An elliptic free boundary arising from the jump of conductivity

作者：Kim Sunghan^*; Lee Ki Ahm; Shahgholian Henrik

来源：Nonlinear Analysis-Theory Methods & Applications, 2017, 161: 1-29.

DOI：10.1016/j.na.2017.05.010

• Conductivity jump
• Free boundary problem
• Quasilinear elliptic equation

In this paper we consider a quasilinear elliptic PDE, div(A(x, u)del u) = 0, where the underlying physical problem gives rise to a jump for the conductivity A(x, u), across a level surface for u. Our analysis concerns Lipschitz regularity for the solution u, and the regularity of the level surfaces, where A(x,u) has a jump and the solution u does not degenerate.
In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACF-monotonicity formula with two different operators, that might be of independent interest, and surely can be applied in other related situations. The proof of the monotonicity formula is done through careful computations, and (as a byproduct) a slight generalization to a specific type of variable matrix-valued conductivity is presented.

• 出版日期2017-9