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Zbl 0702.35052
Ramos Ch., Oswaldo
Regularity property for the nonlinear beam operator.
(English)
[J] An. Acad. Bras. Ciênc. 61, No.1, 15-25 (1989). ISSN 0001-3765; ISSN 1678-2690/e

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 $$\rho(x)\partial\sp 2u/\partial t\sp 2+ \Delta\sp 2u- M(\int\sb{\Omega}\vert \nabla u\vert\sp 2 dx)\Delta u=f\text{ in } \Omega\times]0,T[,$$ $\Omega$ is a bounded open set of ${\bbfR}\sp n$ with regular boundary $\Gamma$ and $\Sigma =\Gamma \times]0,T[$. Here $\rho(x)$ and $M(\lambda)$ are real functions such that $\rho(x)\ge \rho\sb 0>0$ and $M(\lambda)\ge -\xi\sb 1$, for -$\lambda\ge 0$, $\xi\sb 1\ge 0$ appropriately chosen. This type of result was called by {\it J. L. Lions} [Mat. Apl. Comput. 6, 7-16 (1987; Zbl 0656.35097)] of hidden regularity of u.
[A.D.Osborne]
MSC 2000:
*35D10 Regularity of generalized solutions of PDE
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35G30 Boundary value problems for nonlinear higher-order PDE

Keywords: nonlinear beam operator; mixed problem; hidden regularity

Citations: Zbl 0656.35097

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