Smoothness of the gradient of weak solutions of degenerate linear equations
Suppose \(Q\) is a symmetric non-negative semidefinite matrix, the square root of which is Lipschitz continuous. The article considers the equation \[ \nabla \cdot (Q\nabla u) = b \] in a bounded domain. When considered in a suitably weak sense, one can consider solutions \((u,\nabla u) \in W^{1,2}_Q\), where \(\nabla u\) is not the usual weak gradient, but is unique in the sense that if \(u_1 = u_2 \in W^{1,2}_Q\), then \(\sqrt{Q}\nabla u_1 = \sqrt{Q}\nabla u_2\) almost everywhere. Writing the row vectors of \(\sqrt{Q}\) as \(V_i\) and the entries of \(\sqrt{Q}\) as \(w_l\), and approximating the solution of the equation by a sequence of smooth Lipschitz functions \(u_k\), the article presents the following theorem: If \(\sqrt{Q}\in C^{1,1}\) and \[ \limsup_{k \to \infty} \Big(\int \Big| \Big\{ \sum_{l} \left| w_l(x) \Big| \Big\}\nabla u_k(x) \right|^2 d x \Big)^{1/2} = L < \infty, \] then \(V_i \cdot \nabla u \in W^{1,2}_Q\), \(b \in W^{1,2}_Q\) and \[ \| V_i \cdot \nabla u \|_{W^{1,2}_Q} \leq C L, \] where the constant \(C\) depends on \(b\) and \(w_l\). The results are also presented in terms of vector field derivatives \(X = V \cdot \nabla\). The proof uses difference quotients along integral lines associated with the row vectors of \(\sqrt{Q}\).
degenerate elliptic differential equations
scientific article; zbMATH DE number 6839512