INTEGRAL REPRESENTATION OF THE TRACE ON THE SPHERE OF A CERTAIN CLASS OF VECTOR FIELDS, AND UNIFORM ESTIMATES OF STABILITY FOR QUASICONFORMAL MAPPINGS OF THE BALL

The problem of the existence and integral representation of a unique multiperiodic solution of a second-order linear inhomogeneous system with constant coefficients and a differentiation operator on the direction of the main diagonal of the space of time variables and of the vector fields in the form of Lyapunov systems with respect to space variables were considered. The multiperiodicity of zeros of this operator and the condition for the absence of a nonzero multiperiodic and real-analytic solution of the homogeneous system corresponding to the given system are established. An integral representation of solutions of an inhomogeneous linear autonomous system that multiperiodic in time variables and realanalytic in space variables is obtained. The existence theorem of a unique multiperiodic in time variables and real-analytic in space variables solutions of the original linear system in terms of the Green's function under sufficiently general conditions is substantiated.

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Boundary integral representation for Beltrami vector fields

2016 IEEE International Conference on Mathematical Methods in Electromagnetic Theory (MMET) ◽

10.1109/mmet.2016.7544080 ◽

2016 ◽

Author(s):

Mykola Yasko

Keyword(s):

Integral Representation ◽

Vector Fields ◽

Boundary Integral ◽

Boundary Integral Representation

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Linear operators on Köthe spaces of vector fields

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2013-0023 ◽

2013 ◽

Vol 21 (2) ◽

pp. 53-80

Author(s):

Ion Chiţescu ◽

Liliana Sireţchi

Keyword(s):

Integral Representation ◽

Representation Theorem ◽

Vector Fields ◽

Linear Operators ◽

Köthe Spaces

Abstract The study of Köthe spaces of vector fields was initiated by the present authors. In this paper linear operators on these spaces are studied. An integral representation theorem is given and special types of linear operators are introduced and studied.

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Integral representation of local functionals depending on vector fields

Advances in Calculus of Variations ◽

10.1515/acv-2021-0054 ◽

2022 ◽

Vol 0 (0) ◽

Author(s):

Fares Essebei ◽

Andrea Pinamonti ◽

Simone Verzellesi

Keyword(s):

Integral Representation ◽

Sobolev Spaces ◽

Vector Fields ◽

First Order ◽

Bounded Set

Abstract Given an open and bounded set Ω ⊆ ℝ n {\Omega\subseteq\mathbb{R}^{n}} and a family 𝐗 = ( X 1 , … , X m ) {\mathbf{X}=(X_{1},\ldots,X_{m})} of Lipschitz vector fields on Ω, with m ≤ n {m\leq n} , we characterize three classes of local functionals defined on first-order X-Sobolev spaces, which admit an integral representation in terms of X, i.e. F ⁢ ( u , A ) = ∫ A f ⁢ ( x , u ⁢ ( x ) , X ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x , F(u,A)=\int_{A}f(x,u(x),Xu(x))\,dx, with f being a Carathéodory integrand.

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