Portal de Periódicos da CAPES

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

2008; Indiana University; Volume: 57; Issue: 4 Linguagem: Inglês

10.1512/iumj.2008.57.3312

1943-5266

Nguyen Cong Phuc, Monica Torres,

Numerical methods in inverse problems

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 .