Nondivergence elliptic equations with unbounded coefficients
The authors study the Dirichlet problem associated to the second order elliptic equation in nondivergence form \[ L=\sum_{i,j=1}^{n} a_{ij} (x) u_{x_i x_j}=f \] in a domain \(\mathbb R^n, \) \(n \geq 2\), \(f\) in \(L^2(\mathbb R^n) \) and, as in the main theorem, in \(L^2\log^\alpha(\mathbb R^n)\). The coefficients \(a_{ij} \) are assumed to be in the John-Nirenberg class of bounded mean oscillation functions and satisfy the ellipticity condition, not uniformly. The authors also discuss the case that \(a_{ij}\) are in the Sarason class of vanishing mean oscillation functions. We stress that the regularity result obtained do not rely on the representation formula of the derivatives of the solutions but they prove the estimate from below, considering that the second order minors of the Hessian matrix lie in the Hardy space \(H^1,\) and the well known \(H^1\)-BMO duality.
scientific article; zbMATH DE number 5509385