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In this work we consider a nonparametric likelihood approach to multivariate density estimation with a regularization based on the Laplace operator. The complexity of the estimation problem is tackled by means of a finite element formulation, that allows great flexibility and computational tractability. The model is suitable for any type of bounded planar domain and can be generalized to the non-Euclidean settings. Within this framework, we as well discuss a new approach to clustering based on the concept of diffusion in a potential field, and a permutation-based procedure for one and two samples hypothesis testing.