Univalent solutions of elliptic systems of Heinz-Lewy type
The author is concerned with the behavior of homeomorphisms \(u=(u^ 1,u^ 2)\) of a domain \(\Omega \subset {\mathbb{R}}^ 2\) which are solutions to a second order, elliptic quasilinear system of the form \[ - D_{\alpha}(a^{\alpha \beta}(x,u)D_{\beta}u^ k)=f^ k(x,u,Du),\quad k=1,2, \] where the right hand side may have quadratic growth with respect to Du. In the first paragraph the regularity of weak solutions of class \(H^ 1_ 2\cap L_{\infty}(\Omega)\) is investigated. It is shown that if the matrix \(a^{\alpha \beta}\) is uniformly elliptic and Hölder continuous with respect to its variables than any one-to-one weak solution of the above system has Hölder continuous first derivatives and the corresponding Hölder norm can be estimated in terms of the \(L_{\infty}\)-norm of u and the \(L_ 2\)-norm of Du. In the second paragraph the behavior of real-valued solutions \(\phi\) of a linear differential inequality \[ | D_{\alpha}(a^{\alpha \beta}(x)D_{\beta}\phi)| \leq C(| \phi | +| D\phi |) \] at a zero of \(\phi\) is studied and it is shown that if \(\phi (0)=0\) but \(\phi\) \(\not\equiv 0\) then for some integer m the limit \(\lim_{z\to 0}z^{-m}\phi_ z(z)\) exists and is non-zero, where \(z=x_ 1+ix_ 2\) and \(\phi_ z=(1/2)(\phi_{x_ 1}-i\phi_{x_ 2})\). Here the coefficients are again assumed to be Hölder continuous but do not depend on \(\phi\). This result is applied in the third paragraph to demonstrate the non-vanishing of the Jacobian of one-to-one mappings u satisfying a system of the form \[ - (1/a(u))D_{\alpha}(a(u)D_{\alpha}u^ k)=c^{\alpha \beta} h^ k_{i_ j}(u) D_{\alpha}u^ i D_{\beta}u^ j, \] where a and \(h^ k_{ij}\) are bounded and Hölder continuous, a positive, and \(c^{11}=c^{22}\), \(c^{12}=-c^{21}\). A corresponding uniform estimate for the Jacobian from below can also be obtained under suitable normalizations. The results of this paper generalize earlier work of E. Heinz and H. Lewy.
scientific article; zbMATH DE number 4144503
scientific article