On some conformally invariant fully nonlinear equations

By Aobing Li, Yanyan Li. 2003

272 citations recorded by the source corpus.

Abstract We outline proofs of our results in [7] on Liouville type theorems, Harnack type inequalities, and existence and compactness of solutions to some conformally invariant fully nonlinear elliptic equations of second order on locally conformally flat Riemannian manifolds. Details will appear in [7]. To cite this article: A. Li, Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 305–310.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/35de99b3ba0dcf5e041b7d80db0f5e46135d0a5f

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In Applied Mathematics

Citations in the corpus

A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound (2012, methodology): “Extensions, as well as developments of new methods, have been made by Guan and Wang [19], Li and Li [26, 27], Gursky and Viaclovsky [21, 22], Ge and Wang [17], Sheng, Trudinger and Wang [38], Trudinger and Wang [40, 41], among others.”
On Harnack inequalities and singularities of admissible metrics in the Yamabe problem (2005, background): “When k < p < k n+2 n−2 , by a blow-up argument and the Liouville theorem [20], it is known that the set of solutions to (4.”
Recent developments in elliptic partial differential equations of Monge-Ampere type (2006, background): “1 when k = 2, n = 4 was proved in the pioneering work of Chang, Yang and Gursky [6], while the locally conformally flat case was proved by Guan and Wang [15] and Li and Li [19], [20].”
Remark on some conformally invariant integral equations: the method of moving spheres (2003, methodology): “In [24]-[27], all conformally invariant second order fully nonlinear equations are classified and Liouville type theorems are established for the elliptic ones.”
Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities (2019, background): “The main theorems of [5] and [28] were completely extended to solutions of second order fully nonlinear Yamabe equations by Li-Li [30], Y.Y. Li [31], Chang-Han-Yang [10] and Han-Li-Teixeira [23].”
Compactness of solutions to the Yamabe problem (2004, unknown): “Analogous results, extensions, as well as different proofs of (12) can be found in [87], [90], [81], [22], [21], [74], [34], [33], [68], [16], [17], [36], [69], [ 64 ], [14], [78], [37], [65], [15], [88], [89], and others.”
Local gradient estimates of solutions to some conformally invariant fully nonlinear equations (2006, methodology): “1 in [13], there exists some V ∈ R such that λ̄(x)∇v(x) = V ∀ x ∈ Bǫ ∩ D.”
Singular sets of a class of locally conformally flat manifolds (2005, unknown): “For instance, Chang-Gursky-Yang [4], [3], Gursky-Viaclovsky [12], Li-Li [14], Guan-Wang [11].”
Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary (2006, background): “When (M, g) is locally conformally flat and (f,Γ) = (σ 1/k k ,Γk) for 1 ≤ k ≤ n, Guan-Wang [18] 3 and Li-Li [26] independently proved the existence of solutions of (1.8).”
Conformally invariant fully nonlinear elliptic equations and isolated singularities (2005, unknown): “(101) Suppose for some |x| > 0, λ̄(x) < |x|, then we obtain, using the strong maximum principle and the Hopf Lemma as in section 2 of [12] and in view of (14), u(y)− ux,λ̄(x)(y) > 0, ∀ |y − x| > λ̄(x), |y| 6 = 0, (102)”
The Neumann Problem for Hessian Equations (2015, methodology): “ by Y.Y.Li [22], Urbas [44] and Guan [13]. In recent years the Hessian type equation also appears in conformal geometry, which started from Chang-Gursky-Yang [5] and the related development by ([16], [23], [17], [36],[10]). In Kahler geometry, the Hessian equation was studied by Hou-Ma-Wu [18] and D”
A class of fully nonlinear equations arising from conformal geometry (2006, background): “Li obtained the existence for locally conformally flat manifolds ([15]).”
Local estimates for some fully nonlinear elliptic equations (2005, background): “Moreover, it is shown in [12] that F is concave in Γ.”
A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry* (2006, methodology): “ are concentrated on the conformal k-Hessian equation. Just to mention a few, the ﬁrst existence result is by [CGY], where the existence of solutions was proved for the case k = 2,n = 4. Subsequently [LL1], [GW2] proved the existence of solutions for general k and n, but on locally conformally ﬂat m”
Local Analysis of Solutions of Fractional Semi-Linear Elliptic Equations with Isolated Singularities (2013, background): “L i for helpful discussions and constant encouragement, and for informing us a useful lemma in [19] .”
On some conformally invariant fully nonlinear equations, II. Liouville, Harnack and Yamabe (2005, result): “Part of the work of the second author was also supported by NSF-DMS-0100819.”