Conditions for the existence of a fundamental solution of quasilinear elliptic equations and higher-order systems with discontinuous coefficients (Q1896988)

We investigate the existence of a fundamental solution for the elliptic system in \(\mathbb{R}^n\) \({\mathcal L} u \equiv \sum_{|\alpha |= t} (- D)^\alpha A_\alpha (x,u, \ldots, D^su) = \delta (x - y)\), where \(s + t\) is even, \(A_\alpha\), \(u\) are \(N\)-dimensional vector- valued functions, \(y\) is an arbitrary point in \(\mathbb{R}^n\), \(\delta\) is the delta function. The functions \(A_\alpha (x, \xi)\) are assumed to be measurable with respect to \(x\) and continuous with respect to \(\xi\) for almost all \(x\). In particular, \({\mathcal L}\) may be a linear operator with measurable bounded coefficients.