Higher differentiability for solutions to a class of obstacle problems

Abstract

We establish the higher differentiability of integer and fractional order of the solutions to a class of obstacle problems assuming that the gradient of the obstacle possesses an extra (integer or fractional) differentiability property. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form

$$\begin{aligned} \int _{\Omega } \langle \mathcal {A}(x, Du), D(\varphi - u) \rangle \, dx \ge 0 \qquad \forall \varphi \in \mathcal {K}_{\psi }(\Omega ) \end{aligned}$$

where $\mathcal {A}$ is a p-harmonic type operator, $\psi \in W^{1,p}(\Omega )$ is a fixed function called obstacle and $\mathcal {K}_{\psi }=\{w \in W^{1,p}(\Omega ): w \ge \psi \,\,$ a.e. in $\Omega \}$ is the class of the admissible functions. We prove that an extra differentiability assumption on the gradient of the obstacle transfers to Du with no losses in the natural exponent of integrability, provided the partial map $x\mapsto \mathcal {A}(x,\xi )$ possesses a suitable differentiability property measured either in the scale of the Sobolev space $W^{1,n}$ or in that of the critical Besov–Lipschitz spaces $B^\alpha _{\frac{n}{\alpha }, q}$, for a suitable $1\le q\le +\infty$.