  Building upon the dynamic programming principle for set valued functions
arising from many applications, in this paper we propose a new notion of set
valued PDEs. The key component in the theory is a set valued It\^{o} formula,
characterizing the flows on the surface of the dynamic sets. In the contexts of
multivariate control problems, we establish the wellposedness of the set valued
HJB equations, which extends the standard HJB equations in the scalar case to
the multivariate case. As an application, a moving scalarization for certain
time inconsistent problems is constructed by using the classical solution of
the set valued HJB equation.
