A counterexample to Schauder estimates for elliptic operators with unbounded coefficients

Priola, Enrico

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 12 (2001), p. 15-25 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We consider a homogeneous elliptic Dirichlet problem involving an Ornstein-Uhlenbeck operator in a half space $\mathbb{R}^{2}_{+}$ of $\mathbb{R}^{2}$. We show that for a particular initial datum, which is Lipschitz continuous and bounded on $\mathbb{R}^{2}_{+}$, the second derivative of the classical solution is not uniformly continuous on $\mathbb{R}^{2}_{+}$. In particular this implies that the well known maximal Hölder-regularity results fail in general for Dirichlet problems in unbounded domains involving unbounded coefficients.

Résumé

Si considera un problema ellittico di Dirichlet in un semispazio $\mathbb{R}^{2}_{+}$ di $\mathbb{R}^{2}$. In esso compare un operatore di tipo Ornstein-Uhlenbeck. Si dimostra, con calcoli espliciti, che per un particolare dato iniziale lipschitziano la corrispondente soluzione classica non ha la derivata seconda uniformemente continua su $\mathbb{R}^{2}_{+}$. Questo risultato implica in particolare che le ben note stime di Schauder non valgono in generale per problemi di Dirichlet su domini illimitati se i coefficienti sono illimitati.

Détail

Publié le : 2001-03-01

MR 1898445 | Zbl 1072.35521

Bibliographie

[1] Aronson, D.G. - Besala, P., Parabolic Equations with Unbounded Coefficients. J. Diff. Eq., 3, 1967, 1-14. | MR 208160 | Zbl 0149.06804

[2] Besala, P., On the existence of a Foundamental Solution for a Parabolic Equation with Unbounded Coefficients. Ann. Polon. Math., 29, 1975, 403-409. | MR 422862 | Zbl 0305.35047

[3] Bochner, S., Harmonic Analysis and the theory of probability. California Monographs in Mathematical Science, University of California Press, Berkeley1955. | MR 72370 | Zbl 0068.11702

[4] Barucci, E. - Gozzi, F. - Vespri, V., On a semigroup approach to no-arbitrage pricing theory. Proceedings of the Seminar on Stochastic Analysis, Random Fields and Applications, Ascona, Switzerland, 1996. | Zbl 0933.91010

[5] 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

[6] Cannarsa, P. - Vespri, V., Generation of analytic semigroups by elliptic operators with unbounded coefficients. SIAM J. Math. Anal., 18, 1987, 857-872. | MR 883572 | Zbl 0623.47039

[7] Cerrai, S., Elliptic and parabolic equations in $\mathbb{R}^{n}$ with coefficients having polynomial growth. Comm. Part. Diff. Eqns., 21, 1996, 281-317. | MR 1373775 | Zbl 0851.35049

[8] 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

[9] Friedlin, M.I. - Wentzell, A.D., Random perturbation of dynamical systems. Springer-Verlag, Berlin 1983. | Zbl 0522.60055

[10] Gilbarg, D. - Trudinger, N., Elliptic Partial Differential Equations of Second Order. 2nd ed., Springer-Verlag, Berlin 1983. | MR 737190 | Zbl 0361.35003

[11] Krylov, N.V., Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, A.M.S., 1996. | Zbl 0865.35001

[12] Krylov, N.V. - Röckner, M. - Zabczyk, J. - Da Prato, G., Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions. Lect. Notes in Math., 1715, Springer-Verlag, 1999.

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

[14] Lorenzi, L., Schauder estimates for the Ornstein-Uhlenbeck semigroup in spaces of functions with polynomial and exponential growth. Dynamic Syst. and Appl., 9, 2000, 199-220. | MR 1757896 | Zbl 0957.35057

[15] Lunardi, A., Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in $\mathbb{R}^{n}$. Ann. Sc. Norm. Sup. Pisa, s. 4, 24, 1997, 133-164. | MR 1475774 | Zbl 0887.35062

[16] Lunardi, A. - Vespri, V., Optimal $L^{\infty}$ and Schauder estimates for elliptic and parabolic operators with unbounded coefficients. In: G. Caristi - E. Mitidieri (eds.), Proc. Conf. Reaction-diffusion systems. Lecture notes in pure and applied mathematics, 194, M. Dekker, 1998, 217-239. | MR 1472521 | Zbl 0887.47034

[17] Priola, E. - Zambotti, L., New optimal regularity results for infinite dimensional elliptic equations. Boll. Un. Mat. It., (8) 3-B, 2000, 411-429. | MR 1769994 | Zbl 0959.35076

[18] Priola, E., On a Dirichlet Problem Involving an Ornstein-Uhlenbeck Operator. Scuola Normale Superiore di Pisa, settembre 2000, preprint. | MR 1953230 | Zbl 1218.35082

[19] Schilling, R.L., Subordination in the sense of Bochner and a related functional calculus. J. Austral. Math. Soc., Ser. A, 64, n. 3, 1998, 368-396. | MR 1623282 | Zbl 0920.47039

[20] Sheu, S.J., Solution of certain parabolic equations with unbounded coefficients and its application to non-linear filtering. Stochastics, 10, 1983, 31-46. | MR 714706 | Zbl 0533.60068

[21] 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