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

$${\int}_{\Omega}[L\left(\nabla v\left(x\right)\right)+g(x,v\left(x\right))]\mathrm{d}x\phantom{\rule{1em}{0ex}}\text{on}\phantom{\rule{1em}{0ex}}{u}^{0}+{W}_{0}^{1,p}\left(\Omega \right)$$ |

^{p}and ${u}^{0}\in {W}^{1,p}\left(\Omega \right)$. We show that, for $2\le p<3$, under suitable regularity assumptions on $g$, there exists a solution $u$ to the Euler–Lagrange equation associated to the minimization problem, such that

$$\nabla u\in {W}_{\mathrm{loc}}^{1,2}\left(\right).$$ |

$$\mathrm{div}\left(\right|\nabla u{|}^{p-2}\nabla u)=f$$ |

Keywords: Regularity of solutions to variational problems – p-harmonic functions – higher differentiability

@article{COCV_2017__23_4_1543_0, author = {Cellina, Arrigo}, 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}, mrnumber = {3716932}, zbl = {1381.49015}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016064/} }

TY - JOUR AU - Cellina, Arrigo TI - The regularity of solutions to some variational problems, including the $p$-Laplace equation for $2 \leq{} p< 3$ JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1543 EP - 1553 VL - 23 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016064/ DO - 10.1051/cocv/2016064 LA - en ID - COCV_2017__23_4_1543_0 ER -

%0 Journal Article %A Cellina, Arrigo %T The regularity of solutions to some variational problems, including the $p$-Laplace equation for $2 \leq{} p< 3$ %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1543-1553 %V 23 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016064/ %R 10.1051/cocv/2016064 %G en %F COCV_2017__23_4_1543_0

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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