Continuity of a multivalued mapping connected with the problem of minimizing a functional

Let $X$ and $U$ be locally convex spaces, $\varphi(x,u)$ a proper convex lower semicontinuous functional on $X\times U$ and $t=t(u)\geqslant\inf\{\varphi(x,u)\colon x\in X\}$. This paper gives conditions for the multivalued mapping
$$ \Phi_t\colon u\in U\to \Phi_t(u)=\{x\in X\colon\varphi(x,u)\leqslant t\} $$
to be uniformly continuous and satisfy a Lipschitz condition, and determines the relation of $\Phi_t$ with other multivalued mappings, in particular, with a metric projection. On the basis of the functional conjugate to $\varphi$ a mapping conjugate to $\Phi_t$ is introduced and a condition for its upper semicontinuity is presented. The problem of minimizing a homogeneous convex functional on a convex set is considered.
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