Priola, Enrico:
A counterexample to Schauder estimates for elliptic operators with unbounded coefficients (Un controesempio alle stime di Schauder per operatori ellittici con coefficienti illimitati)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni Serie 9 12 (2001), fasc. n.1, p. 15-25, (English)
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Sunto
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.
Referenze Bibliografiche
[1]
D.G. Aronson -
P. Besala,
Parabolic Equations with Unbounded Coefficients.
J. Diff. Eq.,
3,
1967, 1-14. |
MR 208160 |
Zbl 0149.06804[2]
P. Besala,
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]
S. Bochner,
Harmonic Analysis and the theory of probability.
California Monographs in Mathematical Science,
University of California Press, Berkeley
1955. |
MR 72370 |
Zbl 0068.11702[4]
E. Barucci -
F. Gozzi -
V. Vespri,
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]
P. Cannarsa -
G. Da Prato,
Infinite Dimensional Elliptic Equations with Hölder continuous coefficients.
Advances in Differential equations,
1, n. 3,
1996, 425-452. |
MR 1401401 |
Zbl 0926.35153[6]
P. Cannarsa -
V. Vespri,
Generation of analytic semigroups by elliptic operators with unbounded coefficients.
SIAM J. Math. Anal.,
18,
1987, 857-872. |
fulltext (doi) |
MR 883572 |
Zbl 0623.47039[7]
S. Cerrai,
Elliptic and parabolic equations in $\mathbb{R}^{n}$ with coefficients having polynomial growth.
Comm. Part. Diff. Eqns.,
21,
1996, 281-317. |
fulltext (doi) |
MR 1373775 |
Zbl 0851.35049[9]
M.I. Friedlin -
A.D. Wentzell,
Random perturbation of dynamical systems.
Springer-Verlag, Berlin
1983. |
Zbl 0522.60055[10]
D. Gilbarg -
N. Trudinger,
Elliptic Partial Differential Equations of Second Order. 2nd ed.,
Springer-Verlag, Berlin
1983. |
MR 737190 |
Zbl 0361.35003[11]
N.V. Krylov,
Lectures on Elliptic and Parabolic Equations in Hölder Spaces.
Graduate Studies in Mathematics,
A.M.S.,
1996. |
Zbl 0865.35001[12] N.V. Krylov - M. Röckner - J. Zabczyk - G. Da Prato (eds.), Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions. Lect. Notes in Math., 1715, Springer-Verlag, 1999.
[14]
L. Lorenzi,
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[16]
A. Lunardi -
V. Vespri,
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[19]
R.L. Schilling,
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]
S.J. Sheu,
Solution of certain parabolic equations with unbounded coefficients and its application to non-linear filtering.
Stochastics,
10,
1983, 31-46. |
fulltext (doi) |
MR 714706 |
Zbl 0533.60068[21]
L. Zambotti,
A new approach to existence and uniqueness for martingale problems in infinite dimensions.
Prob. Theory and Rel. Fields, to appear. |
Zbl 0963.60059
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