# Disuguaglianze di Harnack alla frontiera per equazioni di Kolmogorov

## DOI:

https://doi.org/10.6092/issn.2240-2829/2672## Abstract

We describe some recent results on the boundary regularity for hypoelliptic Kolmogorov equations. We prove boundary Harnack inequalities of the positive solutions to Kolmogorov equations vanishing on some relatively open subset of the boundary. Sufficient conditions for the boundary Harnack inequality are given in terms of cone conditions, that are satisfied by a wide class of Lipschitz domains. We also prove Carleson type estimates, that are scale-invariant and generalize previous results valid for second order uniformly parabolic equations.

## References

P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints, Ark. Mat., 22 (1984), pp. 153–173.

L. Caffarelli, E. Fabes, S. Mortola, S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J., 30 (1981), pp. 621–640.

L. Carleson, On the existence of boundary values for harmonic functions in several variables, Ark. Mat., 4 (1962), pp. 393–399 (1962).

C. Cinti, K. Nystrom, S. Polidoro, A boundary estimate for non-negative solutions to Kolmogorov operators in non-divergence form, Annali di Matematica Pura ed Applicata, 191(1), pp. 1-23, (2012).

----, A note on Harnack inequalities and propagation sets for a class of hypoelliptic operators, Potential Anal., 33 (2010), pp. 341–354.

----, A Carleson-type estimate in Lipschitz type domains for non-negative solutions to Kolmogorov operators, in corso di stampa su Annali SNS (2011).

D. Danielli, N. Garofalo, A. Petrosyan, The sub-elliptic obstacle problem: C1; regularity of the free boundary in Carnot groups of step two, Adv. Math., 211 (2007), pp. 485–516.

D. Danielli, N. Garofalo, S. Salsa, Variational inequalities with lack of ellipticity. I. Optimal interior regularity and non-degeneracy of the free boundary, Indiana Univ. Math. J., 52 (2003), pp. 361–398.

M. Di Francesco, A. Pascucci, S. Polidoro, The obstacle problem for a class of hypoelliptic ultraparabolic equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), pp. 155–176.

M. Di Francesco, S. Polidoro, Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov type operators in non-divergence form, Advances in Differential Equations, 11 (2006), pp. 1261–1320.

E. Fabes, N. Garofalo, S. Marn-Malave, S. Salsa, Fatou theorems for some nonlinear elliptic equations, Rev. Mat. Iberoamericana, 4 (1988), pp. 227–251.

E. B. Fabes, N. Garofalo, S. Salsa, A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illinois J. Math., 30 (1986), pp. 536–565.

E. B. Fabes, C. E. Kenig, Examples of singular parabolic measures and singular transition probability densities, Duke Math. J., 48 (1981), pp. 845–856.

E. B. Fabes, M. V. Safonov, Behavior near the boundary of positive solutions of second order parabolic equations, in Proceedings of the conference dedicated to Professor Miguel de Guzm´an (El Escorial, 1996), vol. 3, 1997, pp. 871–882.

E. B. Fabes, M. V. Safonov, Y. Yuan, Behavior near the boundary of positive solutions of second order parabolic equations. II, Trans. Amer. Math. Soc., 351 (1999), pp. 4947–4961.

E. B. Fabes, D. W. Stroock, A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal., 96 (1986), pp. 327–338.

F. Ferrari, B. Franchi, Geometry of the boundary and doubling property of the harmonic measure for Grushin type operators, Rend. Sem. Mat. Univ. Politec. Torino, 58 (2000), pp. 281–299 (2002). Partial differential operators (Torino, 2000).

----, A local doubling formula for the harmonic measure associated with subelliptic operators and applications, Comm. Partial Differential Equations, 28 (2003), pp. 1–60.

M. Frentz, K. Nystrom, A. Pascucci, S. Polidoro, Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options, Math. Ann., (2010), pp. 805–838.

N. Garofalo, Second order parabolic equations in nonvariational forms: boundary Harnack principle and comparison theorems for nonnegative solutions, Ann. Mat. Pura Appl. (4), 138 (1984), pp. 267–296.

S. Hofmann, J. L. Lewis, The Dirichlet problem for parabolic operators with singular drift terms, Mem. Amer. Math. Soc., 151 (2001), pp. viii+113.

L. Hormander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), pp. 147– 171.

D. S. Jerison, C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math., 46 (1982), pp. 80–147.

C. E. Kenig, J. Pipher, The Dirichlet problem for elliptic equations with drift terms, Publ. Mat., 45 (2001), pp. 199–217.

N. V. Krylov, M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), pp. 161–175, 239.

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

E. B. Lee, L. Markus, Foundations of optimal control theory, Robert E. Krieger Publishing Co. Inc., Melbourne, FL, second ed., 1986.

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

K. Nystrom, The Dirichlet problem for second order parabolic operators, Indiana Univ. Math. J., 46 (1997), pp. 183–245.

K. Nystrom, A. Pascucci, S. Polidoro, Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators, to appear in J. Differential Equations, (2010).

M. V. Safonov, Y. Yuan, Doubling properties for second order parabolic equations, Ann. of Math. (2), 150 (1999), pp. 313–327.

S. Salsa, Some properties of nonnegative solutions of parabolic differential operators, Ann. Mat. Pura Appl. (4), 128 (1981), pp. 193–206.

## Downloads

## Published

## How to Cite

*Bruno Pini Mathematical Analysis Seminar*,

*2*(1). https://doi.org/10.6092/issn.2240-2829/2672

## Issue

## Section

## License

Copyright (c) 2011 Sergio Polidoro

Copyrights and publishing rights of all the texts on this journal belong to the respective authors without restrictions.

This journal is licensed under a Creative Commons Attribution 4.0 International License. (full legal code)

See also our Open Access Policy.