${\textit H}^1$ regularity of the inviscid total variation and Bingham minimisers for ${\textit H}^1$ data - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2021

${\textit H}^1$ regularity of the inviscid total variation and Bingham minimisers for ${\textit H}^1$ data

Résumé

The Bingham model for viscoplastic materials involves the minimization of a non-differentiable functional. The regularity of the associated solution is investigated here. The simplified scalar case is considered first: The total variation minimization problem seeks the unique minimizer $u ∈$ BV$(Ω)$ of bounded variation of the energy $\frac{1}{2} \|u − f\|^2_{L^2(Ω)} + |u|_{BV(Ω)}$ for data $f ∈ L^2(Ω)$ in a bounded Lipschitz domain $Ω ⊂ \mathbb{R}^n$. Our main result proves for a convex domain $Ω$ that $f ∈ H^1(Ω)$ implies $u ∈ H^1(Ω)$. A modification for homogeneous Dirichlet conditions involves an additional trace term $u \in L^1(∂Ω)$ and then $f ∈ H^1_0(Ω)$ implies $u ∈ H^1_0(Ω)$. In the case of the vector Bingham model without viscosity, the boundary conditions are difficult to handle, but we prove the local $H^1_{loc}(Ω)^n$ regularity of the solution for a right-hand side $f ∈ H^1_{loc}(Ω)^n$. The proofs rely on several generalizations of a lemma due to H. Brézis and on the approximation with small viscosity. As a consequence, 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.
Fichier principal
Vignette du fichier
TVMregularity7.pdf (415.37 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03284591 , version 1 (12-07-2021)

Identifiants

  • HAL Id : hal-03284591 , version 1

Citer

François Bouchut, Carsten Carstensen, Alexandre Ern. ${\textit H}^1$ regularity of the inviscid total variation and Bingham minimisers for ${\textit H}^1$ data. 2021. ⟨hal-03284591⟩
113 Consultations
162 Téléchargements

Partager

Gmail Facebook X LinkedIn More