An Approach to The Numerical Verification of Solutions for Nonlinear Elliptic Problems With Local Uniqueness

Nagatou, Kaori; Yamamoto, Norifumi; Nakao, Mitsuhiro

doi:10.1080/01630569908816910

Cited by 33 publications

(26 citation statements)

References 16 publications

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“…ad ii) For computing bounds to eigenvalues of tJ>-1 L, we choose the parameter a in the HJ-product (22) such that of a> oy (x,w(x)) (x E fl);…”

Section: Making Use Of the Isomorphism Tj>: Hj(fl) --- H-1(fl) Givenmentioning

confidence: 99%

“…For the easier case of a bounded domain, see e.g., [5,26,27,28,32]' and also [22,23,24,25]. Our approach has been applied successfully also to other types of problems; see, e.g., the remarks at the end of Section 2.…”

Section: Introductionmentioning

confidence: 99%

“…An existence and enclosure method similar to ours has been developed by Nakao and his group [22]- [25]. This approach avoids the computation of eigenvalue enclosures for L, which constitutes a significant advantage in some cases.…”

Section: Introductionmentioning

confidence: 99%

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Computer-assisted proofs for semilinear elliptic boundary value problems

Plum

2009

Japan J. Indust. Appl. Math.

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Area (2)For second-order semilinear elliptic boundary value problems on bounded or unbounded domains, a general computer-assisted method for proving the existence of a solution in a "close" and explicit neighborhood of an approximate solution, computed by numerical means, is proposed. To achieve such an existence and enclosure result, we apply Banach's fixed-point theorem to an equivalent problem for the error, i.e., the difference between exact and approximate solution. The verification of the conditions posed for the fixed-point argument requires various analytical and numerical techniques, for example the computation of eigenvalue bounds for the linearization at the approximate solution. The method is used to prove existence and multiplicity results for some specific examples.

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“…ad ii) For computing bounds to eigenvalues of tJ>-1 L, we choose the parameter a in the HJ-product (22) such that of a> oy (x,w(x)) (x E fl);…”

Section: Making Use Of the Isomorphism Tj>: Hj(fl) --- H-1(fl) Givenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computer-assisted proofs for semilinear elliptic boundary value problems

Plum

2009

Japan J. Indust. Appl. Math.

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“…This can be done by the eigenvalue excluding technique which was studied in [9], [4], and is also closely related to the arguments in Section 7. We briefly remark here on the basic principle of the method.…”

Section: Eigenvalue Excluding Methodsmentioning

confidence: 99%

“…Then, the criterion for the proof of unique existence of the trivial solution can be presented as follows (cf. [9] or [4]): The proof of this lemma is quite analogous to that in [9]. Note that PNN(U) can be directly computed or enclosed by solving a linear system of equations with interval righthand side (e.g., see [7], [14] for details).…”

Section: T([a]-r)z := Pnn(z) + (I -Pn)(a["]z +T)mentioning

confidence: 99%

Verified numerical computations for an inverse elliptic eigenvalue problem with finite data

Nakao

Watanabe

Yamamoto

2001

Japan J. Indust. Appl. Math.

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We consider a numerical enclosure method for solutions of an inverse Dirichlet eigenvalue problem. When the finite number of prescribed eigenvalues are given, we reconstruct a potential function, with guaranteed error bounds, for which the corresponding elliptic operator exactly has those eigenvalues including the ordering property. All computations are executed with numerical verifications based upon the finite and infinite fixed point theorems using interval arithmetic. Therefore, the results obtained are mathematically correct. We present numerical examples which confirm us the enclosure algorithm works on real problems.

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A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem

McKenna

Pacella

Plum

et al. 2012

Inequalities and Applications 2010

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A wide variety of articles, starting with the famous paper [11], is devoted to the uniqueness question for the semilinear elliptic boundary value problem −∆u = λu + u p in Ω, u > 0 in Ω, u = 0 on ∂Ω, where λ ranges between 0 and the first Dirichlet Laplacian eigenvalue. So far, this question was settled in the case of Ω being a ball and, for more general domains, in the case λ = 0. In [16], we proposed a computerassisted approach to this uniqueness question, which indeed provided a proof in the case Ω = (0, 1) 2 , and p = 2. Due to the high numerical complexity, we were not able in [16] to treat higher values of p. Here, by a significant reduction of the complexity, we will prove uniqueness for the case p = 3. Dedicated to the memory of Wolfgang Walter2010 Mathematics Subject Classification. 35J25, 35J60, 65N15.

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An Approach to The Numerical Verification of Solutions for Nonlinear Elliptic Problems With Local Uniqueness

Cited by 33 publications

References 16 publications

Computer-assisted proofs for semilinear elliptic boundary value problems

Computer-assisted proofs for semilinear elliptic boundary value problems

Verified numerical computations for an inverse elliptic eigenvalue problem with finite data

A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem

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