  The goal of the paper is four-fold. In the setting of non-smooth spaces with
Ricci curvature lower bounds (more precisely RCD(K,N) metric measure spaces):
  - we develop an intrinsic theory of Laplacian bounds in viscosity sense and
in a pointwise, heat flow related, sense, showing their equivalence also with
Laplacian bounds in distributional sense;
  - relying on these tools, we establish a PDE principle relating lower Ricci
curvature bounds to the preservation of Laplacian lower bounds under the
evolution via the $p$-Hopf-Lax semigroup, for general exponents
$p\in[1,\infty)$. This principle admits a broad range of applications, going
much 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 any
regularity theory; this corresponds to vanishing mean curvature in the smooth
setting and encodes also information about the second variation of the area;
  - we initiate a regularity theory for boundaries of sets (locally) minimizing
the perimeter, obtaining sharp dimension estimates for their singular sets,
quantitative estimates of independent interest even in the smooth setting and
topological regularity away from the singular set.
  The class of RCD(K,N) metric measure spaces includes as remarkable
sub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lower
Ricci curvature bounds and finite dimensional Alexandrov spaces with lower
sectional curvature bounds. Most of the results are new also in these
frameworks. Moreover, the tools that we develop here have applications to
classical questions in Geometric Analysis on smooth, non compact Riemannian
manifolds with lower Ricci curvature bounds.
