On elliptic systems in \(L^ 1\)
type: Item
instance of: scholarly article
title: On elliptic systems in \(L^ 1\)
zbMATH Open document ID: 0797.35037
author: Davide Guidetti
published in: Osaka Journal of Mathematics
publication date: 1994-07-07T00:00:00Z
review text: The author establishes the \(L^ 1\) estimates for the solutions of the boundary value problems of elliptic \(N \times N\) systems in nonvariational form under minimal smoothness hypothesis of the coefficients in a bounded open set \(\Omega\) of \(\mathbb{R}^ n\). It is assumed that the algebraic condition due to S. Agmon holds in order that the realization of the elliptic system under the given boundary conditions in \(L^ p(\Omega)^ N\), \(1<p<\infty\), generates an analytic semigroup. The author establishes very precise estimates of the kernels of the integral operators which give the solutions of the problem in the half space. Based on these estimates he shows that the following inequalities hold for the solution \(u\) of the system of equations \((A_ 1-\lambda) u=f\), \(\text{Re} \lambda \geq 0\), where \(A_ 1\) is the realization in \(L^ 1(\Omega)^ N\) of the elliptic system considered: \[ \| u \|_{L^ 1(\Omega)^ N} \leq C | \lambda |^{-1} \| f \|_{L^ 1(\Omega)^ N},\;\| u \|_{B^ \sigma_{1,\infty} (\Omega)^ N} \leq C | \lambda |^{\sigma/(2m)-1} \| f \|_{L^ 1(\Omega)^ N}\quad \text{ if } \quad 0<\sigma<2m. \] Hence \(A_ 1\) generates an analytic semigroup in \(L^ 1(\Omega)^ N\).
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 579204
zbMATH Keywords: Levi's method
zbMATH Keywords: Besov spaces
zbMATH Keywords: nonvariational form
zbMATH Keywords: analytic semigroup
reviewed by: Hiroki Tanabe
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 579204