On the solvability of the Neumann problem for a planar domain with a peak
Vladimir Gilelevich Maz'ya, S. V. Poborchiĭ · Vestnik St Petersburg University Mathematics · 2008
The Neumann problem for second-order elliptic quasi-linear equations on a planar domain whose boundary contains the vertex of an outward or inward peak. Under certain conditions, the solvability problem for the Neumann problem is reduced to a description of the space dual to the boundary trace space TW 1 (Ω) for functions from the Sobolev class (W 1 (Ω), where 1 < p < ∞. This dual space is characterized in terms of Sobolev classes on Lipschitz curves with negative smoothness exponents and in terms of function spaces on the interval (0, 1) of the real line. The proofs of the main results are essentially based on an explicit description of the space TW 1 (Ω) for a planar domain with a peak due to the author. Necessary and sufficient conditions for q to be such that the Neumann problem is solvable provided that the boundary function belongs to L q (∂Ω) are given.