On the Existence Theorems of Kantorovich, Moore and Miranda

Alefeld, Götz; Potra, Florian A.; Shen, Z.

doi:10.1007/978-3-7091-6217-0_3

Cited by 20 publications

(37 citation statements)

References 6 publications

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“…For the sake of completeness we will prove the existence of at least two non-trivial solutions to the Lane-Emden problem (1). In particular we prove the existence of a ground state solution (non-trivial solution with minimum energy) and a least energy nodal solution (sign-changing solution with minimum energy).…”

Section: Existence Of Solutionsmentioning

confidence: 96%

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On the asymptotics of solutions of the Lane-Emden problem for the p-Laplacian

Grumiau

Parini

2008

Arch. Math.

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ABSTRACT. In this paper we consider the Lane-Emden problem adapted for the p-Laplacianwhere Ω is a bounded domain in R n , n 2, λ > 0 and p < q < p * (with p * = np n−p if p < n, and p * = ∞ otherwise). After some recalls about the existence of ground state and least energy nodal solutions, we prove that, when q → p, accumulation points of ground state solutions or of least energy nodal solutions are, up to a "good" scaling, respectively first or second eigenfunctions of −∆ p .

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Section: Existence Of Solutionsmentioning

confidence: 96%

“…We consider the Lane-Emden equation for the pLaplacian, that is (1) −∆ p u = λ |u| q−2 u, in Ω, u = 0, on ∂ Ω.…”

Section: Introductionmentioning

confidence: 99%

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On the asymptotics of solutions of the Lane-Emden problem for the p-Laplacian

Grumiau

Parini

2008

Arch. Math.

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“…Then well group is a property of Z r ( f ) that captures homological properties common to all zero sets in Z r ( f ). We enhance the theory to include a relative case 2 that is especially convenient in the case when K is a manifold with boundary. Let B ⊆ K be a pair of compact Hausdorff spaces and f : K → R n continuous.…”

Section: Introductionmentioning

confidence: 99%

“…For the moment it is enough to say that theČech homology can be used and that for any computational purposes it behaves like simplicial homology. 2 Authors of [4] develop a different notion of relativity that is based on considering a pair of spaces (Y , Y 0 )…”

Section: Introductionmentioning

confidence: 99%

On Computability and Triviality of Well Groups

Franek

Krčál

2016

Discrete Comput Geom

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The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f : K → R n on a compact space K that are invariant with respect to perturbations of f . The perturbations are arbitrary continuous maps within L ∞ distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R n by f . In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n − 2, our approximation of the (dim K − n)th well group is exact. For the second part, we find examples of maps f, f : K → R n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status. Editor in Charge: Kenneth ClarksonThis is an extended version of a paper that is

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On Existence Theorems for Solutions of Non‐Linear Systems

Mayer

2003

Proc Appl Math and Mech

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An equivalent formulation of a recent result in [1] states that if the conditions of the Theorem of Newton‐Kantorovich are satisfied in a slightly modified (but equivalent) form in the maximum norm for a function g : G → ℝn, G ⊆ ℝn, guaranteeing the existence of a zero in D, then the conditions of Miranda's Theorem are automatically satisfied. We prove that this result holds for arbitrary norms if the conditions of the Theorem of Newton‐Kantorovich are suitable strengthened and Miranda's Theorem is suitably generalized.

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On the Existence Theorems of Kantorovich, Moore and Miranda

Cited by 20 publications

References 6 publications

On the asymptotics of solutions of the Lane-Emden problem for the p-Laplacian

On the asymptotics of solutions of the Lane-Emden problem for the p-Laplacian

On Computability and Triviality of Well Groups

On Existence Theorems for Solutions of Non‐Linear Systems

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