Integral representation of local functionals depending on vector fields
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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1980 ◽
Vol 38
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pp. 118-138
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2017 ◽
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pp. 98-115
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2001 ◽
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pp. 527-563
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pp. 86-91
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