K y(x + ξ).
The value function y∗ is a solution to the QVI above.
There exists a solution to (P) → it can be constructed via a solution y.
![Page 27: Introduction to Quasi-variational Inequalities in Hilbert Spaces …math.gmu.edu/~crautenb/ContentHP/Prints/Lecture1.pdf · 2020-03-04 · Introduction to Quasi-variational Inequalities](https://reader033.vdocuments.us/reader033/viewer/2022050515/5f9ef56218825b2cd92f2eea/html5/thumbnails/27.jpg)
A little history on QVIs
Given k > 0 and α > 0, consider
minν∈U
J(x, ν) := E
∫ ∞
0e−αtf (zx(t)) dt + k
∑i
e−αθi
, (P)
where ν = (θi, ξi), and zx solves (Px).
dz = g(z) dt + σI dw(t) +∑
i
δ(t − θi)ξi
z(0) = x.(Px)
Define the value function
y∗(x) := infν∈U
J(x, ν∗).
Benoussan-Lions(1974)
Under certain conditions, ∃y solution to: Find y ∈ H1(RN) such y ≤ My and⟨−σ
2∆y −
∑i
gi(x)∂y
∂x+ αy − f, v − y
⟩≥ 0, ∀v ≤ My,
where My(x) := k + infξ∈K y(x + ξ). The value function y∗ is a solution to the QVI above.
There exists a solution to (P) → it can be constructed via a solution y.
![Page 28: Introduction to Quasi-variational Inequalities in Hilbert Spaces …math.gmu.edu/~crautenb/ContentHP/Prints/Lecture1.pdf · 2020-03-04 · Introduction to Quasi-variational Inequalities](https://reader033.vdocuments.us/reader033/viewer/2022050515/5f9ef56218825b2cd92f2eea/html5/thumbnails/28.jpg)
A little history on QVIs
Given k > 0 and α > 0, consider
minν∈U
J(x, ν) := E
∫ ∞
0e−αtf (zx(t)) dt + k
∑i
e−αθi
, (P)
where ν = (θi, ξi), and zx solves (Px).
dz = g(z) dt + σI dw(t) +∑
i
δ(t − θi)ξi
z(0) = x.
(Px)
Define the value function
y∗(x) := infν∈U
J(x, ν∗).
Bensoussan-Lions(1974)
Under certain conditions, ∃y solution to: Find y ∈ H1(RN) such y ≤ My and⟨−σ
2∆y −
∑i
gi(x)∂y
∂x+ αy − f, v − u
⟩≥ 0, ∀v ≤ My,
where Mw(x) := k + infξ∈K u(x + ξ). The value function y∗ is a solution to the QVI above.
There exists a solution to (P) → it can be constructed via a solution y.
![Page 29: Introduction to Quasi-variational Inequalities in Hilbert Spaces …math.gmu.edu/~crautenb/ContentHP/Prints/Lecture1.pdf · 2020-03-04 · Introduction to Quasi-variational Inequalities](https://reader033.vdocuments.us/reader033/viewer/2022050515/5f9ef56218825b2cd92f2eea/html5/thumbnails/29.jpg)
QVIs are powerful models
17/33 QVIs
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Stationary QVI models
Stationary Magnetization of a superconductorDetermination of the magnetic field[Rodrigues, Prigozhin, Yousept]
A(y) = −div|∇y|p−2∇y K(y) = v : |∇v| ≤ Φ(y),
Φ is a superposition operator.
Flow trough semi-permeable membranes/anomalousdiffusion. Pressure of a chemical solution of a slightlyincompressible fluid [Antil, R. (2018)].
A = (−∆)s K(y) = v : v ≤ Φ(y),
with s ∈ (0, 1), and Φ solution map of a PDE.0
0.20.4
0.60.8
00.2
0.40.6
0.8
0
0.005
0.01
Thermo-elastic equilibrium of a locking material. Determination of a thedisplacement field [Rodrigues-Santos (2019)].
⟨A(y), v⟩ = µ(Dy, Dv)+λ(∇·y, ∇·v) K(y) = v ∈ H10(Ω) : |Dv| ≤ Φ(y)
18/33 QVIs
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The compactness approach
19/33 QVIs
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Existence theory - Compactness approaches
Find y ∈ K(y) : ⟨A(y) − f, v − y⟩ ≥ 0, ∀v ∈ K(y). (QVI)
Define S(f, K), for a fixed closed, and convex set K as the unique solution to:
Find z ∈ K : ⟨A(z) − f, v − z⟩ ≥ 0, ∀v ∈ K.
Then, for non-constant v 7→ K(v), consider T (v) := S(f, K(v)) so that
y = T (y) ⇐⇒ y solves (QVI).
Proposition
Since V is reflexive, if T : V → V is completely continuous, i.e.,
vn v implies T (vn) → T (v),
then T has fixed point à la Schauder (T (BR(0, V )) ⊂ BR(0, V ) is simple to obtain).
Since T (v) = S(f, K(v)), we need a concept of set convergence so that
S(f, Kn) → S(f, K).
20/33 QVIs
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Existence theory - Compactness approaches
Find y ∈ K(y) : ⟨A(y) − f, v − y⟩ ≥ 0, ∀v ∈ K(y). (QVI)
Define S(f, K), for a fixed closed, and convex set K as the unique solution to:
Find z ∈ K : ⟨A(z) − f, v − z⟩ ≥ 0, ∀v ∈ K.
Then, for non-constant v 7→ K(v), consider T (v) := S(f, K(v)) so that
y = T (y) ⇐⇒ y solves (QVI).
Proposition
Since V is reflexive, if T : V → V is completely continuous, i.e.,
vn v implies T (vn) → T (v),
then T has fixed point à la Schauder (T (BR(0, V )) ⊂ BR(0, V ) is simple to obtain).
Since T (v) = S(f, K(v)), we need a concept of set convergence so that
S(f, Kn) → S(f, K).
20/33 QVIs
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Existence theory - Compactness approaches
Find y ∈ K(y) : ⟨A(y) − f, v − y⟩ ≥ 0, ∀v ∈ K(y). (QVI)
Define S(f, K), for a fixed closed, and convex set K as the unique solution to:
Find z ∈ K : ⟨A(z) − f, v − z⟩ ≥ 0, ∀v ∈ K.
Then, for non-constant v 7→ K(v), consider T (v) := S(f, K(v)) so that
y = T (y) ⇐⇒ y solves (QVI).
Proposition
Since V is reflexive, if T : V → V is completely continuous, i.e.,
vn v implies T (vn) → T (v),
then T has fixed point à la Schauder (T (BR(0, V )) ⊂ BR(0, V ) is simple to obtain).
Since T (v) = S(f, K(v)), we need a concept of set convergence so that
S(f, Kn) → S(f, K).
20/33 QVIs
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Mosco convergence
Find y ∈ K(y) : ⟨A(y) − f, v − y⟩ ≥ 0, ∀v ∈ K(y). (QVI)
Define S(f, K), for a fixed closed, and convex set K as the unique solution to:
Find z ∈ K : ⟨A(z) − f, v − z⟩ ≥ 0, ∀v ∈ K.
Then, for non-constant v 7→ K(v), consider T (v) := S(f, K(v)) so that
y = T (y) ⇐⇒ y solves (QVI).
The sequence of sets Kn is said to converge to K in the sense of Mosco as
n → ∞, denoted by KnM−→ K, if the following two conditions are fulfilled:
(i) For each w ∈ K, ∃ wn′ such that wn′ ∈ Kn′ and wn′ → w in V .
(ii) If wn ∈ Kn and wn w in V along a subsequence, then w ∈ K.
If KnM−−→ K then S(f, Kn) → S(f, K) in V .
If vn v in V implies K(vn) M−−→ K(v) then T is completely continuous.
21/33 QVIs
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Mosco convergence
Find y ∈ K(y) : ⟨A(y) − f, v − y⟩ ≥ 0, ∀v ∈ K(y). (QVI)
Define S(f, K), for a fixed closed, and convex set K as the unique solution to:
Find z ∈ K : ⟨A(z) − f, v − z⟩ ≥ 0, ∀v ∈ K.
Then, for non-constant v 7→ K(v), consider T (v) := S(f, K(v)) so that
y = T (y) ⇐⇒ y solves (QVI).
The sequence of sets Kn is said to converge to K in the sense of Mosco as
n → ∞, denoted by KnM−→ K, if the following two conditions are fulfilled:
(i) For each w ∈ K, ∃ wn′ such that wn′ ∈ Kn′ and wn′ → w in V .
(ii) If wn ∈ Kn and wn w in V along a subsequence, then w ∈ K.
If KnM−−→ K then S(f, Kn) → S(f, K) in V .
If vn v in V implies K(vn) M−−→ K(v) then T is completely continuous.
21/33 QVIs
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Mosco convergence - One Example
The sequence of sets Kn is said to converge to K in the sense of Mosco as
n → ∞, denoted by KnM−→ K, if the following two conditions are fulfilled:
(i) For each w ∈ K, ∃ wn′ such that wn′ ∈ Kn′ and wn′ → w in V .
(ii) If wn ∈ Kn and wn w in V along a subsequence, then w ∈ K.
Example: The obstacle case
Consider Kn = v ∈ H10(Ω) : v ≤ ϕn and ϕn → ϕ in H1
0(Ω).
(i) For each w ∈ K, i.e., w ≤ ϕ. How do we construct wn so that wn ≤ ϕn
and wn → w in H10(Ω)? Consider
wn := w − ϕ + ϕn.
(ii) Suppose wn ∈ Kn and wn w in H10(Ω), since ϕn → ϕ pointwise
a.e. along a subsequence =⇒ w ∈ K.
ϕn → ϕ in H10(Ω) is very strong. How can we relax this?
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Mosco convergence - One Example
The sequence of sets Kn is said to converge to K in the sense of Mosco as
n → ∞, denoted by KnM−→ K, if the following two conditions are fulfilled:
(i) For each w ∈ K, ∃ wn′ such that wn′ ∈ Kn′ and wn′ → w in V .
(ii) If wn ∈ Kn and wn w in V along a subsequence, then w ∈ K.
Example: The obstacle case
Consider Kn = v ∈ H10(Ω) : v ≤ ϕn and ϕn → ϕ in H1
0(Ω).(i) For each w ∈ K, i.e., w ≤ ϕ. How do we construct wn so that wn ≤ ϕn
and wn → w in H10(Ω)?
Consider
wn := w − ϕ + ϕn.
(ii) Suppose wn ∈ Kn and wn w in H10(Ω), since ϕn → ϕ pointwise
a.e. along a subsequence =⇒ w ∈ K.
ϕn → ϕ in H10(Ω) is very strong. How can we relax this?
22/33 QVIs
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Mosco convergence - One Example
The sequence of sets Kn is said to converge to K in the sense of Mosco as
n → ∞, denoted by KnM−→ K, if the following two conditions are fulfilled:
(i) For each w ∈ K, ∃ wn′ such that wn′ ∈ Kn′ and wn′ → w in V .
(ii) If wn ∈ Kn and wn w in V along a subsequence, then w ∈ K.
Example: The obstacle case
Consider Kn = v ∈ H10(Ω) : v ≤ ϕn and ϕn → ϕ in H1
0(Ω).(i) For each w ∈ K, i.e., w ≤ ϕ. How do we construct wn so that wn ≤ ϕn
and wn → w in H10(Ω)? Consider
wn := w − ϕ + ϕn.
(ii) Suppose wn ∈ Kn and wn w in H10(Ω), since ϕn → ϕ pointwise
a.e. along a subsequence =⇒ w ∈ K.
ϕn → ϕ in H10(Ω) is very strong. How can we relax this?
22/33 QVIs
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Mosco convergence - One Example
The sequence of sets Kn is said to converge to K in the sense of Mosco as
n → ∞, denoted by KnM−→ K, if the following two conditions are fulfilled:
(i) For each w ∈ K, ∃ wn′ such that wn′ ∈ Kn′ and wn′ → w in V .
(ii) If wn ∈ Kn and wn w in V along a subsequence, then w ∈ K.
Example: The obstacle case
Consider Kn = v ∈ H10(Ω) : v ≤ ϕn and ϕn → ϕ in H1
0(Ω).(i) For each w ∈ K, i.e., w ≤ ϕ. How do we construct wn so that wn ≤ ϕn
and wn → w in H10(Ω)? Consider
wn := w − ϕ + ϕn.
(ii) Suppose wn ∈ Kn and wn w in H10(Ω), since ϕn → ϕ pointwise
a.e. along a subsequence =⇒ w ∈ K.
ϕn → ϕ in H10(Ω) is very strong. How can we relax this?
22/33 QVIs
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Mosco convergence - One Example
The sequence of sets Kn is said to converge to K in the sense of Mosco as
n → ∞, denoted by KnM−→ K, if the following two conditions are fulfilled:
(i) For each w ∈ K, ∃ wn′ such that wn′ ∈ Kn′ and wn′ → w in V .
(ii) If wn ∈ Kn and wn w in V along a subsequence, then w ∈ K.
Example: The obstacle case
Consider Kn = v ∈ H10(Ω) : v ≤ ϕn and ϕn → ϕ in H1
0(Ω).(i) For each w ∈ K, i.e., w ≤ ϕ. How do we construct wn so that wn ≤ ϕn
and wn → w in H10(Ω)? Consider
wn := w − ϕ + ϕn.
(ii) Suppose wn ∈ Kn and wn w in H10(Ω), since ϕn → ϕ pointwise
a.e. along a subsequence =⇒ w ∈ K.
ϕn → ϕ in H10(Ω) is very strong. How can we relax this?
22/33 QVIs
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Mosco convergence for obstacle constraints
Kn = v ∈ V : v ≤ ϕn
For V = W 1,p0 (Ω), necessary and sufficient conditions on the convergence
ϕn → ϕ for KnM−−→ K are given in terms of capacity
([Attouch-Picard (1979-1982), Dal Maso (1985)])
The above is not so useful in applications: It is of theoretical interest though, but hardto prove unless high regularity is available for obstacles!
A useful result is the following ([Boccardo-Murat (1982)]): For V = W 1,p0 (Ω),
ϕn ϕ in W 1,q(Ω) or W 1,q0 (Ω), for q > p =⇒ Kn
M−−→ K.
The lesson is that you need only an ϵ > 0 more of regularity of your obstacles thanin your state space. This is simple to prove in applications, e.g., in the case whenΦ(v) = ϕ is given by the PDE
−∆ϕ = g(v) + g0.
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Mosco convergence for obstacle constraints
Kn = v ∈ V : v ≤ ϕn
For V = W 1,p0 (Ω), necessary and sufficient conditions on the convergence
ϕn → ϕ for KnM−−→ K are given in terms of capacity
([Attouch-Picard (1979-1982), Dal Maso (1985)])
The above is not so useful in applications: It is of theoretical interest though, but hardto prove unless high regularity is available for obstacles!
A useful result is the following ([Boccardo-Murat (1982)]): For V = W 1,p0 (Ω),
ϕn ϕ in W 1,q(Ω) or W 1,q0 (Ω), for q > p =⇒ Kn
M−−→ K.
The lesson is that you need only an ϵ > 0 more of regularity of your obstacles thanin your state space. This is simple to prove in applications, e.g., in the case whenΦ(v) = ϕ is given by the PDE
−∆ϕ = g(v) + g0.
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Mosco convergence for gradient constraints
Kn = v ∈ H10(Ω) : |∇v| ≤ ϕn
If ϕn → ϕ in C(Ω) if ∂Ω is regular enough then KnM−−→ K. Regularity of
∂Ω plays a role if ϕ is not strictly positive.
If ϕn → ϕ in L∞ν (Ω) then Kn
M−−→ K.([Santos-Assis (2004), Hintermüller-R. (2015)])
Extension from L∞(Ω) or C(Ω) to other spaces is rather complex due to thenon-Lipschitz dependence of the solution w.r.t. the gradient obstacle.
(Open problem) Necessary and sufficient conditions on ϕn so thatKn
M−−→ K holds are not known.
This is a complex task! Let |∇y| ≤ ϕ and ϕn → ϕ in some topology X , try toconstruct yn such that |∇yn| ≤ ϕn and yn → y!
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Mosco convergence for gradient constraints
Kn = v ∈ H10(Ω) : |∇v| ≤ ϕn
If ϕn → ϕ in C(Ω) if ∂Ω is regular enough then KnM−−→ K. Regularity of
∂Ω plays a role if ϕ is not strictly positive.
If ϕn → ϕ in L∞ν (Ω) then Kn
M−−→ K.([Santos-Assis (2004), Hintermüller-R. (2015)])
Extension from L∞(Ω) or C(Ω) to other spaces is rather complex due to thenon-Lipschitz dependence of the solution w.r.t. the gradient obstacle.
(Open problem) Necessary and sufficient conditions on ϕn so thatKn
M−−→ K holds are not known.
This is a complex task! Let |∇y| ≤ ϕ and ϕn → ϕ in some topology X , try toconstruct yn such that |∇yn| ≤ ϕn and yn → y!
24/33 QVIs
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Mosco convergence for mixed constraints
Let L ∈ V ′ and consider the sequence of setsKn = v ∈ V : |v| ≤ ϕn & L(v) = αn
(Open problem) Nontrivial sufficient conditions on ϕn and αn so thatKn
M−−→ K holds are not known.
Unilateral and isoperimetric type constraints do not mix easily.
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Mosco convergence for mixed constraints
Let L ∈ V ′ and consider the sequence of setsKn = v ∈ V : |v| ≤ ϕn & L(v) = αn
(Open problem) Nontrivial sufficient conditions on ϕn and αn so thatKn
M−−→ K holds are not known.
Unilateral and isoperimetric type constraints do not mix easily.
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The contraction approach
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Contractions
Find y ≤ Φ(y) : ⟨A(y) − f, v − y⟩ ≥ 0, ∀v ≤ Φ(y).
If Φ : V → V is Lipschitz with small Lipschitz constant =⇒ Unique solution.
Find y : |∇y| ≤ Φ(y) & ⟨A(y) − f, v − y⟩ ≥ 0, ∀v : |∇v| ≤ Φ(y).
Theorem ([Hintermüller-R.(2013), Rodrigues-Santos(2019)])
Let V = H10(Ω), f ∈ L2(Ω), and suppose that Φ : V → L∞
ν (Ω) is defined as
Φ(y) = λ(y)ϕ
with ϕ ∈ L∞(Ω) and λ a Lipschitz continuous nonlinear functional on V . Then,provided ∥f∥L2 or Lλ are sufficiently small, the QVI of interest has a uniquesolution.
(Open problem) Extension to Φ(y) = λ1(y)ϕ1 + λ2(y)ϕ2 is not direct nor known.
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Contractions
Find y ≤ Φ(y) : ⟨A(y) − f, v − y⟩ ≥ 0, ∀v ≤ Φ(y).
If Φ : V → V is Lipschitz with small Lipschitz constant =⇒ Unique solution.
Find y : |∇y| ≤ Φ(y) & ⟨A(y) − f, v − y⟩ ≥ 0, ∀v : |∇v| ≤ Φ(y).
Theorem ([Hintermüller-R.(2013), Rodrigues-Santos(2019)])
Let V = H10(Ω), f ∈ L2(Ω), and suppose that Φ : V → L∞
ν (Ω) is defined as
Φ(y) = λ(y)ϕ
with ϕ ∈ L∞(Ω) and λ a Lipschitz continuous nonlinear functional on V . Then,provided ∥f∥L2 or Lλ are sufficiently small, the QVI of interest has a uniquesolution.
(Open problem) Extension to Φ(y) = λ1(y)ϕ1 + λ2(y)ϕ2 is not direct nor known.
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The contraction approachSome problems with the QVI literature
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Some problems with the literature
Unfortunately, a significant amount of the literature on QVIs attempts to extendthe famous Lions-Stampacchia result (for existence and design of algorithms)with an extreme assumption on the projection map v 7→ PK(v).
Suppose V is a Hilbert space, and consider
Find y ∈ K(y) : ⟨A(y) − f, v − y⟩ ≥ 0, ∀v ∈ K(y), (QVI)
then...
y solves (QVI) ⇐⇒ y = Bρ(y) := PK(y)(y − ρj(A(y) − f )).
K(v) ≡ K constant
[Lions-Stampacchia (1967)]
For some ρ > 0, Bρ is contractive=⇒ the (VI) has a unique solution.
For a closed, convex C ⊂ V
∥PC(v) − v∥V := infw∈C
∥w − v∥.
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Some problems with the literature
Unfortunately, a significant amount of the literature on QVIs attempts to extendthe famous Lions-Stampacchia result (for existence and design of algorithms)with an extreme assumption on the projection map v 7→ PK(v).
...(QVI) can be equivalently written (ρ > 0) as
Find y ∈ K(y) : (y − j (iy − ρ(A(y) − f ))) , v − y) ≥ 0, ∀v ∈ K(y),
where i : V → V ′ is the duality operator and j := i−1 the Riesz map.
y solves (QVI) ⇐⇒ y = Bρ(y) := PK(y)(y − ρj(A(y) − f )).
K(v) ≡ K constant
[Lions-Stampacchia (1967)]
For some ρ > 0, Bρ is contractive=⇒ the (VI) has a unique solution.
For a closed, convex C ⊂ V
∥PC(v) − v∥V := infw∈C
∥w − v∥.
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Some problems with the literature
Unfortunately, a significant amount of the literature on QVIs attempts to extendthe famous Lions-Stampacchia result (for existence and design of algorithms)with an extreme assumption on the projection map v 7→ PK(v).
...(QVI) can be equivalently written (ρ > 0) as
Find y ∈ K(y) : (y − j (iy − ρ(A(y) − f ))) , v − y) ≥ 0, ∀v ∈ K(y),
where i : V → V ′ is the duality operator and j := i−1 the Riesz map.
y solves (QVI) ⇐⇒ y = Bρ(y) := PK(y)(y − ρj(A(y) − f )).
K(v) ≡ K constant
[Lions-Stampacchia (1967)]
For some ρ > 0, Bρ is contractive=⇒ the (VI) has a unique solution.
For a closed, convex C ⊂ V
∥PC(v) − v∥V := infw∈C
∥w − v∥.
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Some problems with the literature
Unfortunately, a significant amount of the literature on QVIs attempts to extendthe famous Lions-Stampacchia result (for existence and design of algorithms)with an extreme assumption on the projection map v 7→ PK(v).
...(QVI) can be equivalently written (ρ > 0) as
Find y ∈ K(y) : (y − j (iy − ρ(A(y) − f ))) , v − y) ≥ 0, ∀v ∈ K(y),
where i : V → V ′ is the duality operator and j := i−1 the Riesz map.
y solves (QVI) ⇐⇒ y = Bρ(y) := PK(y)(y − ρj(A(y) − f )).
K(v) ≡ K constant
[Lions-Stampacchia (1967)]
For some ρ > 0, Bρ is contractive=⇒ the (VI) has a unique solution.
For a closed, convex C ⊂ V
∥PC(v) − v∥V := infw∈C
∥w − v∥.
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Some problems with the literature
If v 7→ K(v) not constant, to follow the same approach, the assumption used is
∥PK(y)(w) − PK(z)(w)∥V ≤ η∥y − z∥V (@@H)
for some 0 < η < 1 and all y, z, w in some sufficiently large set in V .
Non-expansiveness
∥PC(w1) − PC(w2)∥V ≤ ∥w1 − w2∥V , ∀w1, w2 ∈ V,
holds but it is not the above one!
The 1/2 exponent in the right hand side expression is optimal.
Examples (even in finite dimensions) can be found where equality holds
In Banach spaces like Lp(Ω) or ℓp(N), the exponent further degrades: it is1/p if 2 < p < +∞ and 1/p′ if 1 < p < 2.
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Some problems with the literature
If v 7→ K(v) not constant, to follow the same approach, the assumption used is
∥PK(y)(w) − PK(z)(w)∥V ≤ η∥y − z∥V (@@H)
for some 0 < η < 1 and all y, z, w in some sufficiently large set in V .
THEOREM ([Attouch-Wets])
Let V be a Hilbert space and K1, K2 any two closed, convex, non-empty subsetsof V . For y0 ∈ V , we have that
∥PK1(y0) − PK2(y0)∥V ≤ ρ1/2Hausρ(K1, K2)1/2
with ρ := ∥y0∥ + d(y0, K1) + d(y0, K2).
The 1/2 exponent in the right hand side expression is optimal.
Examples (even in finite dimensions) can be found where equality holds
In Banach spaces like Lp(Ω) or ℓp(N), the exponent further degrades: it is1/p if 2 < p < +∞ and 1/p′ if 1 < p < 2.
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Some problems with the literature
If v 7→ K(v) not constant, to follow the same approach, the assumption used is
∥PK(y)(w) − PK(z)(w)∥V ≤ η∥y − z∥V (@@H)
for some 0 < η < 1 and all y, z, w in some sufficiently large set in V .
THEOREM ([Attouch-Wets])
Let V be a Hilbert space and K1, K2 any two closed, convex, non-empty subsetsof V . For y0 ∈ V , we have that
∥PK1(y0) − PK2(y0)∥V ≤ ρ1/2Hausρ(K1, K2)1/2
with ρ := ∥y0∥ + d(y0, K1) + d(y0, K2).
The 1/2 exponent in the right hand side expression is optimal.
Examples (even in finite dimensions) can be found where equality holds
In Banach spaces like Lp(Ω) or ℓp(N), the exponent further degrades: it is1/p if 2 < p < +∞ and 1/p′ if 1 < p < 2.
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Some problems with the literature
What does the Attouch-Wets result imply for real applications?
Example: Gradient constraints
Let Ω = (0, 1) and V = v ∈ H1(Ω) : v(0) = 0.Let Ki := v ∈ V : |v′| ≤ ϕi with ϕ2, ϕ1 > 0 constant, so that
Hausρ(K1, K2) = |ϕ1 − ϕ2|,
for sufficiently large ρ > 0. So that for
K(w) = z ∈ V : |z′| ≤ Φ(w),
and Attouch-Wets imply if Φ : V → R is Lipschitz continuous
∥PK(y)(w) − PK(z)(w)∥V ≤ ρ1/2L1/2Φ ∥y − z∥1/2
V .
Are there examples for which
∥PK(y)(w) − PK(z)(w)∥V ≤ η∥y − z∥V ,
holds?
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Some problems with the literature
What does the Attouch-Wets result imply for real applications?
Example: Gradient constraints
Let Ω = (0, 1) and V = v ∈ H1(Ω) : v(0) = 0.Let Ki := v ∈ V : |v′| ≤ ϕi with ϕ2, ϕ1 > 0 constant, so that
Hausρ(K1, K2) = |ϕ1 − ϕ2|,
for sufficiently large ρ > 0. So that for
K(w) = z ∈ V : |z′| ≤ Φ(w),
and Attouch-Wets imply if Φ : V → R is Lipschitz continuous
∥PK(y)(w) − PK(z)(w)∥V ≤ ρ1/2L1/2Φ ∥y − z∥1/2
V .
Are there examples for which
∥PK(y)(w) − PK(z)(w)∥V ≤ η∥y − z∥V ,
holds?
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Some problems with the literature
Yes, there are, e.g., some obstacle cases with Φ : V → V Lipschitz with
K(w) := z ∈ V : z ≤ Φ(w).
Q: Is it worthwhile to use (@@H) in the QVI setting?
Let c > 0 and C > 0 be the coercivity and the bound for A : V → V ′, i.e.,
c∥v∥2V ≤ ⟨Av, v⟩ and ∥Av∥V ′ ≤ C∥v∥V , ∀v ∈ V.
The use of (@@H) provides existence,uniqueness and fixed point iterationfor (QVI). As a consequence, wehave that
2C
cLΦ < 1.
Contraction factor is ≥ 2Cc LΦ.
The change of variables y = y −Φ(y), transforms (QVI) to a (VI). Ex-istence, uniqueness, and a fixed pointiteration if
C
cLΦ < 1.
Contraction factor is Cc LΦ.
A: Not exactly...
32/33 QVIs
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Some problems with the literature
Yes, there are, e.g., some obstacle cases with Φ : V → V Lipschitz with
K(w) := z ∈ V : z ≤ Φ(w).
Q: Is it worthwhile to use (@@H) in the QVI setting?
Let c > 0 and C > 0 be the coercivity and the bound for A : V → V ′, i.e.,
c∥v∥2V ≤ ⟨Av, v⟩ and ∥Av∥V ′ ≤ C∥v∥V , ∀v ∈ V.
The use of (@@H) provides existence,uniqueness and fixed point iterationfor (QVI). As a consequence, wehave that
2C
cLΦ < 1.
Contraction factor is ≥ 2Cc LΦ.
The change of variables y = y −Φ(y), transforms (QVI) to a (VI). Ex-istence, uniqueness, and a fixed pointiteration if
C
cLΦ < 1.
Contraction factor is Cc LΦ.
A: Not exactly...
32/33 QVIs
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Some problems with the literature
Yes, there are, e.g., some obstacle cases with Φ : V → V Lipschitz with
K(w) := z ∈ V : z ≤ Φ(w).
Q: Is it worthwhile to use (@@H) in the QVI setting?
Let c > 0 and C > 0 be the coercivity and the bound for A : V → V ′, i.e.,
c∥v∥2V ≤ ⟨Av, v⟩ and ∥Av∥V ′ ≤ C∥v∥V , ∀v ∈ V.
The use of (@@H) provides existence,uniqueness and fixed point iterationfor (QVI). As a consequence, wehave that
2C
cLΦ < 1.
Contraction factor is ≥ 2Cc LΦ.
The change of variables y = y −Φ(y), transforms (QVI) to a (VI). Ex-istence, uniqueness, and a fixed pointiteration if
C
cLΦ < 1.
Contraction factor is Cc LΦ.
A: Not exactly...
32/33 QVIs
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Some problems with the literature
Yes, there are, e.g., some obstacle cases with Φ : V → V Lipschitz with
K(w) := z ∈ V : z ≤ Φ(w).
Q: Is it worthwhile to use (@@H) in the QVI setting?
Let c > 0 and C > 0 be the coercivity and the bound for A : V → V ′, i.e.,
c∥v∥2V ≤ ⟨Av, v⟩ and ∥Av∥V ′ ≤ C∥v∥V , ∀v ∈ V.
The use of (@@H) provides existence,uniqueness and fixed point iterationfor (QVI). As a consequence, wehave that
2C
cLΦ < 1.
Contraction factor is ≥ 2Cc LΦ.
The change of variables y = y −Φ(y), transforms (QVI) to a (VI). Ex-istence, uniqueness, and a fixed pointiteration if
C
cLΦ < 1.
Contraction factor is Cc LΦ.
A: Not exactly...
32/33 QVIs
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Some problems with the literature
Yes, there are, e.g., some obstacle cases with Φ : V → V Lipschitz with
K(w) := z ∈ V : z ≤ Φ(w).
Q: Is it worthwhile to use (@@H) in the QVI setting?
Let c > 0 and C > 0 be the coercivity and the bound for A : V → V ′, i.e.,
c∥v∥2V ≤ ⟨Av, v⟩ and ∥Av∥V ′ ≤ C∥v∥V , ∀v ∈ V.
The use of (@@H) provides existence,uniqueness and fixed point iterationfor (QVI). As a consequence, wehave that
2C
cLΦ < 1.
Contraction factor is ≥ 2Cc LΦ.
The change of variables y = y −Φ(y), transforms (QVI) to a (VI). Ex-istence, uniqueness, and a fixed pointiteration if
C
cLΦ < 1.
Contraction factor is Cc LΦ.
A: Not exactly...
32/33 QVIs
![Page 66: Introduction to Quasi-variational Inequalities in Hilbert Spaces …math.gmu.edu/~crautenb/ContentHP/Prints/Lecture1.pdf · 2020-03-04 · Introduction to Quasi-variational Inequalities](https://reader033.vdocuments.us/reader033/viewer/2022050515/5f9ef56218825b2cd92f2eea/html5/thumbnails/66.jpg)
Thanks for your attention!
33/33 QVIs