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The functional calculus for the Laplacian on Lipschitz domains. (English) Zbl 0725.35021

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1989, Exp. No. 4, 10 p. (1989).
The authors study inequalities between \(\| Af\|_ p\) and \(\| f\|_ p\) where A is the square root of the operator generated by the Dirichlet or Neumann problem from the Poisson equation. For a Lipschitzian domain they give the ranges for p’s, corresponding to inequalities or equivalence between \(\| Af\|_ p\) and \(\| f\|_ p\). The ranges for the power p are shown to be sharp. The Dahlberg apriori estimates for the Dirichlet problem in Lipschitz domain [see B. E. J. Dahlberg, Math. Scand. 44, 149-170 (1979; Zbl 0418.31003)] are extended to the Neumann problem. A counterexample is presented of a \(C^ 1\) domain and a solutions to the Dirichlet and Neumann problem resp. such that \(f\in C_ 0^{\infty}(\Omega)\), but \(\nabla^ 2u\not\in L^ 1(\Omega)\). In particular, a question of J. Nečas about \(\| \nabla u\|_ p\leq c\| f\|_ p\) for the Dirichlet problem for \(-\Delta u=div \vec f\), \(\vec f\in L^ p(\Omega)\), is answered (positively), ranges for p’s are found for the Lipschitz and \(C^ 1\) domains. Applications to the heat equation are given, too.
Reviewer: M.Krbec (Praha)

MSC:

35B45 A priori estimates in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35K05 Heat equation

Citations:

Zbl 0418.31003
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