Regularity property for the nonlinear beam operator. (English) Zbl 0702.35052

Author’s summary: “We study the regularity on the lateral boundary \(\Sigma\) of \(\Delta u\) where u is a weak solution of the mixed problem for the equation \[ \rho(x)\partial^ 2u/\partial t^ 2+ \Delta^ 2u- M(\int_{\Omega}| \nabla u|^ 2 dx)\Delta u=f\text{ in } \Omega\times]0,T[, \] \(\Omega\) is a bounded open set of \({\mathbb{R}}^ n\) with regular boundary \(\Gamma\) and \(\Sigma =\Gamma \times]0,T[\). Here \(\rho(x)\) and \(M(\lambda)\) are real functions such that \(\rho(x)\geq \rho_ 0>0\) and \(M(\lambda)\geq -\xi_ 1\), for -\(\lambda\geq 0\), \(\xi_ 1\geq 0\) appropriately chosen. This type of result was called by J. L. Lions [Mat. Apl. Comput. 6, 7-16 (1987; Zbl 0656.35097)] of hidden regularity of u.
Reviewer: A.D.Osborne


35D10 Regularity of generalized solutions of PDE (MSC2000)
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35G30 Boundary value problems for nonlinear higher-order PDEs


Zbl 0656.35097