On the global solvability for overdetermined systems

Bergamasco, Adalberto P.; Kirilov, Alexandre; Nunes, Wagner Vieira Leite; Zani, Sérgio Luís

doi:10.1090/s0002-9947-2012-05414-6

Cited by 16 publications

(10 citation statements)

References 19 publications

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“…Notice that when we use the induction hypothesis for the first time, Γ 0 is a bounded subset of the plane or of the cylinder, according to the rank of b 0 . By following the proofs of or , respectively, a smooth 1‐form f 0 is constructed in a way that the Fourier coefficients

{false{\overset{v}{true} (t, ξ) false}}_{ξ \in double-struckZ}

of the candidate to a solution does not have tempered growth at a point

t^{*}

. This point can be chosen inside

Π ({normalΓ}_{0})

as the point of maximum or minimum of

{\tilde{B}}_{0}

there.…”

Section: Revisiting a Necessary Conditionmentioning

confidence: 99%

“…In the case when b is closed, but not exact, and M is the torus

T^{2}

, the authors of and of obtained a equivalent condition to the global solvability of

double-struckL

by using a primitive defined on the universal covering space

R^{2}

. Both conditions bear on the sublevel and superlevel sets of the respective primitives.…”

Section: Introductionmentioning

confidence: 99%

“…When is exact, in [15] the authors obtained an equivalent condition to the global solvability of the operator for a closed orientable manifold by using a primitive defined on . In the case when is closed, but not exact, and is the torus 2 , the authors of [3] and of [4] obtained a equivalent condition to the global solvability of by using a primitive defined on the universal covering space ℝ 2 . Both conditions bear on the sublevel and superlevel sets of the respective primitives.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, in Section 5, we furnish another proof of Theorem . Specifically, we reduce the surface's genus coming to a torus, and then we take advantage of some facts from and . It furthermore exhibits a valid second member for which there is not even a distribution solving the system.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Geometrical proofs for the global solvability of systems

Bergamasco¹,

Parmeggiani

Zani³

et al. 2018

Mathematische Nachrichten

Self Cite

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We study a linear operator associated with a closed non‐exact 1‐form b defined on a smooth closed orientable surface M of genus g>1. Here we present two proofs that reveal the interplay between the global solvability of the operator and the global topology of the surface. The first result brings an answer for the global solvability when the system is defined by a generic Morse 1‐form. Necessary conditions for the global solvability bearing on the sublevel and superlevel sets of primitives of a smooth 1‐form b have already been established; we also present a more intuitive proof of this result.

show abstract

{false{\overset{v}{true} (t, ξ) false}}_{ξ \in double-struckZ}

of the candidate to a solution does not have tempered growth at a point

t^{*}

. This point can be chosen inside

Π ({normalΓ}_{0})

as the point of maximum or minimum of

{\tilde{B}}_{0}

there.…”

Section: Revisiting a Necessary Conditionmentioning

confidence: 99%

“…In the case when b is closed, but not exact, and M is the torus

T^{2}

, the authors of and of obtained a equivalent condition to the global solvability of

double-struckL

by using a primitive defined on the universal covering space

R^{2}

. Both conditions bear on the sublevel and superlevel sets of the respective primitives.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Geometrical proofs for the global solvability of systems

Bergamasco¹,

Parmeggiani

Zani³

et al. 2018

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

show abstract

“…With regards to a sufficient condition, we dedicate Chapters 3 through 5. The inspiration comes from [Tr2,BK,BKNZ], but here we do not make use of global coordinates. In order to head off this difficulty, we work with some instructive constructions of forms given by [Far], which have inspired us to furnish a class of globally solvable systems.…”

Section: Abstract Prefacementioning

confidence: 99%

Global solvability of systems on compact surfaces

Zugliani¹

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We are interested in studying an involutive system defined by a closed non-exact 1-form on a closed and orientable surface. Here we present a necessary condition for the global solvability of this system. We also make some particular constructions of globally solvable systems that motivate the equivalence between the global solvability and the necessary condition, for two cases involving 1-forms of the Morse type, namely, when the surface is the bitorus or when the 1-form is generic. v vi form. In Chapter 5, we complete the cases for Morse forms on the bitorus obtaining an equivalence between our condition and the solvability. The final chapter, in turn, ix yields the equivalence for a special case on which b has rank equal to 1 and supplies more examples. Working with Morse forms is far from being particular: the set of Morse forms is open and dense in the set of the smooth closed 1-forms. All the machinery involved in this part enables us to consider, in the future, the problem when b is a real analytic

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Globally hypoelliptic triangularizable systems of periodic pseudo‐differential operators

Silva

2023

Mathematische Nachrichten

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This paper presents an investigation on the global hypoellipticity problem for a class of systems of pseudo‐differential operators on the torus. The approach consists in establishing conditions on the matrix symbol of the system such that it can be transformed into a suitable triangular form involving a nilpotent upper triangular matrix. Hence, the global hypoellipticity is studied by analyzing the behavior of the eigenvalues and their averages.

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On the global solvability for overdetermined systems

Abstract: We consider a class of systems of two smooth vector fields on the 3-torus associated to a closed 1-form. We prove that the global solvability is completely determined by the connectedness of the sublevel and superlevel sets of a primitive of this 1-form in the minimal covering.

Cited by 16 publications

References 19 publications

Geometrical proofs for the global solvability of systems

Geometrical proofs for the global solvability of systems

Global solvability of systems on compact surfaces

Globally hypoelliptic triangularizable systems of periodic pseudo‐differential operators

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