The regularity of solutions to some variational problems, including the p-Laplace equation for 2p<3
ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 4, pp. 1543-1553.

We consider the higher differentiability of solutions to the problem of minimizing

Ω [L(v(x))+g(x,v(x))]dxonu 0 +W 0 1,p (Ω)
where L(|ξ|)=1/p|ξ|p and u 0 W 1,p (Ω). We show that, for 2p<3, under suitable regularity assumptions on g, there exists a solution u to the Euler–Lagrange equation associated to the minimization problem, such that
uW loc 1,2 ().
In particular, for g(x,u)=f(x)u with fW 1,2 (Ω) and 2p<3, any W 1,p (Ω) weak solution to the equation
div (|u| p-2 u)=f
is in W 2,2 loc (Ω).

DOI: 10.1051/cocv/2016064
Classification: 49K10
Keywords: Regularity of solutions to variational problems – p-harmonic functions – higher differentiability
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     title = {The regularity of solutions to some variational problems, including the $p${-Laplace} equation for $2 \leq{} p< 3$},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1543--1553},
     publisher = {EDP-Sciences},
     volume = {23},
     number = {4},
     year = {2017},
     doi = {10.1051/cocv/2016064},
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Cellina, Arrigo. The regularity of solutions to some variational problems, including the $p$-Laplace equation for $2 \leq{} p< 3$. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 4, pp. 1543-1553. doi : 10.1051/cocv/2016064. http://archive.numdam.org/articles/10.1051/cocv/2016064/

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