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A Zaremba-type criterion for hypoelliptic degenerate Ornstein–Uhlenbeck operators
Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica, 13 - 61029 Urbino (PU), Italy |
We prove a cone-type criterion for a boundary point to be regular for the Dirichlet problem related to (possibly) degenerate Ornstein–Uhlenbeck operators in $ \mathbb{R}^N $. Our result extends the classical Zaremba cone criterion for the Laplace operator.
References:
| [1] |
K. Beauchard and K. Pravda-Starov,
Null-controllability of non-autonomous Ornstein–Uhlenbeck equations, J. Math. Anal. Appl., 456 (2017), 496-524.
doi: 10.1016/j.jmaa.2017.07.014. |
| [2] |
J.-M. Bony,
Principe du maximum, inégalité de Harnack et unicité du problèeme de Cauchy pour les opérateurs elliptiques dégénérés, (French)Ann. Inst. Fourier (Grenoble), 19 (1969), 277-304.
doi: 10.5802/aif.319. |
| [3] |
M. Bramanti, G. Cupini, E. Lanconelli and E. Priola,
Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators, Math. Z., 266 (2010), 789-816.
doi: 10.1007/s00209-009-0599-3. |
| [4] |
M. Bramanti, G. Cupini, E. Lanconelli and E. Priola,
Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients, Math. Nachr., 286 (2013), 1087-1101.
doi: 10.1002/mana.201200189. |
| [5] |
C. Cinti and E. Lanconelli,
Riesz and Poisson-Jensen representation formulas for a class of ultraparabolic operators on Lie groups, Potential Anal., 30 (2009), 179-200.
doi: 10.1007/s11118-008-9112-6. |
| [6] |
B. Farkas and A. Lunardi, Maximal regularity for Kolmogorov operators in $L^2$ spaces with respect to invariant measures, J. Math. Pures Appl. (9), 86 (2006), 310–321.
doi: 10.1016/j.matpur.2006.06.002. |
| [7] |
S. Fornaro and A. Rhandi,
On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$-spaces, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 5049-5058.
doi: 10.3934/dcds.2013.33.5049. |
| [8] |
G. R. Goldstein, J. A. Goldstein and A. Rhandi,
Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential., Appl. Anal., 91 (2012), 2057-2071.
doi: 10.1080/00036811.2011.587809. |
| [9] |
A. E. Kogoj,
On the Dirichlet problem for hypoelliptic evolution equations: Perron–Wiener solution and a cone-type criterion, J. Differential Equations, 262 (2017), 1524-1539.
doi: 10.1016/j.jde.2016.10.018. |
| [10] |
A. E. Kogoj and S. Polidoro,
Harnack inequality for hypoelliptic second order partial differential operators, Potential Anal., 45 (2016), 545-555.
doi: 10.1007/s11118-016-9557-y. |
| [11] |
E. Lanconelli and S. Polidoro,
On a class of hypoelliptic evolution operators, Partial differential equations, Ⅱ (Turin, 1993), Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63.
|
| [12] |
P. Negrini, Punti regolari per aperti cilindrici in uno spazio $\beta $-armonico, (Italian) Boll. Un.
Mat. Ital. B (6), 2 (1983), 537–547. |
| [13] |
S. Zaremba,
Sur le Principe de Dirichlet, (French) Acta Math., 34 (1911), 293-316.
doi: 10.1007/BF02393130. |
show all references
References:
| [1] |
K. Beauchard and K. Pravda-Starov,
Null-controllability of non-autonomous Ornstein–Uhlenbeck equations, J. Math. Anal. Appl., 456 (2017), 496-524.
doi: 10.1016/j.jmaa.2017.07.014. |
| [2] |
J.-M. Bony,
Principe du maximum, inégalité de Harnack et unicité du problèeme de Cauchy pour les opérateurs elliptiques dégénérés, (French)Ann. Inst. Fourier (Grenoble), 19 (1969), 277-304.
doi: 10.5802/aif.319. |
| [3] |
M. Bramanti, G. Cupini, E. Lanconelli and E. Priola,
Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators, Math. Z., 266 (2010), 789-816.
doi: 10.1007/s00209-009-0599-3. |
| [4] |
M. Bramanti, G. Cupini, E. Lanconelli and E. Priola,
Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients, Math. Nachr., 286 (2013), 1087-1101.
doi: 10.1002/mana.201200189. |
| [5] |
C. Cinti and E. Lanconelli,
Riesz and Poisson-Jensen representation formulas for a class of ultraparabolic operators on Lie groups, Potential Anal., 30 (2009), 179-200.
doi: 10.1007/s11118-008-9112-6. |
| [6] |
B. Farkas and A. Lunardi, Maximal regularity for Kolmogorov operators in $L^2$ spaces with respect to invariant measures, J. Math. Pures Appl. (9), 86 (2006), 310–321.
doi: 10.1016/j.matpur.2006.06.002. |
| [7] |
S. Fornaro and A. Rhandi,
On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$-spaces, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 5049-5058.
doi: 10.3934/dcds.2013.33.5049. |
| [8] |
G. R. Goldstein, J. A. Goldstein and A. Rhandi,
Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential., Appl. Anal., 91 (2012), 2057-2071.
doi: 10.1080/00036811.2011.587809. |
| [9] |
A. E. Kogoj,
On the Dirichlet problem for hypoelliptic evolution equations: Perron–Wiener solution and a cone-type criterion, J. Differential Equations, 262 (2017), 1524-1539.
doi: 10.1016/j.jde.2016.10.018. |
| [10] |
A. E. Kogoj and S. Polidoro,
Harnack inequality for hypoelliptic second order partial differential operators, Potential Anal., 45 (2016), 545-555.
doi: 10.1007/s11118-016-9557-y. |
| [11] |
E. Lanconelli and S. Polidoro,
On a class of hypoelliptic evolution operators, Partial differential equations, Ⅱ (Turin, 1993), Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63.
|
| [12] |
P. Negrini, Punti regolari per aperti cilindrici in uno spazio $\beta $-armonico, (Italian) Boll. Un.
Mat. Ital. B (6), 2 (1983), 537–547. |
| [13] |
S. Zaremba,
Sur le Principe de Dirichlet, (French) Acta Math., 34 (1911), 293-316.
doi: 10.1007/BF02393130. |
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