Revista Matemática Iberoamericana
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Published online: 2019-02-04
Representation and uniqueness for boundary value elliptic problems via first order systemsPascal Auscher and Mihalis Mourgoglou (1) Université de Picardie-Jules Verne, Amiens, France and Université Paris-Sud, France
(2) Universidad del País Vasco, Leioa, Spain and Ikerbasque, Basque Foundation for Science, Bilbao, Spain
Given any elliptic system with $t$-independent coefficients in the upper-half space, we obtain representation and trace for the conormal gradient of solutions in the natural classes for the boundary value problems of Dirichlet and Neumann types with area integral control or non-tangential maximal control. The trace spaces are obtained in a natural range of boundary spaces which is parametrized by properties of some Hardy spaces. This implies a complete picture of uniqueness vs solvability and well-posedness.
Keywords: First order elliptic systems, Hardy spaces associated to operators, tent spaces, nontangential maximal functions, second order elliptic systems, boundary layer operators, a priori estimates, Dirichlet and Neumann problems, extrapolation
Auscher Pascal, Mourgoglou Mihalis: Representation and uniqueness for boundary value elliptic problems via first order systems. Rev. Mat. Iberoam. 35 (2019), 241-315. doi: 10.4171/rmi/1054