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- Title
THE REGULARITY OF SOLUTIONS TO SOME VARIATIONAL PROBLEMS, INCLUDING THE p-LAPLACE EQUATION FOR 3 ≤ p < 4.
- Authors
Cellina, Arrigo
- Abstract
We consider the higher differentiability of solutions to the problem of minimising Z ∫Ω[L(∇v(x)) + g(x, v(x))]dx on u0 +W10,p Ω ( ) where í ⊂ ⊂N, L(ξ) = l(∣ξ∣) = 1/ p ∣ξ∣p and u0 2 W1,p(Ω) and hence, in particular, the higher differentiability of weak solution to the equation div(∣ru∣p-2∇u) = f: We show that, for 3 ≤ p < 4, under suitable assumptions on g, there exists a solution u* to the Euler-Lagrange equation associated to the minimisation problem, such that ∇u* ∊ Ws,2loc (Ω ) for 0 < s < 4-p. In particular, for p = 3, we show that the solution u* is such that ∇u* ∊ Ws,2loc (Ω) for every s < 1. This result is independent of N. We present an example for N = 1 and p = 3 whose solution u is such that ∇u* is not in W1,2loc (Ω ), thus showing that our result is sharp.
- Subjects
LAPLACE distribution; INVERSE problems; EULER-Lagrange system; MATHEMATICAL models; PRESUPPOSITION (Logic)
- Publication
Discrete & Continuous Dynamical Systems: Series A, 2018, Vol 38, Issue 8, p4071
- ISSN
1078-0947
- Publication type
Article
- DOI
10.3934/dcds.2018177