I will describe the more recent developments starting from the results  contained in two joint works with I. Birindelli and  H. Ishii. In [1] we extend to fully nonlinear operators of the well known result on thin domains of Hale and Raugel ([3]). This result is more general even in the case of the Laplacian. In [2] we consider oblique boundary condition, and find some new phenomena, in particular the limit equations contain ”new terms” in the second, first and zeroth order terms which don’t have an equivalent in the Neumann case. More recently we addressed homogenization problems considering oscillatory boundary, I will discuss some remarks and open problems in this framework.


[1] I. Birindelli, A. Briani, H. Ishii, Test functions approach to fully nonlinear equation in thin domains, Proceedings of the American Mathematical Society,153 (2025), no. 5, 2099–2113 : https://arxiv.org/abs/2404.19577.
[2] I. Birindelli, A. Briani, H. Ishii, Fully nonlinear elliptic PDEs in thin domains with oblique boundary condition, preprint, arXiv: https://arxiv.org/abs/2410.23925.
[3] Jack K. Hale, Geneviève Raugel,  Reaction-diffusion equation on thin domains, J. Math. Pures Appl. (9) 71 (1992), no. 1, 33--95.