Dirichlet and neumann boundary value problems for Yang‐Mills connections

Marini, Antonella

doi:10.1002/cpa.3160450806

Cited by 37 publications

(90 citation statements)

References 15 publications

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“…This problem generalizes the boundary value problem of Neumann type studied in [7]. In [8], the problem is defined using only analytic tools and a good gauge theorem is proven.…”

Section: Introductionmentioning

confidence: 91%

Regularity theory for the generalized Neumann problem for Yang-Mills Connections - Non-trivial examples in dimensions 3 and 4

Marini

2000

Mathematische Annalen

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We develop the existence and regularity theory for the generalized Neumann problem for Yang-Mills connections. This is the most general boundary value problem for connections on a compact manifold with smooth boundary, with geometric meaning. It is obtained by reflecting the base manifold across its boundary and lifting this action non-trivially to the bundle. The prescribed lifting corresponds to a geometric invariant, which is similar to the monopole number. When this invariant is non-zero, there exist non-trivial solutions of the generalized Neumann problem. We prove the existence of non-trivial solutions over the 3-dimensional disk D 3 and over the 4-dimensional manifold D 3 × S 1 . We outline the procedure for finding non-trivial examples of solutions over more general manifolds of dimension 3 and 4.

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“…This problem generalizes the boundary value problem of Neumann type studied in [7]. In [8], the problem is defined using only analytic tools and a good gauge theorem is proven.…”

Section: Introductionmentioning

confidence: 91%

Regularity theory for the generalized Neumann problem for Yang-Mills Connections - Non-trivial examples in dimensions 3 and 4

Marini

2000

Mathematische Annalen

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“…The most basic analytical results needed to achieve the main result is the gauge fixing lemma (see [7]) and the estimate (2.3), both extended by Marini [5] to manifolds with boundary.…”

Section: Basic Set Upmentioning

confidence: 99%

Boundary value problems for the nd-order Seiberg-Witten equations

Doria

2005

Bound Value Probl

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It is shown that the nonhomogeneous Dirichlet and Neuman problems for the 2nd-order Seiberg-Witten equation on a compact 4-manifold X admit a regular solution once the nonhomogeneous Palais-Smale condition Ᏼ is satisfied. The approach consists in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation. The gauge invariance of the functional allows to restrict the problem to the Coulomb subspace Ꮿ C α of configuration space. The coercivity of the ᐃ α -functional, when restricted into the Coulomb subspace, imply the existence of a weak solution. The regularity then follows from the boundedness of L ∞ -norms of spinor solutions and the gauge fixing lemma.

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“…Here * : Λ 2 T * M ⊗Ad(P ) → Λ 2 T * M ⊗ Ad(P ) is the Hodge star operator. Dirichlet and Neumann problems for Yang-Mills connections were first defined and studied by Marini [7]. In [7], Marini showed the existence and regularity of absolute minimum solutions for the Dirichlet problem, and Neumann problem with prescribed class η ∈ H 2 (M ; π 1 (G)), when dim M = 4.…”

Section: Introductionmentioning

confidence: 99%

“…Dirichlet and Neumann problems for Yang-Mills connections were first defined and studied by Marini [7]. In [7], Marini showed the existence and regularity of absolute minimum solutions for the Dirichlet problem, and Neumann problem with prescribed class η ∈ H 2 (M ; π 1 (G)), when dim M = 4. In [4], Isobe and Marini studied the existence of "topologically distinct solutions" to the Dirichlet problem when M = B 4 = {x ∈ R 4 : |x| ≤ 1} and G = SU (2) using the fact that in this case A(A 0 ) is the disjoint union of infinitely many connected components A k indexed by k ∈ Z (specifically, for some fixed B ∈ A(A 0 ),…”

Section: Introductionmentioning

confidence: 99%

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Non-existence and uniqueness results for boundary value problems for Yang-Mills connections

Isobe

1997

Proc. Amer. Math. Soc.

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Abstract. We show uniqueness results for the Dirichlet problem for YangMills connections defined in n-dimensional (n ≥ 4) star-shaped domains with flat boundary values. This result also shows the non-existence result for the Dirichlet problem in dimension 4, since in 4-dimension, there exist countably many connected components of connections with prescribed Dirichlet boundary value. We also show non-existence results for the Neumann problem. Examples of non-minimal Yang-Mills connections for the Dirichlet and the Neumann problems are also given.

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Dirichlet and neumann boundary value problems for Yang‐Mills connections

Cited by 37 publications

References 15 publications

Regularity theory for the generalized Neumann problem for Yang-Mills Connections - Non-trivial examples in dimensions 3 and 4

Regularity theory for the generalized Neumann problem for Yang-Mills Connections - Non-trivial examples in dimensions 3 and 4

Boundary value problems for the nd-order Seiberg-Witten equations

Non-existence and uniqueness results for boundary value problems for Yang-Mills connections

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