By Nazarov S. A., Sweers G. H.
Permit Ω be a site with piecewise tender boundary. regularly, it's most unlikely to procure a generalized answer u ∈ W 2 2 (Ω) of the equation with the boundary stipulations via fixing iteratively a method of 2 Poisson equations less than homogeneous Dirichlet stipulations. one of these method is bought by way of surroundings v = −Δu. within the two-dimensional case, this truth is named the Sapongyan paradox within the conception of easily supported polygonal plates. within the current paper, the 3-dimensional challenge is investigated for a website with a gentle facet Γ. If the variable beginning attitude α ∈ is below π far and wide at the facet, then the boundary-value challenge for the biharmonic equation is akin to the iterated Dirichlet challenge, and its resolution u inherits the positivity holding estate from those difficulties. within the case α ∈ (π, 2π), the approach of fixing the 2 Dirichlet difficulties needs to be converted via allowing infinite-dimensional kernel and co-kernel of the operators and picking the answer u ∈ (Ω) by way of inverting a definite quintessential operator at the contour Γ. If α(s) ∈ (3π/2,2π) for some extent s ∈ Γ, then there exists a nonnegative functionality f ∈ (Ω) for which the answer u adjustments signal contained in the area Ω. relating to crack (α = 2π far and wide on Γ), one must introduce a distinct scale of weighted functionality areas. hence, the positivity conserving estate fails. In a few geometrical events, the issues on well-posedness for the boundary-value challenge for the biharmonic equation and the positivity estate stay open.
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A boundary-value problem for the biharmonic equation and the iterated Laplacian in a 3D-domain with an edge by Nazarov S. A., Sweers G. H.