Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators
The Dirichlet problem for \(Lu=0\) with data in \(L^2(\partial \Omega,d \sigma)\) or the Sobolev space \(L^2_1(\partial \Omega,d \sigma)\) is solved for all real homogeneous constant coefficient elliptic operators \(L=\sum_{|\alpha|=2m} a_\alpha D^\alpha\), \(m\in \mathbb{N}\), in bounded Lipschitz domains \(\Omega\subseteq \mathbb{R}^n\). Then, a perturbation technique due to Dahlberg and Kenig immediately implies solvability in \(L^p\) and \(L^p_1\) for \(2-\varepsilon <p<2+\varepsilon\) where \(\varepsilon>0\) depends on the Lipschitz character of \(\Omega\), \(m,n\), and the ellipticity constant associated with \(L\). The central question we are addressing here is what exactly does ellipticity (in the homogeneous case) imply about dilation invariant estimates near the boundary for solutions to the Dirichlet problem. Our Theorems provide a complete answer in the following sense. For any \(p\neq 2\) it is possible to find an \(m\), an \(n\), and a bounded Lipschitz domain in \(\mathbb{R}^n\) so that the Dirichlet problem is not solvable for data in \(L^p\) (and \(L^p_1)\) for \(L\) of order \(2m\). However, for a given \(m\) or \(n\) the situation is more complicated. All of the previous methods for solving boundary value problems for higher order operators on Lipschitz (or \({\mathcal C}^1)\) boundaries relied on special algebraic properties of the operators in question, namely that they were powers of second order operators. In this paper our main contribution is to overcome the difficulties in relaxing this assumption. Our approach is to obtain a priori dilation invariant estimates for known solutions in smooth domains and then to obtain the same estimates together with point-wise limits at the boundary in Lipschitz domains by an approximation argument. The two main tools used to do this are some new Rellich inequalities and the representation of solutions by means of layer potentials.
scientific article; zbMATH DE number 784170