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The goal of the paper is four-fold. In the setting of non-smooth spaces withRicci curvature lower bounds (more precisely RCD(K,N) metric measure spaces):  - we develop an intrinsic theory of Laplacian bounds in viscosity sense andin a pointwise, heat flow related, sense, showing their equivalence also withLaplacian bounds in distributional sense;  - relying on these tools, we establish a PDE principle relating lower Riccicurvature bounds to the preservation of Laplacian lower bounds under theevolution via the $p$-Hopf-Lax semigroup, for general exponents$p\in[1,\infty)$. This principle admits a broad range of applications, goingmuch beyond the topic of the present paper;  - we prove sharp Laplacian bounds on the distance function from a set(locally) minimizing the perimeter with a flexible technique, not involving anyregularity theory; this corresponds to vanishing mean curvature in the smoothsetting and encodes also information about the second variation of the area;  - we initiate a regularity theory for boundaries of sets (locally) minimizingthe perimeter, obtaining sharp dimension estimates for their singular sets,quantitative estimates of independent interest even in the smooth setting andtopological regularity away from the singular set.  The class of RCD(K,N) metric measure spaces includes as remarkablesub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lowerRicci curvature bounds and finite dimensional Alexandrov spaces with lowersectional curvature bounds. Most of the results are new also in theseframeworks. Moreover, the tools that we develop here have applications toclassical questions in Geometric Analysis on smooth, non compact Riemannianmanifolds with lower Ricci curvature bounds.

Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds