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Decomposition of L2-vector fields on Lipschitz surfaces: characterization via null-spaces of the scalar potential

Abstract : For dU the boundary of a bounded and connected strongly Lipschitz domain in R^n with n >= 3, we prove that any field f in L^2(dU, R^n) decomposes, in a unique way, as the sum of three invisible vector fields -- fields whose magnetic potential vanishes in one or both components of R^n \ dU. Moreover, this decomposition is orthogonal if and only if dU is a sphere. We also show that any f in L^2(dU, R^n) is uniquely the sum of two invisible fields and a Hardy function, in which case the sum is orthogonal regardless of dU ; we express the corresponding orthogonal projections in terms of layer potentials. When dU is a sphere, both decompositions coincide and match what has been called the Hardy-Hodge decomposition in the literature.
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https://hal.inria.fr/hal-03086446
Contributor : Laurent Baratchart Connect in order to contact the contributor
Submitted on : Thursday, December 9, 2021 - 2:27:05 PM
Last modification on : Thursday, December 9, 2021 - 5:34:52 PM

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Laurent Baratchart, Christian Gerhards, Alexander Kegeles. Decomposition of L2-vector fields on Lipschitz surfaces: characterization via null-spaces of the scalar potential. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2021, 53 (4), pp.4096 - 4117. ⟨10.1137/20M1387754⟩. ⟨hal-03086446⟩

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