On the torsion function with Robin or Dirichlet boundary conditions

Dorin Bucur (1) , Michiel van den Berg (2)

1 LAMA - Laboratoire de Mathématiques
2 School of mathematics

For p epsilon (1, + infinity) and b epsilon (0, + infinity] the p-torsion function with Robin boundary conditions associated to an arbitrary open set Omega subset of R-m satisfies formally the equation -Delta(p) = 1 in Omega and vertical bar del u vertical bar(p-2) partial derivative u/partial derivative u + b vertical bar u vertical bar(p-2)u = 0 on partial derivative Omega. We obtain bounds of the L-infinity norm of u only in terms of the bottom of the spectrum (of the Robin p-Laplacian), b and the dimension of the space in the following two extremal cases: the linear framework (corresponding to p = 2) and arbitrary b > 0, and the non-linear framework (corresponding to arbitrary p > 1) and Dirichlet boundary conditions (b = +infinity). In the general case, p not equal 2, p epsilon (1, +infinity) and b > 0 our bounds involve also the Lebesgue measure of Omega

Domaines: Equations aux dérivées partielles [math.AP]

Dorin Bucur, Michiel van den Berg. On the torsion function with Robin or Dirichlet boundary conditions. Journal of Functional Analysis, 2014, 266 (3), pp.1647-1666. ⟨10.1016/j.jfa.2013.07.007⟩. ⟨hal-00956613⟩