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December  2020, 13(12): 3491-3494. doi: 10.3934/dcdss.2020112

A Zaremba-type criterion for hypoelliptic degenerate Ornstein–Uhlenbeck operators

Alessia E. Kogoj

Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica, 13 - 61029 Urbino (PU), Italy

Dedicated to Gisèle Ruiz Goldstein on the occasion of her 60th birthday

Received  January 2019 Revised  March 2019 Published  December 2020 Early access  October 2019

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.

Keywords: Ornstein–Uhlenbeck operators, Dirichlet problem, Perron–Wiener solution, boundary regularity, exterior cone criterion, hypoelliptic operators, axiomatic potential theory.
Mathematics Subject Classification: Primary: 35H10, 35J70, 35D99, 35J25; Secondary: 31D05.
Citation: Alessia E. Kogoj. A Zaremba-type criterion for hypoelliptic degenerate Ornstein–Uhlenbeck operators. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3491-3494. doi: 10.3934/dcdss.2020112
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

P. Negrini, Punti regolari per aperti cilindrici in uno spazio $\beta $-armonico, (Italian) Boll. Un. Mat. Ital. B (6), 2 (1983), 537–547.  Google Scholar

[13]

S. Zaremba, Sur le Principe de Dirichlet, (French) Acta Math., 34 (1911), 293-316.  doi: 10.1007/BF02393130.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

P. Negrini, Punti regolari per aperti cilindrici in uno spazio $\beta $-armonico, (Italian) Boll. Un. Mat. Ital. B (6), 2 (1983), 537–547.  Google Scholar

[13]

S. Zaremba, Sur le Principe de Dirichlet, (French) Acta Math., 34 (1911), 293-316.  doi: 10.1007/BF02393130.  Google Scholar

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