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Harnack inequality and principal eigentheory for general infinity Laplacian operators with gradient in $mathbb{R}^N$ and applications

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 Added by Anup Biswas
 Publication date 2020
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and research's language is English




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Under the lack of variational structure and nondegeneracy, we investigate three notions of textit{generalized principal eigenvalue} for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality and boundary Harnack inequality are proved to support our analysis. This is a continuation of our first work [3] and a substantial contribution in the development of the theory of textit{generalized principal eigenvalue} beside the works [8, 13, 12, 9, 29]. We use these notions to characterize the validity of maximum principle and study the existence, nonexistence and uniqueness of positive solutions of Fisher-KPP type equations in the whole space. The sliding method is intrinsically improved for infinity Laplacian to solve the problem. The results are related to the Liouville type results, which will be meticulously explained.



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We study the generalized eigenvalue problem in $mathbb{R}^N$ for a general convex nonlinear elliptic operator which is locally elliptic and positively $1$-homogeneous. Generalizing article of Berestycki and Rossi in [Comm. Pure Appl. Math. 68 (2015), no. 6, 1014-1065] we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.
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239 - Seick Kim , Soojung Kim , 2012
We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabre proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabres result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.
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