Lp - Liouville theorems for invariant evolution equations

Alessia E. Kogoj


Some Liouville-type theorems in Lebesgue spaces for several classes of evolution equations are presented. The involved operators are left invariant with respect to Lie group composition laws.
Results for both solutions and sub-solutions are given.


Liouville Theorems; Invariant Partial Dierential Operators; Evolution Operators on Lie groups

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J. August and S.W. Zucker. Sketches with curvature: the curve indicator random field and markov processes. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 25(4):387-400, April 2003.

G. Caristi, L. D'Ambrosio, and E. Mitidieri. Liouville theorems for some nonlinear inequalities. Tr. Mat. Inst. Steklova, 260(Teor. Funkts. i Nelinein. Uravn. v Chastn. Proizvodn.):97-118, 2008.

G. Da Prato. Kolmogorov equations for stochastic PDEs. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2004.

B. Helffer and F. Nier. Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, volume 1862 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2005.

A.E. Kogoj and E. Lanconelli. Lp-Liouville theorems for invariant partial differential operators in Rn. Nonlinear Anal., in press. http://dx.doi.org/10.1016/j.na.2014.12.004

E. Lanconelli and A. Pascucci. Superparabolic functions related to second order hypoelliptic operators. Potential Anal., 11(3):303-323, 1999.

D. Mumford. Elastica and computer vision. In Algebraic geometry and its applications (West Lafayette, IN, 1990), pages 491-506. Springer, New York, 1994.

A. Pascucci. Kolmogorov equations in physics and in finance. In Elliptic and parabolic problems, volume 63 of Progr. Nonlinear Differential Equations Appl., pages 353-364. Birkhäuser, Basel, 2005.

E. Priola and J. Zabczyk. Liouville theorems for non-local operators. J. Funct. Anal., 216(2):455-490, 2004.

Yunfeng Wang, Yu Zhou, D.K. Maslen, and G.S. Chirikjian. Solving phase-noise fokkerplanck equations using the motion-group fourier transform. Communications, IEEE Transactions on, 54(5):868-877, May 2006.

DOI: 10.6092/issn.2240-2829/4674


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