Contractive properties for the heat equation in Sobolev spaces (Q1066052)

Let S(t) denote the heat semigroup with zero Dirichlet data, acting on functions in a bounded smooth domain \(\Omega\). For \(1<p<\infty\), let \(q=p/(p-1)\), and let \(\lambda_ p\) be the smallest eigenvalue of - 4/pq\(\cdot \Delta\) with the boundary condition \[ 4/pq\cdot \partial_{\nu}u+(n-1)\cdot H\cdot u=0, \] where H is the mean curvature of \(\partial \Omega\). It is shown that \(e^{\lambda_ pt}\cdot S(t)\) is a contraction in \(W_ 0^{1,p}(\Omega)\). Thus, if \(H\geq 0\) everywhere on \(\partial \Omega\), then the well-known result follows that the gradient of \(u(\cdot,t)=S(t)u_ 0\) attain its maximum for \(t=0\), by sending p to infinity. A related result is shown for the case of zero Neumann boundary conditions. For the case of arbitrary geometries, equivalent norms to the \(W^{1,p}\)-norm are constructed explicitly for which S(t) is a contraction semigroup.