Relaxation and Attainment Results for an Integral Functional with Unbounded Energy Well

نویسنده

  • BAISHENG YAN
چکیده

Consider functional I u R jjDuj L detDuj dx whose energy well consists of matrices satisfying j j L det We show that the relaxations of this functional in various Sobolev spaces are signi cantly di erent We also make several remarks concerning various p growth semiconvex hulls of the energy well set and prove an attainment result for a special Hamilton Jacobi equation jDuj L detDu in the so called grand Sobolev space W q R with q nL L Introduction and main results Given L consider function fL j j n L det on the space M n n of n n matrices and integral functional I u Z jfL Du x j dx Z jjDu x j L detDu x j dx where u is a mapping from domain R to R and Du x is the Jacobi matrix of u In general j j denotes the operator norm of m n matrix M m n de ned by j j maxh Rn jhj j hj The absolute energy minimizers of I u can be characterized as map pings satisfying the Hamilton Jacobi equation Du x ZL f M n n j j j L det g a e x In the terminology used for phase transition problems see e g the set ZL is the energy well of energy functional I u Note that ZL is also the boundary of the so called L quasiconformal set KL de ned by KL f M n n j j j L det g

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تاریخ انتشار 2002