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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