Given a bounded domain $\Omega$ $h:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ defined by $$ h(u)=\int_{\Omega}|Du|^{p-2}(Du,Dh)_{\mathbb{R}^N}dz$$ why is it continuous and maximal monotone ?

  • $\begingroup$ because it is the subgradient of a convex and continuous function $\endgroup$ – daw May 22 at 6:34
  • $\begingroup$ How can we show it is continuous $\endgroup$ – Mono May 22 at 6:44
  • $\begingroup$ Continuity follows from Hölder's inequality. $\endgroup$ – gerw May 22 at 17:22

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