One Dimensional Infinite-horizon Variational Problems arising in Continuum Mechanics

Abstract

In this paper we study a variational problem for real valued functions defined on an infinite semiaxis of the line. To wit, given x ∈ ℝ² we seek a “minimal solution” to the problem Minimize the functional given by

$$I\left( {w\left( \cdot \right)} \right) = \int\limits_{0}^{\infty } {f\left( {w\left( s \right),\ddot{w}\left( s \right)} \right)} ds,w \in Ax = \left\{ {\upsilon \in W_{{loc}}^{{2,1}}\left( {0,\infty } \right):\left( {\upsilon \left( 0 \right),\dot{\upsilon }\left( 0 \right) = x} \right)} \right\}.$$

((P∞))