Minimization Problems and Linear and Nonlinear Spline Functions. II. Convergence

Abstract

The nonconvex problem of minimum curvature in $L^2 (a,b)$ generalizes naturally to the minimization in $L^p (a,b),\, 1 < p \leqq \infty $, of constrained nonlinear differential expressions of order $m \geqq 1$. Separating the cases $1 < p < \infty $ and $p = \infty $, we demonstrate the existence of extremal solutions for wide classes of such differential expressions. These extremal solutions, in the case of interpolation constraints, are used as approximants to develop a convergence theory for the approximation of $W^{m,p} (a,b)$ functions.