In this work, we propose a nonparametric density estimation technique for space-time inhomogeneous Poisson point processes. We employ a penalized likelihood framework able to handle event data occurring over spatial regions with complex shape. The regularization term, guided by partial differential equations, ensures smoothness in the estimate. To substantiate our method, we provide theoretical validation. For the estimation procedure, we rely on advanced numerical techniques. Moreover, we incorporate uncertainty quantification tools into our methodology. Finally, we demonstrate the effectiveness of our proposed approach through simulation studies and an application to epidemiological data.