Global weighted Orlicz estimates for irregular obstacle problems with general growth over bounded nonsmooth domains

We study a generalized variational inequality with an irregular obstacle in the frame of Orlicz–Sobolev spaces. Over a bounded nonsmooth domain having a sufficiently flat boundary in the Reifenberg sense, a global weighted Orlicz estimate is established for the gradient of the solution to the obstacle problem assumed BMO smallness of a coefficient.

An extension of a norm inequality for a semi-discrete g*λ function

A norm inequality for a semi-discrete g*?(f) function is obtained for functions, f, that can be written as a sum whose terms consist of a numerical coefficient multiplying a member of a family of functions that have properties of geometric decay, minimal smoothness and almost orthogonality condition. The theorem is applied to the rate of change of u, a solution to Lu = div?f in a bounded, nonsmooth domain ? Rd, d?3, u = 0 on ??.

The Griffith formula and the Rice–Cherepanov integral for crack problems with unilateral conditions in nonsmooth domains

As a paradigm for non-interpenetrating crack models, the Poisson equation in a nonsmooth domain in R2 is considered. The geometrical domain has a cut (a crack) of variable length. At the crack faces, inequality type boundary conditions are prescribed. The behaviour of the energy functional is analysed with respect to the crack length changes. In particular, the derivative of the energy functional with respect to the crack length is obtained. The associated Griffith formula is derived, and properties of the solution are investigated. It is shown that the Rice–Cherepanov integral defined for the solutions of the unilateral problem defined in the nonsmooth domain is path-independent. Finally, a non-negative measure characterising interaction forces between the crack faces is constructed.