Minimization without weak lower semicontinuity

Abstract

The problemm of minimizing a real valued function ƒ on a subset M of a reflexive Banach space E is reconsidered. Under fairly weak restrictions on the pair (ƒM) we present a necessary and sufficient condition for the existence of a minimum. This necessary and sufficient condition is a restricted monotonicity condition for the Gâteaux–derivative ƒ' of ƒ, called weak K–monotonicity of ƒ'. The set depends in a subtle way on the pair (ƒM). We give some elementary properties and examples of weakly K–monotone maps and discuss their relation with previously used notions of monotonicity. The concept of weak K–monotonicity allows to show under which circumstances the standard assumption of weak sequential lower semicontinuity of ƒ occurs as natural restriction and when various weaker assumptions suffice to ensure the existence of a minimizer.

Keywords:

• Minimization
• weak lower semicontinuity
• non
• linear non
• elliptic PDE's