H1 regularity of the minimizers for the inviscid total variation and Bingham fluid problems for H1 data

The Bingham fluid model for viscoplastic materials involves the minimization of a nondifferentiable functional. The regularity of the associated solution is investigated here. The simplified scalar case is considered first: the total variation minimization problem. Our main result proves for a convex domain $\Omega$ that a right-hand side $f\in H^1\left(\Omega \right)$ gives a solution $u\in H^1\left(\Omega \right)$. Homogeneous Dirichlet conditions involve an additional trace term, then $f\in H_0^1\left(\Omega \right)$ implies $u\in H_0^1\left(\Omega \right)$. In the case of the inviscid vector Bingham fluid model, boundary conditions are difficult to handle, but we prove the local $H_{\text{loc}}^1{\left(\Omega \right)}^n$ regularity of the solution for $f\in H_{\text{loc}}^1{\left(\Omega \right)}^n$. The proofs rely on several generalizations of a lemma due to Brézis and on the viscous approximation. We obtain Euler–Lagrange characterizations of the solution. Homogeneous Dirichlet conditions on the viscous problem lead in the vanishing viscosity limit to relaxed boundary conditions of frictional type.