2010
DOI: 10.1016/j.jde.2010.07.002
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Abstract: In this paper we extend the ideas of the so-called validated continuation technique to the context of rigorously proving the existence of equilibria for partial differential equations defined on higherdimensional spatial domains. For that effect we present a new set of general analytic estimates. These estimates are valid for any dimension and are used, together with rigorous computations, to construct a finite number of radii polynomials. These polynomials provide a computationally efficient method to prove, … Show more

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Cited by 62 publications
(115 citation statements)
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“…Now, there are some analytic convolution estimates (e.g. the ones developed in [11,13,15]) that allow computingZ M (r) satisfying (16). These explicit estimates essentially follow from the fact that the Banach space Ω q given in (4) is an algebra under discrete convolutions.…”
Section: The Radii Polynomial Approachmentioning
confidence: 99%
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“…Now, there are some analytic convolution estimates (e.g. the ones developed in [11,13,15]) that allow computingZ M (r) satisfying (16). These explicit estimates essentially follow from the fact that the Banach space Ω q given in (4) is an algebra under discrete convolutions.…”
Section: The Radii Polynomial Approachmentioning
confidence: 99%
“…It can be shown that looking for solutions of (40) is equivalent to looking for solutions of f (x) = 0 in the Banach space X = R 2 × Ω q , where Ω q = {a = (a k ) k≥0 : a q < ∞} is the Banach space of infinite sequences algebraically decaying to 0 at least as fast as k −q with decay rate q > 1 (e.g. see [13]). The regularity estimates given by Lemma A.4 in Appendix A can be used to show that if a ∈ Ω q , then a 3 ∈ Ω q .…”
Section: Two-dimensional Manifold Of Equilibria Of Cahn-hilliardmentioning
confidence: 99%
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“…For instance, as mentioned in [13], "it is an interesting open problem to prove that both symmetric and asymmetric solutions coexist in S 1 ∩ A 2 ." The goal of the present paper is to prove these open questions for specific parameter values using the rigorous computational methods of [20][21][22][23][24] and more specifically with the approach as introduced in [25].…”
Section: (A))mentioning
confidence: 99%
“…All proofs can be found in [23,28]. Consider a decay rate s > 2, a computational parameter M > 6 and define, for k > 3,…”
Section: Coexistence Of Nontrivial Solutionsmentioning
confidence: 99%