Olli Martio: Effective estimates for quasiminimizers

Abstract: Let $p>1$. A function $u \in W^{1,p}_{loc}(\Omega)$, $\Omega$ open in $\mathbf{R}^n$, is a $K$--quasiminimizer if for all open sets $D \subset \subset \Omega$ $$ \int_{D} |\nabla u|^p \, dx \leq K \int_{D}
|\nabla v|^p \, dx $$ for all functions $v - u \in W^{1,p}_{0}(D)$. For
a condenser $(C,\Omega)$, $C$ compact in $\Omega$, there is a unique $p$--potential $u$ such that $\int_{\Omega} |\nabla u|^p \, dx = cap_p(C, \Omega).$ Here $cap_p(C, \Omega)$ refers to the variational $p$--capacity of the condenser $(C, \Omega)$. Estimates for the $p$--potential $u$ are well known. In the lecture a method to derive similar estimates for a $K$--quasiminimizing potential is described.