An optimization problem with volume constraint for a degenerate quasilinear operator

Abstract We consider the optimization problem of minimizing ∫ Ω | ∇ u | p d x with a constraint on the volume of { u > 0 } . We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution u is locally Lipschitz continuous and that the free boundary, ∂ { u > 0 } ∩ Ω , is smooth.

[1]  A free boundary problem with a volume penalization , 1996 .

[2]  Krerley Oliveira,et al.  AN OPTIMIZATION PROBLEM WITH FREE BOUNDARY GOVERNED BY A DEGENERATE QUASILINEAR OPERATOR , 2006 .

[3]  M. Flucher.  An asymptotic formula for the minimal capacity among sets of equal area , 1993 .

[4]  T. O’Neil.  Geometric Measure Theory , 2002 .

[5]  Neil S. Trudinger,et al.  On harnack type inequalities and their application to quasilinear elliptic equations , 1967 .

[6]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[7]  Andrew Acker,et al.  An Extremal Problem Involving Current Flow Through Distributed Resistance , 1981 .

[8]  L. Caffarelli,et al.  Existence and regularity for a minimum problem with free boundary. , 1981 .

[9]  Bernhard Kawohl,et al.  Rearrangements and Convexity of Level Sets in PDE , 1985 .

[10]  A. Petrosyan,et al.  Full regularity of the free boundary in a Bernoulli-type problem in two dimensions , 2006 .

[11]  Arshak Petrosyan,et al.  A minimum problem with free boundary for a degenerate quasilinear operator , 2005 .

[12]  Petri Juutinen,et al.  On the Equivalence of Viscosity Solutions and Weak Solutions for a Quasi-Linear Equation , 2001, SIAM J. Math. Anal..

[13]  Julián Fernández Bonder,et al.  Regularity of the Free Boundary in an Optimization Problem Related to the Best Sobolev Trace Constant , 2005, SIAM J. Control. Optim..

[14]  A. Petrosyan,et al.  A singular perturbation problem for the p-Laplace operator , 2003 .

[15]  Luis A. Caffarelli,et al.  An optimization problem with volume constraint , 1986 .

[16]  Luis A. Caffarelli,et al.  An optimization problem in heat conduction , 1987 .

[17]  Eduardo V. Teixeira,et al.  The nonlinear optimization problem in heat conduction , 2005 .