Strict convexity and the regularity of solutions to variational problems∗
Dipartimento di Matematica e Applicazioni, Università degli Studi
di Milano-Bicocca, Via R. Cozzi
We consider the problem of minimizing where Ω is a bounded open subset of ℝN and L is a convex function that grows quadratically outside the unit ball, while, when | ∇v | < 1, it behaves like | ∇v | p with 1 <p< 2. We show that, for each ω ⊂ ⊂ Ω, there exists a constant H, depending on ω but not on p, such that both in particular, for every i = 1,...N, we have .
Mathematics Subject Classification: 49K10
Key words: Regularity of solutions / higher differentiability / strict convexity
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