Semi-Lagrangian (SL) schemes are a class of numerical schemes that try to mimic the method of characteristics at the numerical level, looking for the foot of the characteristic curve passing through every node, and following this curve for a single time step [1]. Differently to other conventional schemes like, e.g., those based on finite difference, SL are free from the CFL conditions allowing stability for large time steps, becoming powerful and preferable to other numerical methods in different application fields. Several image processing problems can be described by nonlinear partial differential equations (PDEs), which require a numerical solution. To name just a few, e.g., the image segmentation, image denoising, 3D reconstruction or image inpainting are among these. All the above mentioned problems have been successfully solved via SL schemes. In this talk we will focus on the use of such SL schemes to solve the image inpainting problem, which consists in reconstructing one or more missing parts of an image using information taken from the known part. More in details, a new semi-Lagrangian scheme for the game ∞-Laplacian is proposed in [2]. The convergence of such a scheme to the viscosity solution of the given problem (a Dirichlet problem associated with the game ∞-Laplacian) is demonstrated, proving that the scheme is consistent, and, thanks to the addition of an artificial viscosity term, it is also stable and monotone [2]. Considering the game p-Laplacian operator instead of the game ∞-Laplacian, a generalization of the Dirichlet problem is obtained, which has been explored in [3] to address the image inpainting problem. The numerical tests show the reliability of the proposed method and the advantages of taking a p > 1 in terms of execution time and accuracy. REFERENCES [1] Falcone, M., and Ferretti, R. Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. Society for Industrial and Applied Mathematics (SIAM) (2014). [2] Carlini, E., Tozza, S. A Convergent Semi-Lagrangian Scheme for the Game ∞-Laplacian. Dyn Games Appl (2024). https://doi.org/10.1007/s13235-024-00596-1 [3] Carlini, E., Tozza, S. A scheme for the game p-Laplacian and its application to image inpainting. Appl. Math. Comput. (2024) 461:128299. https://doi.org/10.1016/j.amc.2023.128299