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Cannarsa, Piermarco:
Funzioni semiconcave, singolarità e pile di sabbia
Bollettino dell'Unione Matematica Italiana Serie 8 8-B (2005), fasc. n.3, p. 549-567, Unione Matematica Italiana (Italian)
pdf (278 Kb), djvu (236 Kb). | MR2182415 | Zbl 1182.49011

Sunto

La semiconcavità è una nozione che generalizza quella di concavità conservandone la maggior parte delle proprietà ma permettendo di ampliarne le applicazioni. Questa è una rassegna dei punti più salienti della teoria delle funzioni semiconcave, con particolare riguardo allo studio dei loro insiemi singolari. Come applicazione, si discuterà una formula di rappresentazione per la soluzione di un modello dinamico per la materia granulare.
Referenze Bibliografiche
[1] P. ALBANO - P. CANNARSA, Structural properties of singularities of semiconcave functions, Annali Scuola Norm. Sup. Pisa Sci. Fis. Mat., 28 (1999), 719-740. | fulltext mini-dml | MR 1760538 | Zbl 0957.26002
[2] P. ALBANO - P. CANNARSA, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal., 162 (2002), 1-23. | MR 1892229 | Zbl 1043.35052
[3] G. ALBERTI - L. AMBROSIO - P. CANNARSA, On the singularities of convex functions, Manuscripta Math., 76 (1992), 421-435. | MR 1185029 | Zbl 0784.49011
[4] L. AMBROSIO, Optimal transport maps in Monge-Kantorovich problem, in Proceedings of the International Congress of Mathematicians, vol. III (Beijing 2002), Higher Ed. Press, Beijing, 2002, 131-140. | MR 1957525 | Zbl 1005.49030
[5] D. G. ARONSON, The porous medium equation, in Some problems on nonlinear diffusion (FasanoA. and PrimicerioM., Eds.), Lect. Notes Math. 1224, Springer, 1986, 1-46. | MR 877986 | Zbl 0626.76097
[6] M. BARDI - I. CAPUZZO DOLCETTA, Optimal control and viscosity solutions of Hamilton-Jacobi equations, Birkhäuser, Boston, 1997. | MR 1484411 | Zbl 0890.49011
[7] T. BHATTACHARYA - E. DI BENEDETTO - J. MANFREDI, (1991) Limits as $p \to \infty$ of $\Delta_p u_{p} = f$ and related extremal problems, Some topics in nonlinear PDEs (Turin, 1989). Rend. Sem. Mat. Univ. Politec. Torino, 1989, 15-68. | MR 1155453
[8] G. BOUCHITTÉ - G. BUTTAZZO, Characterization of optimal shapes and masses through Monge-Kantorovich equation, J. Eur. Math. Soc., 3, No. 2 (2001), 139-168. | MR 1831873 | Zbl 0982.49025
[9] T. BOUTREUX - P.-G. DE GENNES, Surface flows of granular mixtures, I. General principles and minimal model, J. Phys. I France, 6 (1996), 1295-1304.
[10] P. CANNARSA - P. CARDALIAGUET, Representation of equilibrium solutions to the table problem for growing sandpile, J. Eur. Math. Soc., 6 (2004), 1-30. | MR 2094399 | Zbl 1084.35015
[11] P. CANNARSA - P. CARDALIAGUET - G. CRASTA - E. GIORGIERI, A boundary value problem for a PDE model in mass transfer theory: representation of solutions and applications, pre-print. | Zbl 1089.35076
[12] P. CANNARSA - P. CARDALIAGUET - E. GIORGIERI, The table problem for granular matter: regularity of solutions, pre-print.
[13] P. CANNARSA - C. SINESTRARI, Semiconcave functions, Hamilton-Jacobi equations and optimal control, Birkhäuser, Boston, 2004. | MR 2041617 | Zbl 1095.49003
[14] H. I. CHOI - S. W. CHOI - H. P. MOON, Mathematical theory of medial axis transform, Pac. J. Math., 181 (1997), 57-88. | MR 1491036 | Zbl 0885.53004
[15] F. H. CLARKE, Optimization and nonsmooth analysis, Wiley, New York, 1983. | MR 709590 | Zbl 0582.49001
[16] A. DOUGLIS, The continuous dependence of generalized solutions of non-linear partial differential equations upon initial data, Comm. Pure Appl. Math., 14 (1961), 267-284. | MR 139848 | Zbl 0117.31102
[17] P. ERDÖS, Some remarks on the measurability of certain sets, Bull. Amer. Math. Soc., 51 (1945), 728-731. | fulltext mini-dml | MR 13776 | Zbl 0063.01269
[18] L. C. EVANS, Partial Differential Equations, A.M.S., Providence, 1998. | Zbl 1194.35001
[19] L. C. EVANS - M. FELDMAN - R. GARIEPY, Fast/slow diffusion and collapsing sandpiles, J. Differential Equations, 137, no. 1 (1997), 166-209. | MR 1451539 | Zbl 0879.35019
[20] L. C. EVANS - W. GANGBO, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137, no. 653, 1999. | MR 1464149 | Zbl 0920.49004
[21] FLEMING W. H., The Cauchy problem for a nonlinear first order partial differential equation, J. Diff. Eq., 5 (1969), 515-530. | MR 235269 | Zbl 0172.13901
[22] FLEMING W. H. - MCENEANEY W. M., A max-plus based algorithm for an HJB equation of nonlinear filtering, SIAM J. Control Optim., 38 (2000), 683-710. | MR 1741434 | Zbl 0949.35039
[23] W. H. FLEMING - H. M. SONER, Controlled Markov processes and viscosity solutions, Springer Verlag, Berlin, 1993. | MR 1199811 | Zbl 1105.60005
[24] J. H. G. FU, Tubular neighborhoods in Euclidean spaces, Duke Math. J., 52 (1985), 1025-1046. | fulltext mini-dml | MR 816398 | Zbl 0592.52002
[25] K. P. HADELER - C. KUTTLER, Dynamical models for granular matter, Granular Matter, 2 (1999), 9-18.
[26] J. ITOH - M. TANAKA, The Lipschitz continuity of the distance function to the cut locus, Trans. Am. Math. Soc., 353, No. 1 (2001), 21-40. | MR 1695025 | Zbl 0971.53031
[27] U. JANFALK, Behaviour in the limit, as $p \to +\infty$, of minimizers of functionals involving p-Dirichlet integrals, SIAM J. Math. Anal., 27, no. 2 (1996), 341-360. | MR 1377478 | Zbl 0853.35028
[28] S. N. KRUZHKOV, The Cauchy problem in the large for certain nonlinear first order differential equations, Soviet. Math. Dokl., 1 (1960), 474-477. | MR 121575 | Zbl 0128.32303
[29] S. N. KRUZHKOV, The Cauchy problem in the large for nonlinear equations and for certain quasilinear systems of the first order with several variables, Soviet. Math. Dokl., 5 (1964), 493-496. | Zbl 0138.34702
[30] S. N. KRUZHKOV, Generalized solutions of the Hamilton–Jacobi equations of the eikonal type I, Math. USSR Sb., 27 (1975), 406-445. | Zbl 0369.35012
[31] X. J. LI - J. M. YONG Optimal control theory for infinite-dimensional systems, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, 1995. | MR 1312364 | Zbl 0817.49001
[32] P. L. LIONS, Generalized solutions of Hamilton-Jacobi equations, Pitman, Boston, 1982. | MR 667669 | Zbl 0497.35001
[33] T. MOTZKIN, Sur quelques propriétés caractéristiques des ensembles convexes, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 21 (1935), 562-567. | Zbl 0011.41105
[34] Y. Y. LI - L. NIRENBERG, The distance function to the boundary, Finsler geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations, pre-print. | fulltext mini-dml | Zbl 1062.49021
[35] L. PRIGOZHIN, Variational model of sandpile growth, European J. Appl. Math., 7, no. 3 (1996), 225-235. | MR 1401168 | Zbl 0913.73079
[36] RIFFORD L., Existence of Lipschitz and semiconcave control-Lyapunov functions, SIAM J. Control Optim., 39 (2000), 1043-1064. | MR 1814266 | Zbl 0982.93068
[37] RIFFORD L., Semiconcave control-Lyapunov functions and stabilizing feedbacks, SIAM J. Control Optim., 41 (2002), 659-681. | MR 1939865 | Zbl 1034.93053
[38] ROCKAFELLAR R. T., Favorable classes of Lipschitz continuous functions in subgradient optimization, in Progress in Nondifferential Optimization (NurminskiE., Ed.), IIASA Collaborative Proceedings Series, Laxenburg, 125 (1982). | MR 704977 | Zbl 0511.26009
[39] L. ZAJAÍČEK, On the points of multiplicity of monotone operators, Comment. Math. Univ. Carolin., 19 (1978), 179-189. | fulltext mini-dml | MR 493541 | Zbl 0404.47025
[40] L. ZAJAÍČEK, On the differentiation of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J., 29 (1979), 340-348. | fulltext mini-dml | MR 536060 | Zbl 0429.46007

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