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Polyhedricity of convex sets in sobolev space

Published online by Cambridge University Press:  22 January 2016

Murali Rao  and
Jan Sokołowski
Murali Rao
Affiliation:
Department of Mathematics, University of Florida, 201 Walker Hall, Gainesville, FL 32611, USA
Jan Sokołowski
Affiliation:
Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland
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We provide results on differential stability of metric projection in Sobolev space onto convex set

(1.1)

where Ω ⊂ Rd is open, bounded domain.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 130 , June 1993 , pp. 101 - 110
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

[1] Adams, R. A., Sobolev Spaces, Academic Press, New York (1975).Google Scholar
[2] Ancona, A., Une propriete des espaces de Sobolev, C. R. Acad. Sc. Paris, t. 292, Serie I. 477–480.Google Scholar
[3] Brezis, H. and Browder, F., Some properties of higher order Sobolev spaces, J. Math. Pures Appl., 61 (1982), 245–259.Google Scholar
[4] Coffman, C. V. and Grover, C. L., Obtuse cones in Hilbert spaces and applications to partial differential equations, J. Funct. Anal., 35 (1980), 369–396.Google Scholar
[5] Haraux, A., How to differentiate the projection on a convex set in Hilbert space, Some applications to variational inequalities, J. Math. Soc. Japan, 29 (1977), 615–631.Google Scholar
[6] Haug, E. J. and Cea, J. (EDS.), Optimization of Distributed Parameter Structures, Sijthoff and Noordhoff, Alpen aan den Rijn, The Netherlands, (1981).Google Scholar
[7] Hedberg, L. L., Spectral synthesis in Sobolev spaces, and uniqueness of solutions of Dirichlet problem, Acta Math., 147 (1981), 237–264.Google Scholar
[8] Mignot, F., Controle dans les inequations variationelles elliptiques, J. Funct. Anal., 22 (1976), 25–39.CrossRefGoogle Scholar
[9] Rao, M. and Sokołowski, J., Sensitivity of unilateral problems in and applications, to appear.Google Scholar
[10] Rao, M. and Sokołowski, J., Shape sensitivity analysis of state constrained optimal control problems for distributed parameter systems, Lecture Notes in Control and Information Sciences, Vol. 114, Springer Veriag, (1989), 236–245.Google Scholar
[11] Rao, M. and Sokołowski, J., Differential stability of solutions to parametric optimization problems, to appear.Google Scholar
[12] Sokołowski, J., Differential stability of solutions to constrained optimization problems, Appl. Math. Optim., 13 (1985), 97–115.Google Scholar
[13] Sokołowski, J., Sensitivity analysis of control constrained optimal control problems for distributed parameter systems, SIAM J. Control Optim., 25 (1987), 1542–1556.Google Scholar
[14] Sokołowski, J., Shape sensitivity analysis of boundary optimal control problems for parabolic systems, SIAM J. Control Optim., 26 (1988), 763–787.Google Scholar
[15] Sokołowski, J., Stability of solutions to shape optimization problems, to appear.Google Scholar
[16] Sokołowski, J. and Zolesio, J. P., Shape sensitivity analysis of unilateral problems. SIAM J. Math. Anal, 18 (1987), 1416–1437.Google Scholar
[17] Sokołowski, J., Introduction to Shape Optimization. Shape sensitivity analysis, to appear.Google Scholar
[18] Zarantonello, F. H., Projections on convex sets in Hilbert space and spectral theory, In: Contributions to Nonlinear Functional Analysis, Publ. No. 27, Math. Res. Center. Univ. Wisconsin, Madison, Academic Press, New York, (1971), 237–424.Google Scholar
[19] Ziemer, P. W., Weakly Differentiable Functions, Springer Verlag, New York, 1989.Google Scholar
[20] Ziemer, P. W., Private communication.Google Scholar