Liouville theorem for X-elliptic operators

Kogoj, Alessia E.; Lanconelli, Ermanno

doi:10.1016/j.na.2008.12.029

Cited by 32 publications

(24 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof Using Lemmas 11.4, 11.5 and Proposition 11.2 , we find With (11.22) in hands, we can now invoke the Colding-Minicozzi type theorem at the end of the paper [30] by Kogoj and Lanconelli to conclude that P α,κ (R N ) is finite dimensional.…”

Section: Lemma 113mentioning

confidence: 70%

Properties of a frequency of Almgren type for harmonic functions in Carnot groups

Garofalo

Rotz

2015

Calc. Var.

View full text Add to dashboard Cite

The celebrated frequency function of Almgren (Proceedings of Japan-United States Sem., Tokyo. North-Holland, Amsterdam, 1979), and its local and global properties, play a fundamental role in several questions in partial differential equations and geometric measure theory. In this paper we introduce a notion of Almgren's frequency functional in any Carnot group G, and we analyze some local and global consequences of the boundedness of the frequency. Although our results are the counterpart of by now well-known classical ones, their proof is much more delicate and involved than their elliptic predecessors, and serious new obstructions arise. A central motivation for our study is the fundamental open question whether harmonic functions (i.e., solutions of a sub-Laplacian) in a Carnot group G of step r ≥ 2 possess the strong unique continuation property (scup). Among the results in this paper, in Theorem 4.3 we show that a quantitative answer to such question is in fact equivalent to proving the local boundedness of the frequency. Mathematics Subject Classification

show abstract

Section: Lemma 113mentioning

confidence: 70%

Properties of a frequency of Almgren type for harmonic functions in Carnot groups

Garofalo

Rotz

2015

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…We refer to Chapter 5.8 of the monograph [15] and the references therein for a survey about Liouville properties for sublaplacians, mostly obtained by Harnack-type inequalities for solutions. We refer also to [17] for results on inequalities of the form Lu + h(x)u p ≤ 0 with L linear degenerate elliptic, and to [35,36,38] for more recent results on linear subelliptic equations.…”

Section: Introductionmentioning

confidence: 99%

Liouville properties and critical value of fully nonlinear elliptic operators

Bardi

Cesaroni

2016

Journal of Differential Equations

View full text Add to dashboard Cite

Abstract. We prove some Liouville properties for sub-and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in OrnsteinUhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.

show abstract

“…(Ω), for some constant c ≥ 0, with q = 2Q Q−2 was proved in Remark 9 in [21]. Since the domain Ω is bounded, the inequalities are satisfied if L q (Ω) is replaced by L p (Ω) for any p ∈ [1, q], and the Poincaré inequality (P ) follows by the X-ellipticity of the operator L.…”

Section: Resultsmentioning

confidence: 95%

“…However, several families of operators that fall into this class were already present in literature; see, e.g., [13], [32], [33], [36], [24] and [25]. More recently, X-elliptic operators were widely studied in [15], where a maximum principle, a non-homogeneous Harnack inequality and a Liouville theorem were obtained, and in [21], where a one-sided Liouville-type property was proved, which extends the previous result by Gutierrez and Lanconelli in [15] and a celebrated Liouville-type theorem by Colding and Minicozzi in [11].…”

Section: Introductionmentioning

confidence: 99%

Attractors met X-elliptic operators

Kogoj

Sonner

2014

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Abstract. We consider degenerate parabolic and damped hyperbolic equations involving an operator L, that is X-elliptic with respect to a family of locally Lipschitz continuous vector fields X = {X 1 , . . . , Xm}. The local wellposedness is established under subcritical growth restrictions on the nonlinearity f , which are determined by the geometry and functional setting naturally associated to the family of vector fields X. Assuming additionally that f is dissipative, the global existence of solutions follows, and we can characterize their longtime behavior using methods from the theory of infinite dimensional dynamical systems.

show abstract

Liouville theorem for X-elliptic operators

Cited by 32 publications

References 15 publications

Properties of a frequency of Almgren type for harmonic functions in Carnot groups

Properties of a frequency of Almgren type for harmonic functions in Carnot groups

Liouville properties and critical value of fully nonlinear elliptic operators

Attractors met X-elliptic operators

Contact Info

Product

Resources

About