Removable singularities for div v = f in weighted Lebesgue spaces

2015/10/13 by Moonens, Laurent, Russ, Emmanuel, Tuominen, Heli
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1510.03544

Abstract

Let w∈ L¹_loc(\Rⁿ) be apositive weight. Assuming that a doubling condition and an L¹ Poincaré inequality on balls for the measure w(x)dx, as well as a growth condition on w, we prove that the compact subsets of \Rⁿ which are removable for the distributional divergence in L^∞_1/w are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for L^p_1/w, 1\textlessp\textless+∞, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.

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