New optimal regularity results for infinite-dimensional elliptic equations
Priola, Enrico ; Zambotti, Lorenzo
Bollettino dell'Unione Matematica Italiana, Tome 3-A (2000), p. 411-429 / Harvested from Biblioteca Digitale Italiana di Matematica

In questo articolo si ottengono stime di Schauder di tipo nuovo per equazioni ellittiche infinito-dimensionali del secondo ordine con coefficienti Hölderiani a valori nello spazio degli operatori Hilbert-Schmidt. In particolare si mostra che la derivata seconda delle soluzioni è Hilbert-Schmidt.

Publié le : 2000-06-01
@article{BUMI_2000_8_3B_2_411_0,
     author = {Enrico Priola and Lorenzo Zambotti},
     title = {New optimal regularity results for infinite-dimensional elliptic equations},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {3-A},
     year = {2000},
     pages = {411-429},
     zbl = {0959.35076},
     mrnumber = {1769994},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2000_8_3B_2_411_0}
}
Priola, Enrico; Zambotti, Lorenzo. New optimal regularity results for infinite-dimensional elliptic equations. Bollettino dell'Unione Matematica Italiana, Tome 3-A (2000) pp. 411-429. http://gdmltest.u-ga.fr/item/BUMI_2000_8_3B_2_411_0/

[1] Bererezansky, Y. M.-Kondratiev, Y. G., Spectral Methods in Infinite-Dimensional Analysis, Voll. 1-2, Kluwer Academic Publishers (1995). | MR 1340626 | Zbl 0832.47001

[2] Cannarsa, P.-Da Prato, G., Infinite Dimensional Elliptic Equations with Hölder continuous coefficients, Advances in Differential Equations, 1, n. 3 (1996), 425-452. | MR 1401401 | Zbl 0926.35153

[3] Cannarsa, P.-Da Prato, G., Potential Theory in Hilbert Spaces, Sympos. Appl. Math., 54, Amer. Math. Soc., Providence (1998), 27-51. | MR 1492691 | Zbl 0898.31008

[4] Dalecky, Y. L.-Fomin, S. V., Measures and differential equations in infinite-dimensional space, Mathematics and its applications, Kluwer Academic Publishers, Dordrecht, Boston, London (1991). | MR 1140921 | Zbl 0753.46027

[5] Da Prato, G.-Lunardi, A., On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal., 131 (1995), 94-114. | MR 1343161 | Zbl 0846.47004

[6] Da Prato, G.-Zabczyk, J., Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications44, Cambridge University Press (1992). | MR 1207136 | Zbl 0761.60052

[7] Dunford, N.-Schwartz, J., Linear operators, Part II, Interscience, New York (1958). | Zbl 0084.10402

[8] Gross, L., Potential theory on Hilbert space, J. Funct. Anal., 1 (1967), 123-181. | MR 227747 | Zbl 0165.16403

[9] Ishii, H., Viscosity solutions of nonlinear second-order partial differential equations in Hilbert spaces, Comm. in Part. Diff. Eq., 18 (1993), 601-651. | MR 1214874 | Zbl 0812.35153

[10] Krylov, N. V.-Röckner, M.-Zabczyk, J., Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions, Lect. Notes in Math.1715, Springer (1999). | MR 1730228 | Zbl 0927.00037

[11] Kuo, H. H., Gaussian Measures in Banach Spaces, Lect. Notes in Math., 463, Springer Verlag (1975). | MR 461643 | Zbl 0306.28010

[12] Lions, P. L., Viscosity Solutions of fully nonlinear second-order equations and optimal stochastic control in infinite-dimentions. Part III. Uniqueness of viscosity solutions of general second order equations, J. Funct. Anal., 86 (1989), 1-18. | MR 1013931 | Zbl 0757.93084

[13] Lunardi, A., Analytic semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel (1995). | MR 1329547 | Zbl 0816.35001

[14] Ma, Z. M.-Rockner, M., Introduction to the Theory of (Non Symmetric) Dirichlet FormsSpringer-Verlag (1992). | Zbl 0826.31001

[15] Piech, A., A fundamental solution of the parabolic equation on Hilbert spaces, J. Funct. Anal., 3 (1969), 85-114. | MR 251588 | Zbl 0169.47103

[16] Priola, E., Schauder estimates for a homogeneous Dirichlet problem in a half space of a Hilbert space, Nonlinear Analysis, T.M.A., to appear. | MR 1822238 | Zbl 0992.35108

[17] Priola, E., Partial Differential Equations with infinitely many variables, Ph.D. Thesis. in Mathematics, Università di Milano (1999).

[18] Priola, E., On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions, Studia Mathematica, 136 (3) (1999), 271-295. | MR 1724248 | Zbl 0955.47024

[19] Stroock, D. W., Logarithmic Sobolev Inequalities for Gibbs States, Corso CIME, Springer-Verlag (1994). | MR 1292280 | Zbl 0801.60056

[20] Swiech, A., «Unbounded» second order partial differential equations in infinite dimensional Hilbert spaces, Comm. in Part. Diff. Eq., 19 (1994), 1999-2036. | MR 1301180 | Zbl 0812.35154

[21] Van Neerven, J. M. A. M.-Zabczyk, J., Norm discontinuity of Ornstein-Uhlenbeck semigroups, Semigroup Forum, to appear. | MR 1847653 | Zbl 0960.47024

[22] Triebel, H., Interpolation Theory, Function spaces, Differential Operators, North-Holland, Amsterdam (1986). | MR 503903 | Zbl 0387.46032

[23] Zambotti, L., Infinite-Dimensional Elliptic and Stochastic Equations with Hölder-Continuous Coefficients, Stochastic Anal. Appl., 17 (3) (1999), 487-508. | MR 1686963 | Zbl 0938.60051

[24] Zambotti, L., A new approach to existence and uniqueness for martingale problems in infinite dimensions, Prob. Theory and Rel. Fields., to appear. | Zbl 0963.60059