Communications in Mathematical Sciences

Volume 16 (2018)

Number 7

A penalty method for some nonlinear variational obstacle problems

Pages: 1757 – 1777



Hayden Schaeffer (Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)


We formulate a penalty method for the obstacle problem associated with a nonlinear variational principle. It is proven that the solution to the relaxed variational problem (in both the continuous and discrete settings) is exact for finite parameter values above some calculable quantity. To solve the relaxed variational problem, an accelerated forward-backward method is used, which ensures convergence of the iterates, even when the Euler–Lagrange equation is degenerate and nondifferentiable. Several nonlinear examples are presented, including quasi-linear equations, degenerate and singular elliptic operators, discontinuous obstacles, and a nonlinear two-phase membrane problem.


nonlinear obstacle problem, sparsity, penalty method, variational methods

2010 Mathematics Subject Classification

Primary 35A15. Secondary 35J60, 35R35.

H.S. acknowledges the support of AFOSR, FA9550-17-1-0125.

Received 9 January 2017

Accepted 20 March 2018

Published 7 March 2019