Discontinuous solutions of variational problems
Derek F. Lawden · Journal of the Australian Mathematical Society · 1959
The most elementary problem of the calculus of variations consists in finding a single-valued function y(x) , defined over an interval [ a, b ] and taking given values at the end points, such that the integral is stationary relative to all small weak variations of the function y ( x ) consistent with the boundary conditions. Since y′ occurs in the integrand, it is clear that I is only defined when y ( x ) is differentiable and accordingly when y ( x ) is continuous. Usually y ′( x ) is also continuous. Occasionally, however, the boundary conditions can only be satisfied and a stationary value of I found, by permitting y ′ ( x ) to be discontinuous at a finite number of points. The arc y = y ( x ) will then possess ‘corners’ and the well-known Weierstrass-Erdmann corner conditions [1]must be satisfied at all such points by any function y (x) for which I is stationary. Arcs y = y ( x ) for which y ′( x ) is continuous except at a finite number of points, are referred to as admissible arcs . In this paper, we shall extend the range of admissible arcs to include those for which y ( x ) is discontinuous at a finite number of points.