Partial regularity for an exponential PDE in crystal surface models

We study regularity properties of weak solutions to the boundary value problem for the equation −Δ ρ + au = f in a bounded domain Ω ⊂ R N , where ρ = e − div | ∇ u | p − 2 ∇ u + β 0 | ∇ u | − 1 ∇ u . This problem is derived from the mathematical modelling of crystal surfaces. It is known that the exponential term can be a measure-valued function. In this paper we obtain a partial regularity result, which asserts that there exists an open subset Ω 0 ⊂ Ω such that |Ω\Ω 0 | = 0 and the exponential term is locally bounded in Ω 0 . Furthermore, if x 0 ∈ Ω\Ω 0 , then ρ vanishes of N + 2 − ɛ order at x 0 for each ɛ ∈ (0, 2).