In recent years, various results showed that second-order regularity of solutions to the p-Laplace equation can be properly formulated in terms of the expression under the divergence, the so-called stress field, see [3]. I will discuss the extension of these results to the anisotropic p-Laplace problem, namely equations of the kind \(-\mathrm{div}\,\big(\mathcal{A}(\nabla u)\big)=f\,, \) in which the stress field is given by \(\mathcal{A}(\nabla u)=H^{p-1}(\nabla u)\,\nabla_\xi H(\nabla u)\), where \(H(\xi)\) is a norm on \(\mathbb{R}^n\) satisfying suitable ellipticity assumptions. \(W^{1,2}\)-Sobolev regularity of \(\mathcal{A}(\nabla u)\) is established when \(f\) is square integrable, and both local and global estimates are obtained. The latter apply to solutions to homogeneous Dirichlet problems on sufficiently regular domains. A key point in our proof is an extension of Reilly's identity to the anisotropic setting. This is joint work with A. Cianchi, G. Ciraolo, A. Farina and V.G. Maz'ya. References [1] C.A. Antonini, G. Ciraolo, A. Farina, \textit{Interior regularity results for inhomogeneous anisotropic quasilinear equations}, Math. Ann. (2023). [2] C.A. Antonini, A. Cianchi, G. Ciraolo, A. Farina, V.G. Maz'ya, \textit{Global second-order estimates in anisotropic elliptic problems}, arXiv preprint (2023) arXiv:2307.03052. [3] A. Cianchi, V.G. Maz'ya, \textit{Second-order two-sided estimates in nonlinear elliptic problems}, Arch. Ration. Mech. Anal. 229 (2018), no. 2, 569-599.