## 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

### MSC:

 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

### Keywords:

nonlinear beam operator; mixed problem; hidden regularity

Zbl 0656.35097