Maximal \(L^{p}\)-regularity for the Laplacian on Lipschitz domains
The author considers the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains \(\Omega,\) both with the following two domains of definition: \[ D_1(\Delta)=\{u\in W^{1,p}(\Omega): \;\Delta u\in L^p(\Omega),\;Bu=0\} \] or \[ D_2(\Delta)=\{u\in W^{2,p}(\Omega): \;Bu=0\} \] where \(B\) is the boundary operator. It is proved that, under certain restrictions on the range of \(p,\) these operators generate positive analytic contraction semigroups on \(L^p(\Omega)\) which implies maximal regularity for the corresponding Cauchy problems. In particular, if \(\Omega\) is bounded and convex and \(1<p\leq 2\) the Laplacian with domain \(D_2(\Delta)\) has the maximal regularity property, as in the case of smooth domains. Moreover, it is constructed an example that proves that, in general, the Dirichlet-Laplacian with domain \(D_1(\Delta)\) is not even a closed operator.
scientific article; zbMATH DE number 5189585
scientific article