Area integral estimates for higher order elliptic equations and systems
Let \(\Omega\) be a domain in \(\mathbb{R}^n\), and \(L\) an elliptic equation (or system). When one studies boundary value problems for \(L\) in \(\Omega\), and the given data are in some \(L^p\), the boundary is to be understood in the sense of nontangential \(L^p\) estimates. When the boundary \(\partial \Omega\) of \(\Omega\) is just Lipschitz, there are additional difficulties. In this situation, it is known that the nontangential maximal function gives a very precise control over the growth of solutions of the boundary value problem. But there is another approach, via the square function of solutions, which presents some advanges. The goal of this article is to show the \(L^p\) equivalence (for \(0<p <\infty)\) between the nontangential maximal function and the square function of a solution \(u\). Namely, let \(\Gamma(Q) = \Gamma_a (Q) =\{x\in \Omega,| X- Q| \leq(i+a) d(X,\partial \Omega)\}\) be the nontangential approach region when \(Q\in \partial \Omega\), \(a>0\) depends only on the Lipschitz constant of \(\Omega\). Then the maximal nontangential function \(N(u)\) is defined by \(N(u)(Q)= \sup_{x\in \Gamma(Q)} | u(X) |\). The square function for a function \(u\) in \(\Omega\) is defined by \[ S(u)(Q)= \left(\int_{\Gamma (Q)} {\bigl| \nabla u(X) \bigr|^2 \over | X- Q|^{n-2}} dX \right)^{1/2}. \] The elliptic systems considered by the authors, called elliptic symmetric \(K\)-systems are of the form \(Lu= \sum^K_{l=1} L^{ql}u^l\) for \(u= (u^1, \dots, u^K)\), where \(L^{kl}= \sum_{| \alpha| =|\beta |=m} D^\alpha a^{kl}_{\alpha \beta} D^\beta\) for \(m,k,l\) position integers, \(a^{kl}_{\alpha \beta}\) real constants, with \(L^{kl}= L^{lk}\), and the system is strongly elliptic, i.e. \(\text{Re} \sum^k_{k,l=1} L^{kl} (\xi) \zeta_k \overline \zeta_lz \geq| \xi |^{2w} |\zeta |^2\) for \(\zeta\in \mathbb{C}^n\), \(\xi\in \mathbb{R}^n) \). Because of the fact that the boundary of \(\Omega\) is not necessarily smooth, the authors have to use an adapted distance function \(\delta\) (a variant of Dahlberg's adapted distance function [Stud. Math. 66, 13-24 (1979; Zbl 0422.31008)] used already by C. Kenig - E. Stein), whose merit is that \(\delta (X)\approx \text{dist} (X,\partial \Omega)\), and that \(\delta(X) |\nabla \nabla \delta(X) |^2dX\) is a Carleson measure; the transverse differentiation of \(\delta\) has also good properties. To give a feeling of the proofs, the authors show first that, for harmonic \(u\), \[ \int_{\partial \Omega} S^2(u)d \sigma \approx \int_{\partial \Omega}N^2 (u)d \sigma. \] Now, in the general case of higher order elliptic systems things are more complicated; the idea is to approximate from the inside \(\Omega\) by Lipschitz domains, and to use the properties of being a Charleson measure of \(\delta\), and a trick that allows on the boundary of a domain to change the indices of components of the unit normal vector with the indices of spatial derivatives by introducing tangential derivatives. This gives, for bounded Lipschitz domains with connected boundary, and elliptic \(K\)-systems of order \(2m\) an inequality which shows the fact that the nontangential maximal function \(N(u)\) of a solution is dominated by the square function \(S(u)\). The reciprocal uses in an essential way results of \textit{J. Pipher} and \textit{G. C. Verchota} [Ann. Math., II. Ser. 142, No. 1, 1-38 (1995; Zbl 0878.35035)], a result of Habicht on strictly positive polynomials with real coefficients, and a reduction to elliptic operators of a special form (i.e. of type \(\sum_{| \alpha | =p} a_\alpha (D^\alpha)^2\), \(a_\alpha> 0\) \(\forall\) \(\alpha\), \(|\alpha |=p)\). For the latter, tedions but elementary calculations give the result (here the adapted distance function is essential). Finally, the authors show how to modify the previous proofs for proving the \(L^p\) equivalence of \(N(u)\) and \(S(u)\).
scientific article; zbMATH DE number 1092700
scientific article