Potential theory for the sum of the 1-Laplacian and p-Laplacian. (January 2015) - Krügel, Florian
Abstract: The variational problem of minimizing the functional u ↦ ∫ Ω | D u | + 1 p ∫ Ω | D u | p − ∫ Ω a u on a domain Ω ⊂ R n under zero boundary values, which appears for example in the theory of Bingham fluids, shows interesting phenomena such as the formation of a subset of Ω with positive measure where the minimizer is constant (a "plateau"). The corresponding Euler–Lagrange equation should be − Δ 1 u − Δ p u = a, but the 1-Laplacian is not defined in points where D u = 0 . We show that it is nevertheless possible to define subsolutions and supersolutions for this equation. We show that the comparison principle is valid, and we begin developing a potential theory analogous to what is known about the p -Laplacian. In addition we apply those notions to rotationally invariant solutions, and we discuss properties of the plateau and a possible connection to the theory.