A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners. (English) Zbl 1108.35043

Nazarov, Sergueï A.; Sweers, Guido

J. Differ. Equations 233, No. 1, 151-180 (2007).

Fourth order hinged plate type problems are usually solved via a mixed system of two second order equations. For smooth domains such an approach can be justified. However, when the domain has a concave corner the bi-Laplace problem with Navier boundary conditions may have two different types of solutions. A striking difference is that in general only the first solution, obtained by decoupling into a system, preserves positivity, that is, a positive source implies that the solution is positive.
The main part of this paper is concerned with the analytical treatment of the problem. A bounded uniformly Lipschitz domain is studied. The authors consider domains with one ‘concave’ boundary point in second section of this paper. It is assumed that the domain near this one point is like a cone. The Dirichlet problem for the Poisson equation is investigated in third section. It is shown that typical four and higher-dimensional concave boundary points do not destroy the positivity preserving property. The analytical results are illustrated by some numerical experiments for planar domains. The authors used FreeFem++ software. It is shown, firstly, that on a standard L-shaped domain both solutions are crucially different. Secondly, sign-change occurs for angles just below 270 degrees in sectorial domains.

Reviewer: V. Leontiev (Ulyanovsk)

MSC:

35J40	Boundary value problems for higher-order elliptic equations
74K20	Plates

Keywords:

higher order elliptic p.d.e.; positivity; hinged plate; non-smooth boundary

Software:

FreeFem++

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