Boundary Value Problems
Volume 2005 (2005), Issue 1, Pages 73-91

Boundary value problems for the 2nd-order Seiberg-Witten equations

Celso Melchiades Doria

Departamento de Matemática, Universidade Federal de Santa Catarina, Campus Universitario, Trindade 88040900 Florianópolis - SC, Brazil

Received 8 June 2004

Copyright © 2005 Celso Melchiades Doria. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


It is shown that the nonhomogeneous Dirichlet and Neuman problems for the 2nd-order Seiberg-Witten equation on a compact 4-manifold X admit a regular solution once the nonhomogeneous Palais-Smale condition is satisfied. The approach consists in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation. The gauge invariance of the functional allows to restrict the problem to the Coulomb subspace 𝒞α of configuration space. The coercivity of the 𝒮𝒲α-functional, when restricted into the Coulomb subspace, imply the existence of a weak solution. The regularity then follows from the boundedness of L-norms of spinor solutions and the gauge fixing lemma.