JournalsaihpcVol. 33, No. 3pp. 829–848

A Dirichlet problem involving the divergence operator

  • G. Csató

    Tata Institute of Fundamental Research, Centre for Applicable Mathematics, 560065 Bangalore, India
  • B. Dacorogna

    Section de Mathématiques, EPFL, 1015 Lausanne, Switzerland
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Abstract

We consider the problem

{divu+a;u=finΩu=u0onΩ.\left\{\begin{matrix} \mathrm{div}\:u + 〈a;u〉 = f & \text{in}\Omega \\ u = u_{0} & \text{on}\partial \Omega . \\ \end{matrix}\right.

We show that if curla(x0)0\mathrm{curl}\:a(x_{0}) \neq 0 for some x0Ωx_{0} \in \Omega , then the problem is solvable without restriction on f. We also discuss the regularity of the solution.

Cite this article

G. Csató, B. Dacorogna, A Dirichlet problem involving the divergence operator. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 3, pp. 829–848

DOI 10.1016/J.ANIHPC.2015.01.006