Boundary value problems for the systems of elastostatics in Lipschitz domains

Area Integral Estimates for the Biharmonic Operator in Lipschitz Domains

“…However, C. Kenig has informed us that the techniques in Dahlberg and Kenig [7] for solutions to systems of equations should also give the area integral estimates we obtain here. Their idea is to build from the case of small Lipschitz constant via an argument like that of G. David [9].…”

“…In general, solutions of the bi-Laplacian on nonsmooth domains will not satisfy a maximal principle or Harnack principle, and there is no positive "harmonic" measure associated with the operator. Finally, the area integral results, together with an argument of Dahlberg and Kenig [7], lead to another proof of solvability of the regularity problem for A2 in Lp(Ri), 1 < p < 2. In Pipher and Verchota [13], it was shown that the Dirichlet problem for A2 in Lp (p > 2) could be reduced to solvability of the regularity problem in the dual range of p. This regularity problem was then shown to have Lp solutions for all 1 < p < 2 only in R3 and not in R", n > 4 (see also [14]).…”

“…In Pipher and Verchota [7] the Dirichlet problem with Lp data (p > 2) was solved in R and shown to be unsolvable for some Lipschitz domains in R", n > 4. Because the operator T defined above is not an Lp -> Lp mapping when p > 2, a different approach was required.…”

An Elastic Model for Volcanology

“…The following theorem about the jump relations of single and double potentials for Lipschitz domains is due to Dahlberg, Kenig and Verchota [68].…”

“…For Lipschitz domains the analysis is much more involved and, as for the Laplace operator, based on Rellich formulas. These results are contained in [68] and its companion article [79]. We recall here only the main aspects for the three-dimensional case.…”