Optimization in problems involving the p-Laplacian

• Main Author: Monica Marras
• Format: Article
• Language: English
• Published: Texas State University 2010-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: p-Laplacian; energy integral; eigenvalues; rearrangements; shape optimization
• Online Access: http://ejde.math.txstate.edu/Volumes/2010/02/abstr.html

We minimize the energy integral $int_Omega | abla u|^p,dx$, where $g$ is a bounded positive function that varies in a class of rearrangements, $p>1$, and $u$ is a solution of $$displaylines{ -Delta_p u=g quadhbox{in } Omegacr u=0quad hbox{on } partialOmega,. }$$ Also we maximize the first eigenvalue $lambda=lambda_g$, where $$ -Delta_p u=lambda g u^{p-1}quadhbox{in }Omega,. $$ For both problems, we prove existence, uniqueness, and representation of the optimizers.