Another theorem of classical solvability `in small' for one-dimensional variational problems
A direct method for studying local minimizers of the variation problem \[ \int_a^bL(x,u(x),\dot{u}(x))dx\rightarrow \min,\;\; L(x,u,v):[a,b]\times R^{2}\rightarrow R, \] \[ u(a)=A,u(b)=B,\;\; u:[a,b]\rightarrow R \] is suggested on the basis of a regularity theory (a-priori estimates and compactness in \(C^1\)). The following aspects are investigated: 1. Is each weak local minimizer also a strong local minimizer? 2. Let \(u(\cdot)\) be a local minimizer and \(L\in C^{\infty},L_{vv}>0.\) Let \(L_n\in C^{\infty},(L_n)_{vv}>0\) and let \(\|L_n-L\|\rightarrow 0\) as \(n\rightarrow \infty .\) Is it correct that there exist local minimizers \(u_n(\cdot)\) associated with \(L_n\) such that \(\|u_n-u\|\rightarrow 0?\) 3. What are necessary and sufficient conditions to admit a local minimizer?
necessary and sufficient conditions for local minimizer
scientific article; zbMATH DE number 6101952
scientific article