By Nazarov S. A., Sweers G. H.
Permit Ω be a website with piecewise tender boundary. usually, it really is most unlikely to procure a generalized resolution u ∈ W 2 2 (Ω) of the equation with the boundary stipulations by means of fixing iteratively a method of 2 Poisson equations lower than homogeneous Dirichlet stipulations. this type of approach is received via atmosphere v = −Δu. within the two-dimensional case, this truth is named the Sapongyan paradox within the concept of easily supported polygonal plates. within the current paper, the three-d challenge is investigated for a website with a tender aspect Γ. If the variable starting attitude α ∈ is under π in every single place at the side, then the boundary-value challenge for the biharmonic equation is such as the iterated Dirichlet challenge, and its resolution u inherits the positivity maintaining estate from those difficulties. within the case α ∈ (π, 2π), the approach of fixing the 2 Dirichlet difficulties needs to be transformed via allowing infinite-dimensional kernel and co-kernel of the operators and deciding upon the answer u ∈ (Ω) through inverting a definite indispensable operator at the contour Γ. If α(s) ∈ (3π/2,2π) for some degree s ∈ Γ, then there exists a nonnegative functionality f ∈ (Ω) for which the answer u adjustments signal contained in the area Ω. on the subject of crack (α = 2π all over the place on Γ), one must introduce a different scale of weighted functionality areas. consequently, the positivity conserving estate fails. In a few geometrical occasions, the issues on well-posedness for the boundary-value challenge for the biharmonic equation and the positivity estate stay open.
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