American Institute of Mathematical Sciences

Advanced
July  2009, 8(4): 1159-1201. doi: 10.3934/cpaa.2009.8.1159

On a class of hypoelliptic operators with unbounded coefficients in $R^N$

Bálint Farkas 1, and  Luca Lorenzi 2,

1. 

Technische Universität Darmstadt, Fachbereich Mathematik, AG Analysis, Schloßgartenstraße 7, D-64289, Darmstadt, Germany
2. 

Dipartimento di Matematica, Universitá degli Studi di Parma, Viale G. Usberti 85/A, 43100 Parma

Received  June 2008 Revised  December 2008 Published  March 2009

We consider second order linear partial differential operators $A$ on $R^N$ which are not assumed to be uniformly elliptic and whose coefficients in the second order part may grow quadratically, while the drift part has essentially linear growth. Instead of uniform ellipticity, we require a much weaker hypothesis of uniform hypoellipticity, which in an equivalent formulation connects the behaviour of the diffusion and the drift coefficients, by requiring that a Kalman-type condition is satisfied for them. By refining Bernstein's method we prove the existence of a semigroup {$T(t)$} of bounded linear operators (in the space of bounded and continuous functions) associated to the operator $A$. We also show uniform estimates for the spatial derivatives of the semigroup {$T(t)$} in (an)isotropic spaces of (Hölder-) continuous functions. As a consequence, we obtain Hölder estimates for the solutions of some elliptic and parabolic problems associated to the operator $A$.
Keywords: anisotropic Hölder estimates., distributional solutions to elliptic and parabolic problems, uniform estimates, Degenerate elliptic operators with unbounded coefficients in $R^N$.
Mathematics Subject Classification: Primary: 35K65; Secondary: 35J70, 35B65, 35K1.
Citation: Bálint Farkas, Luca Lorenzi. On a class of hypoelliptic operators with unbounded coefficients in $R^N$. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1159-1201. doi: 10.3934/cpaa.2009.8.1159
[1]

Giorgio Metafune, Chiara Spina. Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2285-2299. doi: 10.3934/dcds.2012.32.2285
[2]

N. V. Krylov. Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2073-2090. doi: 10.3934/dcdsb.2012.17.2073
[3]

Sallah Eddine Boutiah, Abdelaziz Rhandi, Cristian Tacelli. Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 803-817. doi: 10.3934/dcds.2019033
[4]

Agnese Di Castro, Mayte Pérez-Llanos, José Miguel Urbano. Limits of anisotropic and degenerate elliptic problems. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1217-1229. doi: 10.3934/cpaa.2012.11.1217
[5]

Jianguo Huang, Jun Zou. Uniform a priori estimates for elliptic and static Maxwell interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 145-170. doi: 10.3934/dcdsb.2007.7.145
[6]

Hannes Meinlschmidt, Joachim Rehberg. Hölder-estimates for non-autonomous parabolic problems with rough data. Evolution Equations & Control Theory, 2016, 5 (1) : 147-184. doi: 10.3934/eect.2016.5.147
[7]

Giorgio Metafune, Chiara Spina, Cristian Tacelli. On a class of elliptic operators with unbounded diffusion coefficients. Evolution Equations & Control Theory, 2014, 3 (4) : 671-680. doi: 10.3934/eect.2014.3.671
[8]

Tommaso Leonori, Ireneo Peral, Ana Primo, Fernando Soria. Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6031-6068. doi: 10.3934/dcds.2015.35.6031
[9]

Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315
[10]

Lucio Boccardo, Alessio Porretta. Uniqueness for elliptic problems with Hölder--type dependence on the solution. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1569-1585. doi: 10.3934/cpaa.2013.12.1569
[11]

Francois van Heerden, Zhi-Qiang Wang. On a class of anisotropic nonlinear elliptic equations in $\mathbb R^N$. Communications on Pure & Applied Analysis, 2008, 7 (1) : 149-162. doi: 10.3934/cpaa.2008.7.149
[12]

Giuseppina Barletta, Gabriele Bonanno. Multiplicity results to elliptic problems in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 715-727. doi: 10.3934/dcdss.2012.5.715
[13]

Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure & Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043
[14]

Feng Zhou, Zhenqiu Zhang. Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3137-3160. doi: 10.3934/cpaa.2019141
[15]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
[16]

Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801
[17]

Luigi Greco, Gioconda Moscariello, Teresa Radice. Nondivergence elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 131-143. doi: 10.3934/dcdsb.2009.11.131
[18]

Chunrong Chen, Shengji Li. Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria. Journal of Industrial & Management Optimization, 2012, 8 (3) : 691-703. doi: 10.3934/jimo.2012.8.691
[19]

Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri. Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electronic Research Announcements, 2006, 12: 95-99.

[20]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences