Abstract: I will describe the more recent developments starting from results contained in two joint works with I. Birindelli and H. Ishii. In [1] we extend the well-known result on thin domains of Hale and Raugel ([3]) to fully nonlinear operators. 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- and first-order terms that have no equivalent in the Neumann case. Our approach differs from all subsequent works in that we use an "Evans-style" technique, often called the test function approach in the context of viscosity solutions.
More recently, we have addressed homogenization problems by considering oscillating boundary conditions; I will discuss some observations and open problems in this area.
[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
[2] I. Birindelli, A. Briani, H. Ishii, Fully nonlinear elliptic PDEs in thin domains with oblique boundary condition, SIAM Journal on Mathematical Analysis, 58, (2026), no.2 1232-1256.
[3] Jack K. Hale, Geneviève Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures Appl. (9) 71 (1992), no. 1, 33--95.
