The regularity of solutions to some variational problems, including the $p$-Laplace equation for $2\le p<3$

Cellina, Arrigo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1543-1553.

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

$${\int }_{\Omega }[L\left(\nabla v\left(x\right)\right)+g(x,v\left(x\right))]\mathrm{d}x\text{on}{u}^0+W_0^{1,p}\left(\Omega \right)$$

where L(|ξ|)=1/p|ξ|^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

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

In particular, for $g(x,u)=f\left(x\right)u$ with $f\in W^{1,2}\left(\Omega \right)$ and $2\le p<3$, any $W^{1,p}\left(\Omega \right)$ weak solution to the equation

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

is in $W^{2,2}$${}_{loc}\left(\Omega \right).$

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