Green matrices and continuity of the weak solutions for the elliptic systems with lower order terms (Q2797894)

The paper studies existence, uniqueness and pointwise bounds for Green's matrix of divergence form, second order uniformly elliptic systems \(\mathcal{L}\) posed in bounded domains \(\Omega\subset\mathbb R^n\), \(n\geq 3\). Such result is proved when \(\mathcal{L}\) is coercive and its lower order coefficients belong to some Kato-Stummel type classes, provided that a unique Green's matrix \(G_0\) for the principal part of the operator exists and satisfies the same pointwise bounds as the desired ones.NEWLINENEWLINENEWLINEUsing some existence result for \(G_0\) of [\textit{S. Hofmann} and \textit{S. Kim}, Manuscr. Math. 124, No. 2, 139--172 (2007; Zbl 1130.35042)], [\textit{K. Kang} and \textit{S. Kim}, J. Differ. Equations 249, No. 11, 2643--2662 (2010; Zbl 1202.35079)], which relies on a vanishing mean oscillation property for the coefficients of the principal part, the authors can apply their result on Green's matrix and, proving a global Hölder estimate for \(G_0\) and a local boundedness of the weak solutions, they obtain the continuity of the weak solution \(u\in H^1(\Omega)\) of \(\mathcal{L}u=0\).NEWLINENEWLINENEWLINEFinally, under better conditions for the zero order coefficient, they prove interior and boundary Schauder estimates for the weak solution \(u\in H^1(\Omega)\) of the nonhomogeneous problem \(\mathcal{L}u=\nabla\cdot f\).