Integral Functionals on L^p-Spaces: Infima Over Sublevel Sets

Abstract

In this article, we establish the following result: Let (T, ℱ, μ) be a σ-finite measure space, let Y be a reflexive real Banach space, and let ϕ, ψ: Y → R be two sequentially weakly lower semicontinuous functionals such that

for some p > 0. Moreover, assume that ϕ has no global minima, while ϕ + λψ is coercive and has a unique global minimum for each λ > 0. Then, for each γ ∈L ^∞(T) ∩ L ¹(T)∖{0}, with γ ≥0, and for each r > inf  _Y ψ, if we put

we have

Keywords:

• Coercivity
• Global minimum
• Integral functional
• L ^p -space
• Uniqueness

Mathematics Subject Classification:

• 49J99
• 49K99

ACKNOWLEDGMENTS

This article is dedicated to Professor Alfonso Villani, with esteem and friendship, on his 60th birthday. The author has been supported by the GNAMPA of the Istituto Nazionale di Alta Matematica (INdAM).

Part of the special issue, “Variational Analysis and Applications.”