Half century ago, Umberto Mosco was the ``relatore di tesi (tesi about the Mosco- convergence) di laurea'' of the first author; a quart of century ago, the first author was the ``relatore di tesi di laurea'' of the second author. The roots of this paper are the Mosco-convergence of convex sets and the minimization of integral functionals of the Calculus of Variations. We consider integral functionals of the type J(v)=∫_Ω j(x,Dv)-∫_Ω f(x)v(x). We study the existence of T-minima (infinite energy minima) on convex sets of the Sobolev space W₀^1,p(Ω) and the stability of the T-minima under the Mosco-convergence of the convex sets.