Maximal L p -regularity for the Laplacian on Lipschitz domains

Abstract

Abstract. We consider the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains Ω, both with the following two domains of definition:, where B is the boundary operator. We prove that, under certain restrictions on the range of p, these operators generate positve analytic contraction semigroups on L p (Ω) which implies maximal regularity for the corresponding Cauchy problems. In particular, if Ω is bounded and convex and 1 < p ≤ 2, the Laplacian with domain D 2 (∆) has the maximal regularity property, as in the case of smooth domains. In the last part, we construct an example that proves that, in general, the Dirichlet-Laplacian with domain D 1 (∆) is not even a closed operator.

Global regularity for divergence form elliptic equations on quasiconvex domains

“…In [16], Wood applied the semigroup theory to derive the maximal L p regularity for the Laplacian operator on Lipschitz domains, some counterexamples were constructed to support his results. As an application of a special variational argument based on Nirenberg difference quotient technique, Savaré [13] got the regularity of problem (1.…”

Data Assimilation for Linear Parabolic Equations: Minimax Projection Method

“…Existence and uniqueness of I ∈ B(T ) solving (A.1) was proved in [21]. The estimate (A.2) can be verified projecting (A.1) on the span of {ψ k := λ (Ω) with respect to the inner product u, v 1 := ∇u, ∇v L 2 (Ω) , and estimating the projection coefficients by applying the CauchySchwatrz-Bunyakovsky inequality.…”

Mathematical and numerical analysis of the time-dependent Ginzburg–Landau equations in nonconvex polygons based on Hodge decomposition

“…We need the following lemma concerning the maximal L p regularity of parabolic equations in a Lipscthiz domain [27].…”