A free boundary problem for the p(x)-Laplacian

作者：Fernandez Bonder Julian; Martinez Sandra; Wolanski Noemi

来源：Nonlinear Analysis-Theory Methods & Applications, 2010, 72(2): 1078-1103.

DOI：10.1016/j.na.2009.07.048

• Free boundaries
• Variable exponent spaces
• Minimization

We consider the optimization problem of minimizing integral(Omega) 1/p(x)|del u|(p(x)) + lambda(x)chi({u>0}) dx in the class of functions W-1,W-p(.)(Omega) with u - phi(0) is an element of W-0(1,p(.)) (Omega), for a given phi(0) >= 0 and bounded, W-1,W-p(.)(Omega) is the class of weakly differentiable functions with integral(Omega) |del u|(p(x)) dx < infinity. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Omega boolean AND partial derivative{u > 0}, is a regular surface.

• 出版日期2010-1-15