×

Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term. (English) Zbl 1175.35081

This article deals with regularity results for solutions of ultraparabolic equations. More precisely the following equation in \(\mathbb{R}^{N+1}\) is considered: \[ {\mathcal L}_vu:-{\mathcal L}_0u+{\mathcal V}u= 0\tag{1} \] where \({\mathcal L}_0\) is a linear second order operator of the form \[ {\mathcal L}_0=\sum^m_{k=1}X^2_k+X_0-\partial_t \tag{2} \] the terms \(X_k\) denote smooth vector fields on \(\mathbb{R}^N\) i.e. \[ X_k(x)= \sum^N_{j-1}a^k_j(x)\partial_{c_j}\;k=0,1,\dots,m.\tag{3} \] Finally \({\mathcal V}\) is a singular potential belonging to a Stummel-Kato class defined by the fundamental solution \(\Gamma_0\) of \({\mathcal L}_0\). The main result is an invariant Harnack inequality for positive solutions of (1) that extends previous results of A. Kogoj and E. Lanconelli for the solutions of the equation \({\mathcal L}_0u=0\). The proof, inspired by some arguments by Safonov, is based on pointwise lower bounds for the Green function \(G_0\) of \({\mathcal L}_0\) related to suitable “cylindrical” open sets. The Green function for the operator \({\mathcal L}_\nu\) is also constructed by the Levi parametrix method and \(L^p\) estimates and pointwise lower bounds are proved. An uniqueness result for the Cauchy-Dirichlet problem is also obtained.

MSC:

35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35J10 Schrödinger operator, Schrödinger equation
35K20 Initial-boundary value problems for second-order parabolic equations
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
35B65 Smoothness and regularity of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI Euclid EuDML

References:

[1] Alexopoulos, G. K.: Sub-Laplacians with drift on Lie groups of polynomial volume growth. Mem. Amer. Math. Soc. 155 , no. 739, (2002). · Zbl 0994.22006
[2] Bonfiglioli, A., Lanconelli, E. and Uguzzoni, F.: Stratified Lie groups and potential theory for their sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin, 2007. · Zbl 1128.43001
[3] Bony, J. M.: Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble) 19 (1969), 277-304. · Zbl 0176.09703 · doi:10.5802/aif.319
[4] Bramanti, M. and Brandolini, L.: Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDEs. Rev. Mat. Iberoamericana 21 (2005), no. 2, 511-556. · Zbl 1082.35060 · doi:10.4171/RMI/428
[5] Citti, G., Garofalo, N. and Lanconelli, E.: Harnack’s inequality for sum of squares of vector fields plus a potential. Amer. J. Math. 115 (1993), no. 3, 699-734. JSTOR: · Zbl 0795.35018 · doi:10.2307/2375077
[6] Fabes, E. B. and Stroock, D. W.: The \(L^p-\)integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations. Duke Math. J. 51 (1984), no. 4, 997-1016. · Zbl 0567.35003 · doi:10.1215/S0012-7094-84-05145-7
[7] Fabes, E. B. and Stroock, D. W.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. 96 (1986), no. 4, 327-338. · Zbl 0652.35052 · doi:10.1007/BF00251802
[8] Folland, G. B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13 (1975), no. 2, 161-207. · Zbl 0312.35026 · doi:10.1007/BF02386204
[9] Garofalo, N. and Lanconelli, E.: Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type. Trans. Amer. Math. Soc. 321 (1990), no. 2, 775-792. JSTOR: · Zbl 0719.35007 · doi:10.2307/2001585
[10] Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119 (1967), 147-171. · Zbl 0156.10701 · doi:10.1007/BF02392081
[11] Kogoj, A. and Lanconelli, E.: An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations. Med. J. Math. 1 (2004), no. 1, 51-80. · Zbl 1150.35354 · doi:10.1007/s00009-004-0004-8
[12] Kogoj, A. and Lanconelli, E.: One-side Liouville theorems for a class of hypoelliptic ultraparabolic equations. In Geometric Analysis of PDE and Several Complex Variables , 305-312. Contemp. Math. 368 . Amer. Math. Soc., Providence, RI, 2005. · Zbl 1073.35068
[13] Kogoj, A. and Lanconelli, E.: Link of groups and homogeneous Hörmander operators. Proc. Amer. Math. Soc. 135 (2007), no. 7, 2019-2030. · Zbl 1170.35034 · doi:10.1090/S0002-9939-07-08646-7
[14] Kuptsov, L. P.: On parabolic means. Dokl. Acad. Nauk SSSR 252 (1980), no. 2, 296-301. · Zbl 0484.35051
[15] Kusuoka, S. and Stroock, D. W.: Applications of the Malliavin calculus, III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 391-442. · Zbl 0633.60078
[16] Lanconelli, E., Pascucci, A. and Polidoro, S.: Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance. In Nonlinear problems in mathematical physics and related topics, II , 243-265. Int. Math. Ser. 2 . Kluwer/Plenum, New York, 2002. · Zbl 1032.35114
[17] Lanconelli, E. and Pascucci, A.: On the fundamental solution for hypoelliptic second order partial differential equations with non-negative characteristic form. Ricerche Mat. 48 (1999), no. 1, 81-106. · Zbl 0965.35005
[18] Lanconelli, E. and Polidoro, S.: On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), no. 1, 29-63. · Zbl 0811.35018
[19] Lu, G.: On Harnack’s inequality for a class of strongly degenerate Schrödinger operators formed by vector fields. Differential Integral Equations 7 (1994), no. 1, 73-100. · Zbl 0827.35032
[20] Montanari, A.: Harnack inequality for totally degenerate Kolmogorov-Fokker-Planck operators. Boll. Un. Mat. Ital. (7) 10 (1996), no. 4, 903-926. · Zbl 0884.35072
[21] Nagel, A., Stein, E. M. and Wainger, S.: Balls and metrics defined by vector fields. I. Basic properties. Acta Math. 155 (1985), no. 1-2, 103-147. · Zbl 0578.32044 · doi:10.1007/BF02392539
[22] Pascucci, A. and Polidoro, S.: Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators. Trans. Amer. Math. Soc. 358 (2006), no. 11, 4873-4893. · Zbl 1172.35339 · doi:10.1090/S0002-9947-06-04050-5
[23] Polidoro, S. and Ragusa, M. A.: A Green function and regularity results for an ultraparabolic equation with a singular potential. Adv. Differential Equations 7 (2002), no. 11, 1281-1314. · Zbl 1207.35200
[24] Rothschild, L. P. and Stein, E. M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1977), no. 3-4, 247-320. · Zbl 0346.35030 · doi:10.1007/BF02392419
[25] Safanov, M. N.: Harnack’s inequality for elliptic equations and the Hölder property of their solutions. J. Soviet Mathematics 21 (1983), 851-863. · Zbl 0511.35029 · doi:10.1007/BF01094448
[26] Schechter, M.: Spectra of partial differential operators. North-Holland Series in Applied Mathematics and Mechanics 14 . North-Holland Publishing Co., Amsterdam, 1971. · Zbl 0225.35001
[27] Sturm, K. T.: Harnack’s inequality for parabolic operators with singular low order terms. Math. Z. 216 (1994), no. 4, 593-611. · Zbl 0808.35044 · doi:10.1007/BF02572341
[28] Varopoulos, N. Th., Saloff-Coste, L. and Coulhon, T.: Analysis and geometry on groups Cambridge Tracts in Mathematics 100 . Cambridge University Press, Cambridge, 1992. · Zbl 0813.22003
[29] Zhang, Q.: On a parabolic equation with a singular lower order term. Trans. Amer. Math. Soc. 348 (1996), no. 7, 2811-2844. JSTOR: · Zbl 0860.35044 · doi:10.1090/S0002-9947-96-01675-3
[30] Zhang, Q.: On a parabolic equation with a singular lower order term, II. The Gaussian bounds. Indiana Univ. Math. J. 46 (1997), no. 3, 989-1020. · Zbl 0909.35054 · doi:10.1512/iumj.1997.46.1112
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.