zbMATH — the first resource for mathematics

Periodic solution for a plate operator. (English) Zbl 1173.35087
The author investigates the existence and uniqueness of periodic solutions problem for the nonlinear plate problem:
\[ w''+\Delta^2w+|w'|^{p-2}w'=f\quad \text{in }Q, \]
\[ w=\frac{\partial w}{\partial \nu}=0\quad \text{on }\Sigma,\quad w(0)=w(T),\quad w'(0)=w'(T), \] where \(Q=\Omega\times (0,T)\), \(\Omega\subset \mathbb{R}^2\), \(\Sigma=\Gamma\times (0,T)\), \(\Gamma=\partial\Omega\). A solution \(w\) has the form \(w=u+u_0\), where \(u\) is a weak solution of the problem
\[ \frac d{dt}\left(u''+\Delta^2u+|u'|^{p-2}u'\right)=\frac {df}{dt}\quad \text{in }Q, \] \[ u=\frac{\partial u}{\partial \nu}=\quad \text{on }\Sigma,\quad u(0)=u(T),\quad u'(0)=u'(T),\quad \int_0^Tu(s)\,ds=0\;\text{ in }\Omega \] and \(u_0\) solves the Dirichlet-Neumann problem \(\Delta^2u_0=g_0\) in \(\Omega\), \(u_0=\frac{\partial u_0}{\partial \nu}=\;\text{on }\Gamma.\)
The main result is the existence and the uniqueness of a periodic solution \(w=u+u_0.\)
35L75 Higher-order nonlinear hyperbolic equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35B10 Periodic solutions to PDEs
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI
[1] Agmon, S., The \(L^p\) approach to the Dirichlet problem I, Ann. sc. norm. sup. Pisa, 405-448, (1959)
[2] Browder, F.E., ()
[3] Lions, J.L., ()
[4] Lions, J.L., (), (Nouvelle Présentation Dunod 2002)
[5] Prodi, G., Soluzioni periodiche de l’equazioni Della onde con terme dissipative non linéaire, Rend. sem. mat. Padova, 35, 38-49, (1968)
[6] Tanabe, H., Evolution equation, (1979), Pitman London
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.