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May  2018, 38(5): 2467-2485. doi: 10.3934/dcds.2018102

Wiener-Landis criterion for Kolmogorov-type operators

Alessia E. Kogoj 1,, , Ermanno Lanconelli 2, and  Giulio Tralli 3,

1. 

Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino "Carlo Bo", Piazza della Repubblica, 13 - 61029 Urbino (PU), Italy
2. 

Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato, 5 - 40126 Bologna, Italy
3. 

Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5 - 00185 Roma, Italy

* Corresponding author: Alessia E. Kogoj

Received  February 2017 Revised  November 2017 Published  March 2018

We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Gariepy's Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials.

Keywords: Kolmogorov operators, potential analysis, Perron-Wiener solution, boundary regularity, Wiener test.
Mathematics Subject Classification: Primary: 35H10, 31C15; Secondary: 35K65.
Citation: Alessia E. Kogoj, Ermanno Lanconelli, Giulio Tralli. Wiener-Landis criterion for Kolmogorov-type operators. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2467-2485. doi: 10.3934/dcds.2018102
References:
[1]

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

[2]

C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, New York-Heidelberg, 1972, With a preface by H. Bauer, Die Grundlehren der mathematischen Wissenschaften, Band 158.  Google Scholar

[3]

L. C. Evans and R. F. Gariepy, Wiener's criterion for the heat equation, Arch. Rational Mech. Anal., 78 (1982), 293-314.  doi: 10.1007/BF00249583.  Google Scholar

[4]

E. B. Fabes, N. Garofalo and E. Lanconelli, Wiener's criterion for divergence form parabolic operators with C1-Dini continuous coefficients, Duke Math. J., 59 (1989), 191-232.  doi: 10.1215/S0012-7094-89-05906-1.  Google Scholar

[5]

N. Garofalo and E. Lanconelli, Wiener's criterion for parabolic equations with variable coefficients and its consequences, Trans. Amer. Math. Soc., 308 (1988), 811-836.   Google Scholar

[6]

N. Garofalo and E. Lanconelli, Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc., 321 (1990), 775-792.  doi: 10.1090/S0002-9947-1990-0998126-5.  Google Scholar

[7]

N. Garofalo and F. Segàla, Estimates of the fundamental solution and Wiener's criterion for the heat equation on the Heisenberg group, Indiana Univ. Math. J., 39 (1990), 1155-1196.  doi: 10.1512/iumj.1990.39.39053.  Google Scholar

[8]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.  Google Scholar

[9]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990.  Google Scholar

[10]

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

[11]

L.P. Kuptsov, Fundamental solutions for a class of second-order elliptic-parabolic equations, Differentcial'nye Uravnenija, 8 (1972), 1649-1660,1716.   Google Scholar

[12]

L.P. Kuptsov, Fundamental solutions of certain second-order degenerate parabolic equations, Math. Notes, 31 (1982), 283-289.   Google Scholar

[13]

E. Lanconelli, Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl. (4), 97 (1973), 83-114.  doi: 10.1007/BF02414910.  Google Scholar

[14]

E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63, Partial differential equations, Ⅱ (Turin, 1993).  Google Scholar

[15]

E. Lanconelli, G. Tralli and F. Uguzzoni, Wiener-type tests from a two-sided Gaussian bound, Ann. Mat. Pura Appl. (4), 196 (2017), 217-244.  doi: 10.1007/s10231-016-0570-y.  Google Scholar

[16]

E. Lanconelli and F. Uguzzoni, Potential analysis for a class of diffusion equations: A Gaussian bounds approach, J. Differential Equations, 248 (2010), 2329-2367.  doi: 10.1016/j.jde.2010.01.007.  Google Scholar

[17]

E.M. Landis, Necessary and sufficient conditions for the regularity of a boundary point for the Dirichlet problem for the heat equation, Dokl. Akad. Nauk SSSR, 185 (1969), 517-520.   Google Scholar

[18]

M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations, Adv. Differential Equations, 2 (1997), 831-866.   Google Scholar

[19]

V. Scornazzani, The Dirichlet problem for the Kolmogorov operator, Boll. Un. Mat. Ital. C (5), 18 (1981), 43-62.   Google Scholar

show all references

References:
[1]

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

[2]

C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, New York-Heidelberg, 1972, With a preface by H. Bauer, Die Grundlehren der mathematischen Wissenschaften, Band 158.  Google Scholar

[3]

L. C. Evans and R. F. Gariepy, Wiener's criterion for the heat equation, Arch. Rational Mech. Anal., 78 (1982), 293-314.  doi: 10.1007/BF00249583.  Google Scholar

[4]

E. B. Fabes, N. Garofalo and E. Lanconelli, Wiener's criterion for divergence form parabolic operators with C1-Dini continuous coefficients, Duke Math. J., 59 (1989), 191-232.  doi: 10.1215/S0012-7094-89-05906-1.  Google Scholar

[5]

N. Garofalo and E. Lanconelli, Wiener's criterion for parabolic equations with variable coefficients and its consequences, Trans. Amer. Math. Soc., 308 (1988), 811-836.   Google Scholar

[6]

N. Garofalo and E. Lanconelli, Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc., 321 (1990), 775-792.  doi: 10.1090/S0002-9947-1990-0998126-5.  Google Scholar

[7]

N. Garofalo and F. Segàla, Estimates of the fundamental solution and Wiener's criterion for the heat equation on the Heisenberg group, Indiana Univ. Math. J., 39 (1990), 1155-1196.  doi: 10.1512/iumj.1990.39.39053.  Google Scholar

[8]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.  Google Scholar

[9]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990.  Google Scholar

[10]

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

[11]

L.P. Kuptsov, Fundamental solutions for a class of second-order elliptic-parabolic equations, Differentcial'nye Uravnenija, 8 (1972), 1649-1660,1716.   Google Scholar

[12]

L.P. Kuptsov, Fundamental solutions of certain second-order degenerate parabolic equations, Math. Notes, 31 (1982), 283-289.   Google Scholar

[13]

E. Lanconelli, Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl. (4), 97 (1973), 83-114.  doi: 10.1007/BF02414910.  Google Scholar

[14]

E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63, Partial differential equations, Ⅱ (Turin, 1993).  Google Scholar

[15]

E. Lanconelli, G. Tralli and F. Uguzzoni, Wiener-type tests from a two-sided Gaussian bound, Ann. Mat. Pura Appl. (4), 196 (2017), 217-244.  doi: 10.1007/s10231-016-0570-y.  Google Scholar

[16]

E. Lanconelli and F. Uguzzoni, Potential analysis for a class of diffusion equations: A Gaussian bounds approach, J. Differential Equations, 248 (2010), 2329-2367.  doi: 10.1016/j.jde.2010.01.007.  Google Scholar

[17]

E.M. Landis, Necessary and sufficient conditions for the regularity of a boundary point for the Dirichlet problem for the heat equation, Dokl. Akad. Nauk SSSR, 185 (1969), 517-520.   Google Scholar

[18]

M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations, Adv. Differential Equations, 2 (1997), 831-866.   Google Scholar

[19]

V. Scornazzani, The Dirichlet problem for the Kolmogorov operator, Boll. Un. Mat. Ital. C (5), 18 (1981), 43-62.   Google Scholar

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