On the First Boundary Value Problem for Hypoelliptic Evolution Equations: Perron-Wiener Solutions and Cone-Type Criteria


  • Alessia E. Kogoj Università degli Studi di Salerno




Dirichlet problem, Perron-Wiener solution, Boundary behavior of Perron-Wiener solutions, Exterior cone criterion, Hypoelliptic operators, Potential theory


For every bounded open set Ω in RN+1, we study the first boundary problem for a wide class of hypoelliptic evolution operators. The operators are assumed to be endowed with a well behaved global fundamental solution that allows us to construct a generalized solution in the sense of Perron-Wiener of the Dirichlet problem. Then, we give a criterion of regularity for boundary points in terms of the behavior, close to the point, of the fundamental solution of the involved operator. We deduce exterior conetype criteria for operators of Kolmogorov-Fokker-Planck-type, for the heat operators and more general evolution invariant operators on Lie groups. Our criteria extend and generalize the classical parabolic-cone condition for the classical heat operator due to Effros and Kazdan. The results presented are contained in [K16].


H. Bauer. Harmonische Räume und ihre Potentialtheorie. Ausarbeitung einer im Sommersemester 1965 an der Universität Hamburg gehaltenen Vorlesung. Lecture Notes in Mathematics, No. 22. Springer-Verlag, Berlin-New York, 1966.

A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni. Stratified Lie groups and potential theory for their sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin, 2007.

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.

E. G. Effros and J. L. Kazdan. Applications of Choquet simplexes to elliptic and parabolic boundary value problems. J. Differential Equations, 8:95-134, 1970.

E. G. Effros and J. L. Kazdan. On the Dirichlet problem for the heat equation. Indiana Univ. Math. J., 20:683-693, 1970/1971.

N. Fabes E. B., Garofalo and E. Lanconelli. Wiener's criterion for divergence form parabolic operators with C1-Dini continuous coefficients. Duke Math. J., 59(1):191-232, 1989.

N. Garofalo and E. Lanconelli. Wiener's criterion for parabolic equations with variable coefficients and its consequences. Trans. Amer. Math. Soc., 308(2):811-836, 1988.

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(4):1155-1196, 1990.

A.E. Kogoj, On the Dirichlet Problem for hypoelliptic evolution equations: Perron-Wiener solution and a cone-type criterion. J. Differential Equations 2016, http://dx.doi.org/10.1016/j.jde.2016.10.018

H. Lebesgue. Sur des cas dimpossibilité du problème de Dirichlet ordinaire. Vie de la société (in the part C. R. Séances Soc. Math. France (1912)) Bull. Soc. Math. Fr. 41(17):1-62, 1913 (supplément éspecial).

E. Lanconelli and S. Polidoro. On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Politec. Torino, 52(1):29-63, 1994. Partial differential equations, II (Turin, 1993).

E. Lanconelli, G. Tralli, and F. Uguzzoni. Wiener-type tests from a two-sided gaussian bound. Annali di Matematica Pura ed Applicata, pages 1-28, 2016. Article in Press.

E. Lanconelli and F. Uguzzoni. Potential analysis for a class of diffusion equations: a Gaussian bounds approach. J. Differential Equations, 248(9):2329-2367, 2010.

M. Manfredini. The Dirichlet problem for a class of ultraparabolic equations. Adv. Differential Equations, 2(5):831-866, 1997.

P. Negrini and V. Scornazzani. Superharmonic functions and regularity of boundary points for a class of elliptic-parabolic partial differential operators. Boll. Un. Mat. Ital. C (6), 3(1):85-107, 1984.

O. Perron. Eine neue Behandlung der ersten Randwertaufgabe für u = 0. Math. Z., 18(1):42-54, 1923.

V. Scornazzani. The Dirichlet problem for the Kolmogorov operator. Boll. Un. Mat. Ital. C (5), 18(1):43-62, 1981.

Francesco Uguzzoni. Cone criterion for non-divergence equations modeled on Hörmander vector fields. In Subelliptic PDE's and applications to geometry and finance, volume 6 of Lect. Notes Semin. Interdiscip. Mat., pages 227-241. Semin. Interdiscip. Mat. (S.I.M.), Potenza, 2007.

S. Zaremba. Sur le principe de Dirichlet. Acta Math., 34(1):293-316, 1911.




How to Cite

Kogoj, A. E. (2016). On the First Boundary Value Problem for Hypoelliptic Evolution Equations: Perron-Wiener Solutions and Cone-Type Criteria. Bruno Pini Mathematical Analysis Seminar, 7(1), 116–128. https://doi.org/10.6092/issn.2240-2829/6694