The direct method takes a minimizing sequence, extracts a subsequence using compactness, and uses lower semicontinuity to show that its limit attains the infimum.
Boundary data satisfy the bounded slope condition with constant when every boundary point admits affine lower and upper supporting functions of Lipschitz constant at most . Comparison with these barriers gives a global Lipschitz bound for convex variational problems.
For suitable convex integral functionals, ordered boundary values of two minimizers imply the same ordering in the domain. Affine minimizers can therefore serve as upper and lower barriers.
Articles by others on the same topic
The Direct Method in the calculus of variations is a powerful approach used to find the extrema (minima or maxima) of functionals, which are mappings from a space of functions to the real numbers. This method primarily involves establishing the existence of a solution to a variational problem and typically uses concepts from analysis, compactness, and weak convergence.