The author studies two variants of the Stefan problem: First he considers the one-phase one-dimensional problem \[ \theta_ t- \theta_{xx}=0,\quad \theta (x,0)=\phi (x),\int^{s(t)}_{0}\theta (x,t)dx=E(t),\quad \theta (s(t),t)=0,\quad \theta_ x(s(t),t)=-s'(t). \] Here \(\phi\) and E are given \(\theta\) and s are unknown. He proves that the corresponding (weak) formulation for the function \(z(x,t)=\int_{x}\int^{t}\theta (\xi,\tau) d\tau d\xi\) has a solution. Second, he considers a multi-phase multi-dimensional problem for the enthalpy density u and temperature \(\theta =\beta (u)\), namely \(u_ t- \Delta \beta (u)=0\), \(u(x,0)=u_ 0(x)\), \(\beta (u(x,t))=h(x,t)+\gamma (t)\) on the boundary, \(\int_{\Omega}u(x,t)dx=E(t)\). He obtains a variational equality as well as a variational inequality formulation and proves in both cases, that a unique weak solution exists.