Geodesic
Approximate smoothings of locally Lipschitz functionals
Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 289-320.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

The paper deals with approximation of locally Lipschitz functionals. A concept of approximation, based on the idea of graph approximation of the generalized gradient, is discussed and the existence of such approximations for locally Lipschitz functionals, defined on open domains in $\mathbb{R}^{N}$, is proved. Subsequently, the procedure of a smooth normal approximation of the class of regular sets (containing e.g. convex and/or epi-Lipschitz sets) is presented.
L'articolo tratta il problema dell'approssimazione di funzionali localmente Lipschitziani. Viene proposto un concetto di approssimazione che si basa sull'idea dell'approssimazione in grafico del gradiente generalizzato. Si prova l'esistenza di tali approssimazioni per funzionali localmente Lipschitziani definiti in domini aperti di $\mathbb{R}^{N}$. Infine, si presenta un procedimento di approssimazione normale regolare di insiemi regolari (introdotti in [13]). 
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Ćwiszewski, Aleksander; Kryszewski, Wojciech. Approximate smoothings of locally Lipschitz functionals. Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 289-320. http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_2_a2/

[1] J.-P. Aubin, Optima and equilibria, Springer-Verlag, Berlin, Heidelberg 1993. | MR | Zbl

[2] J.-P. Aubin-I. Ekeland, Applied Nonlinear Analysis, Wiley, New York 1986. | MR | Zbl

[3] J.-P. Aubin-H. Frankowska, Set-valued Analysis, Birkhäuser, Boston 1991. | MR | Zbl

[4] R. Bader-W. Kryszewski, On the solution sets of differential inclusions and the periodic problem, Set-valued an 9, no. 3 (2001), 289-313. | MR | Zbl

[5] J. Benoist, Approximation and regularization of arbitrary sets in finite dimension, Set Valued Anal., 2 (1994), 95-115. | MR | Zbl

[6] J.-M. Boniseau-B. Cornet, Fixed point theorem and Morse's lemma for Lipschitzian functions, J. Math. Anal. Appl., 146 (1990), 318-322. | MR | Zbl

[7] J. M. Borwein-Q. J. Zhu, Multifunctional and functional analytic techniques in nonsmooth analysis, in Nonlinear Analysis, Differential Equations and Control (F. H. Clarke and R. J. Stern, eds.), Kluwer Acad. Publ. (1999), 61-157. | MR | Zbl

[8] A. Cellina, Approximation of set-valued functions and fixed point theorems, Ann. Mat. Pura Appl., 82 (1969), 17-24. | MR | Zbl

[9] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York 1983. | MR | Zbl

[10] F. H. Clarke-Yu. S. Ledyaev-R. J. Stern, Complements, approximations, smoothings and invariance properties, J. Convex Anal., 4 (1997), 189-219. | MR | Zbl

[11] B. Cornet-M.-O. Czarnecki, Représentations lisses de sous-ensemble épi-lipschitziens de $\mathbb{R}^n$, C. R. Acad. Paris Sèr. I, 325 (1997), 475-480. | MR | Zbl

[12] B. Cornet-M.-O. Czarnecki, Smooth normal approximations of epi-Lipschitzian subsets of $\mathbb{R}^N$, to appear in SIAM J. Control Opt. | MR | Zbl

[13] A. Ćwiszewski-W. Kryszewski, Equilibria of Set-Valued Maps: a variational approach, Nonlinear Anal., 48 (2002), 707-746. | MR | Zbl

[14] A. Ćwiszewski-W. Kryszewski, Partial differential equations with discontinuous nonlinearities – approximation approach, in preparation.

[15] R. E. Edwards, Functional analysis, Theory and applications, Holt, Rinehart and Winston, New York 1965. | MR | Zbl

[16] L. C. Evans, Partial differential equations, Graduate Studies in Math., Vol. 19, American Math. Soc. 1998. | Zbl

[17] L. Górniewicz, Topological approach to differential inclusions, Topological Methods in Differential Equations and Inclusions, (eds. A. Granas, M. Frigon), NATO ASI Series, Kluwer Acad. Publ. 1995, 129-190. | MR | Zbl

[18] H. Hartman, Ordinary differential equations, Birkhäuser, Boston 1982. | MR | Zbl

[19] W. Kryszewski, Graph-approximation of set-valued maps on noncompact domains, Topology and Appl., 83 (1998), 1-21. | MR | Zbl

[20] W. Kryszewski, Graph approximation of set-valued maps. A survey, Differential Inclusions and Optimal Control, Lecture Notes in Nonlinear Analysis 2, J. P. Schauder Center for Nonlinear Studies Publ., Toruń 1998, 223-235. | Zbl

[21] W. Kryszewski, Homotopy properties of set-valued mappings, The Nicholas Copernicus University, Toruń 1997.

[22] R. Narasimhan, Analysis on Complex Manifolds, Masson & Cie (Paris), North Holland, Amsterdam 1968. | Zbl

[23] S. Plaskacz, Periodic solutions of differential inclusions on compact subsets of $\mathbb{R}^n$, J. Math. Anal. Appl., 148 (1990), 202-212. | MR | Zbl

[24] R. T. Rockafellar, Clarke's tangent cones and boundaries of closed sets in Rn, Nonlinear Anal., 3 (1979), 145-154. | MR | Zbl

[25] J. Schwartz, Nonlinear functional analysis, Gordon & Breach, New York 1969. | Zbl

[26] J. Warga, Derivate containers, inverse functions, and controllability, in Calculus of Variations and Control Theory, D. L. Russell, ed., Academic Press, New York 1976, 13-46. | MR | Zbl

[27] J. Warga, Optimal Control of Differential and Functional Equations; Chap. XI (Russian Translation) Nauka, Moscow 1978. | Zbl

[28] J. Warga, Fat homeomorphisms and unbounded derivare containers, J. Math. Anal. Appl., 81 (1981), 545-560. | MR | Zbl

[29] M. Willem, Minimax Theorems, Birkhäuser, Boston 1996. | MR | Zbl

[30] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89. | MR | Zbl