On elliptic systems in \(L^ 1\)
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 \(L^ 1(\Omega)^ N\).
scientific article; zbMATH DE number 579204