One-dimensional infinite-horizon variational problems arising in continuum mechanics
The authors study a variational problem for real valued functions defined on an infinite semiaxis of the line. They seek a ``minimal solution'' to the following problem: minimize the functional \[ (1)\quad I(w(\cdot))=\int^{\infty}_{0}f(w(s),\dot w(s),\quad \ddot w(s))ds, \] \[ (2)\quad w\in A_ x=\{v\in W^{2,1}_{loc}(0,\infty):\quad (v(0),\dot v(0))=\underline x\}. \] Here \(W^{2,1}_{loc}\subset C^ 1\) denotes the Sobolev space of functions possessing a locally integrable second derivative, and \(f=f(w,p,r)\) is a smooth function satisfying \(f_{rr}\geq 0,\) \(f(w,p,r)\geq a| w|^{\alpha}-b| p|^{\beta}+c| r|^{\gamma}-d;\) \(a,b,c,d>0;\) \(\alpha,\gamma \in (1,\infty),\) \(\beta \in [1,\infty),\) \(\alpha >\beta,\) \(\gamma >\beta\) and \(\underline x\) is a given value. The organization of the paper is as follows: In Section 1 the connection of the variational problem (1),(2) with the equilibrium problem of a long slender bar of polymetric material under tension is discussed. In Section 2 a fixed endpoint variational problem is analyzed. In Section 3 a criterion is given for the solution of (1),(2) to be minimal. Section 4 gives the proof the existence of a minimal energy solution. The Section 5 is containing the main result of the paper: there always exists a periodic minimal solution of (1),(2). In Section 7 an interesting analytic result is established.
scientific article; zbMATH DE number 4100677