Regularity properties of solutions of a class of ellipticparabolic nonlinear Levi type equations
Authors:
G. Citti and A. Montanari
Journal:
Trans. Amer. Math. Soc. 354 (2002), 28192848
MSC (2000):
Primary 35J70, 35K65; Secondary 22E30
Published electronically:
February 14, 2002
MathSciNet review:
1895205
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper we prove the smoothness of solutions of a class of ellipticparabolic nonlinear Levi type equations, represented as a sum of squares plus a vector field. By means of a freezing method the study of the operator is reduced to the analysis of a family of left invariant operators on a free nilpotent Lie group. The fundamental solution of the operator is used as a parametrix of the fundamental solution of the Levi operator, and provides an explicit representation formula for the solution of the given equation. Differentiating this formula and applying a bootstrap method, we prove that the solution is .
 [BG]
Eric
Bedford and Bernard
Gaveau, Envelopes of holomorphy of certain 2spheres in
𝐶², Amer. J. Math. 105 (1983),
no. 4, 975–1009. MR 708370
(84k:32016), http://dx.doi.org/10.2307/2374301
 [BK]
Eric
Bedford and Wilhelm
Klingenberg, On the envelope of holomorphy of a
2sphere in 𝐶², J. Amer. Math.
Soc. 4 (1991), no. 3, 623–646. MR 1094437
(92j:32034), http://dx.doi.org/10.1090/S08940347199110944370
 [CS]
E.
M. Chirka and N.
V. Shcherbina, Pseudoconvexity of rigid domains and foliations of
hulls of graphs, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)
22 (1995), no. 4, 707–735. MR 1375316
(98f:32016)
 [C]
G.
Citti, 𝐶^{∞} regularity of solutions of a quasilinear
equation related to the Levi operator, Ann. Scuola Norm. Sup. Pisa Cl.
Sci. (4) 23 (1996), no. 3, 483–529. MR 1440031
(98b:35072)
 [Csem]
G. Citti, Regolarità di grafici non Levipiatti, Seminario di Analisi, Dip. Mat. Univ. Bologna (A.A. 1996/97), Tecnoprint, Bologna, 186195.
 [CM1]
G. Citti, A. Montanari, Analytic estimates of solutions of the Levi equation, J. Differential Equations, 173, (2001), 356389.
 [CM2]
G. Citti, A. Montanari, regularity of solutions of an equation of Levi's type in , Annali di Matematica Pura Appl. (4), 180, (2001), 2758 CMP 2001:11
 [CPP]
G. Citti, A. Pascucci, S. Polidoro, On the regularity of solutions to a nonlinear ultraparabolic equation in mathematical finance, Diff. Int. Eq, 14 (2001), 701738.
 [F]
G.
B. Folland, Subelliptic estimates and function spaces on nilpotent
Lie groups, Ark. Mat. 13 (1975), no. 2,
161–207. MR 0494315
(58 #13215)
 [FS]
G.
B. Folland and E.
M. Stein, Estimates for the ∂_{𝑏} complex and analysis
on the Heisenberg group, Comm. Pure Appl. Math. 27
(1974), 429–522. MR 0367477
(51 #3719)
 [FL]
Bruno
Franchi and Ermanno
Lanconelli, Hölder regularity theorem for a class of linear
nonuniformly elliptic operators with measurable coefficients, Ann.
Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 4,
523–541. MR
753153 (85k:35094)
 [H]
Lars
Hörmander, Hypoelliptic second order differential
equations, Acta Math. 119 (1967), 147–171. MR 0222474
(36 #5526)
 [HK]
Gerhard
Huisken and Wilhelm
Klingenberg, Flow of real hypersurfaces by the trace of the Levi
form, Math. Res. Lett. 6 (1999), no. 56,
645–661. MR 1739222
(2001f:53141), http://dx.doi.org/10.4310/MRL.1999.v6.n6.a5
 [K]
W. Klingenberg, Real hypersurfaces in Kähler manifolds preprint.
 [M]
A. Montanari, Real hypersurfaces evolving by Levi curvature: smooth regularity of solutions to the parabolic Levi equation, to appear in Comm. Partial Differential Equations 26 (9&10) (2001) 16331664.
 [NSW]
Alexander
Nagel, Elias
M. Stein, and Stephen
Wainger, Balls and metrics defined by vector fields. I. Basic
properties, Acta Math. 155 (1985), no. 12,
103–147. MR
793239 (86k:46049), http://dx.doi.org/10.1007/BF02392539
 [RS]
Linda
Preiss Rothschild and E.
M. Stein, Hypoelliptic differential operators and nilpotent
groups, Acta Math. 137 (1976), no. 34,
247–320. MR 0436223
(55 #9171)
 [Sc]
N.
V. Shcherbina, On the polynomial hull of a graph, Indiana
Univ. Math. J. 42 (1993), no. 2, 477–503. MR 1237056
(95e:32017), http://dx.doi.org/10.1512/iumj.1993.42.42022
 [ST1]
Zbigniew
Slodkowski and Giuseppe
Tomassini, Weak solutions for the Levi equation and envelope of
holomorphy, J. Funct. Anal. 101 (1991), no. 2,
392–407. MR 1136942
(93c:32018), http://dx.doi.org/10.1016/00221236(91)90164Z
 [ST2]
Yongjing
Lü and Zhongtai
Ma, 𝐿^{𝑠}norm estimates of solutions for the
inhomogeneous CauchyRiemann equation, Chinese Quart. J. Math.
13 (1998), no. 2, 10–15. MR 1696510
(2000c:32013)
 [ST3]
Z.
Slodkowski and G.
Tomassini, Evolution of special subsets of 𝐂²,
Adv. Math. 152 (2000), no. 2, 336–358. MR 1764108
(2001i:32019), http://dx.doi.org/10.1006/aima.1999.1905
 [ST4]
Z. Slodkowski, G. Tomassini, Evolution of a graph by Levi form, Gulliver, Robert (ed.) et al., Differential geometric methods in the control of partial differential equations. (Boulder, CO, 1999), Providence, RI: American Mathematical Society (AMS). Contemp. Math. 268, 373382 (2000). CMP 2001:07
 [BG]
 E. Bedford, B. Gaveau, Envelopes of holomorphy of certain 2spheres in , Amer. J. Math., 105, 1983, 9751009. MR 84k:32016
 [BK]
 E. Bedford, W. Klingenberg, On the envelopes of holomorphy of a spheres in , Journal of the A.M.S. , 4, 3, 1991, 623646.MR 92j:32034
 [CS]
 E. M. Chirka, Shcherbina, Pseudoconvexity of rigid domains and foliation of hulls of graphs, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21, 1995, 707735.MR 98f:32016
 [C]
 G. Citti, regularity of solutions of a quasilinear equation related to the Levi operator, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 23, 3, (1996), 483529. MR 98b:35072
 [Csem]
 G. Citti, Regolarità di grafici non Levipiatti, Seminario di Analisi, Dip. Mat. Univ. Bologna (A.A. 1996/97), Tecnoprint, Bologna, 186195.
 [CM1]
 G. Citti, A. Montanari, Analytic estimates of solutions of the Levi equation, J. Differential Equations, 173, (2001), 356389.
 [CM2]
 G. Citti, A. Montanari, regularity of solutions of an equation of Levi's type in , Annali di Matematica Pura Appl. (4), 180, (2001), 2758 CMP 2001:11
 [CPP]
 G. Citti, A. Pascucci, S. Polidoro, On the regularity of solutions to a nonlinear ultraparabolic equation in mathematical finance, Diff. Int. Eq, 14 (2001), 701738.
 [F]
 G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv Mat., 13, 1975, 161207.MR 58:13215
 [FS]
 G. B. Folland, E. M. Stein, Estimates for the Complex and Analysis on the Heisenberg Group, Comm. Pure Appl. Math. 20, 1974, 429  522.MR 51:3719
 [FL]
 B. Franchi, E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 10, 1983, 523541. MR 85k:35094
 [H]
 L. Hörmander Hypoelliptic second order differential equations Acta Math., 119, 1967, 147171. MR 36:5526
 [HK]
 G. Huisken, W. Klingenberg, Flow of real hypersurfaces by the trace of the Levi form, Mathematical Research Letters 6 (1999), 645662. MR 2001f:53141
 [K]
 W. Klingenberg, Real hypersurfaces in Kähler manifolds preprint.
 [M]
 A. Montanari, Real hypersurfaces evolving by Levi curvature: smooth regularity of solutions to the parabolic Levi equation, to appear in Comm. Partial Differential Equations 26 (9&10) (2001) 16331664.
 [NSW]
 A. Nagel, E. M. Stein, S. Wainger Balls and metrics defined by vector fields I: Basic properties. Acta Math. 155, 1985, 103147. MR 86k:46049
 [RS]
 L. Rothschild, E. M. Stein Hypoelliptic differential operators and nilpotent groups Acta Math. 137, 1977, 247320.MR 55:9171
 [Sc]
 N. V. Shchcerbina, On the polynomial hull of a graph. Indiana Univ. Math J., 42, 1993, 477503. MR 95e:32017
 [ST1]
 Z. Slodkowski, G. Tomassini, Weak solutions for the Levi equation and Envelope of Holomorphy, J. Funct. Anal, 101, no. 4, 1991, 392407. MR 93c:32018
 [ST2]
 Z. Slodkowski, G. Tomassini, Evolution of subsets in and a parabolic problem for the Levi equation, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 25, (1997), 757784. MR 2000c:32013
 [ST3]
 Z. Slodkowski, G. Tomassini, Evolution of special subsets of , Adv. Math. 152, (2000), no. 2, 336358. MR 2001i:32019
 [ST4]
 Z. Slodkowski, G. Tomassini, Evolution of a graph by Levi form, Gulliver, Robert (ed.) et al., Differential geometric methods in the control of partial differential equations. (Boulder, CO, 1999), Providence, RI: American Mathematical Society (AMS). Contemp. Math. 268, 373382 (2000). CMP 2001:07
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
35J70,
35K65,
22E30
Retrieve articles in all journals
with MSC (2000):
35J70,
35K65,
22E30
Additional Information
G. Citti
Affiliation:
Dipartimento di Matematica, Universita di Bologna, Piazza di Porta S. Donato 5, 40127, Bologna, Italy
Email:
citti@dm.unibo.it
A. Montanari
Affiliation:
Dipartimento di Matematica, Universita di Bologna, Piazza di Porta S. Donato 5, 40127, Bologna, Italy
Email:
montanar@dm.unibo.it
DOI:
http://dx.doi.org/10.1090/S0002994702029288
PII:
S 00029947(02)029288
Keywords:
Levi equation,
ellipticparabolic nonlinear equation,
freezing method,
Lie groups,
fundamental solution,
regularity properties
Received by editor(s):
May 3, 2000
Received by editor(s) in revised form:
August 8, 2001
Published electronically:
February 14, 2002
Additional Notes:
Investigation supported by University of Bologna, founds for selected research topics.
Article copyright:
© Copyright 2002
American Mathematical Society
