Characterizations of the existence and removable singularities of divergence-measure vector fields

• Autores: Nguyen Cong Phuc, Mónica Torres
• Localización: Indiana university mathematics journal, ISSN 0022-2518, Vol. 57, Nº 4, 2008, págs. 1573-1598
• Idioma: inglés
• DOI: 10.1512/iumj.2008.57.3312
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• Resumen

  • We study the solvability and removable singularities of the equation divF=µ , with measure data µ , in the class of continuous or Lp vector fields F , where 1=p=8 . In particular, we show that, for a signed measure µ , the equation divF=µ has a solution F?L8(Rn) if and only if |µ(U)|=CHn-1(?U) for any open set U with smooth boundary. For non-negative measures µ , we obtain explicit characterizations of the solvability of divF=µ in terms of potential energies of µ for p?8 , and in terms of densities of µ for continuous vector fields. These existence results allow us to characterize the removable singularities of the corresponding equation divF=µ with signed measures µ .