Regularity of solutions in the fundamental problem of the classical calculus of variations (Q1324911)

This note considers the fundamental problem of the classical calculus of variations, which means the problem of minimizing the functional \(\int^ b_ a L(t,x(t),x'(t))dt\) in the class of absolutely continuous functions \(x: [a,b]\to\mathbb{R}^ n\) satisfying the boundary conditions \(x(a)= A\) and \(x(b)= B\) with \(A,B\in \mathbb{R}^ n\). Assume that the Lagrangian \(L(t,x,y)\) is of \(C^ 1\) and strictly convex with respect to \(y\). The author proves that if for \(| y|> \lambda\) sufficiently large, 1. \(L(t,x,y)> \Lambda(b- a)^{-1}\), where \(\Lambda\) is the value of the problem; 2. \(L^ 2_ y(t,x,y)> N> p= \max_{| y|\leq \lambda} L^ 2_ y(t,x,y)\); 3. \(|(L_ x,L_ y)|< K\varphi(L^ 2_ y)L\) with \(\int^ N_ p ds/\varphi(s)> 2K\Lambda\); then every solution of the problem is smooth.