if u be a harmonic function in sobolev space W^(1,p) (Omega) with zero trace (i.e; T(u)=0) and {Laplace operator} (u) =0, then u identically zero.
Gianluca Argentini
Dear Shatha, see Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. If in W1,p(Omega) you can establish a unicity result for the laplacian, then from your hypothesis u = 0 in the only solution. Gianluca
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Shatha S Alhily
Dear Gianluca
Thank you so much for guiding me through that book , really it is a good reference.
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Fabrizio Morlando
Hi, I want to point out that in the case p=1 the conclusion holds for the ball and it doesn't hold in general for less regular domain Omega.
See "Function Spaces and Partial Differential Equations: Contemporary Analysis"
by A. Taheri
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