Regularity and analyticity of solutions in a direction for elliptic equations (Q2516658)

The authors study solutions to the equation \[ \sum\limits_{i,j=1}^n a_{ij}(x) u_{x_i x_j}\sum\limits_{i=1}^n a_{i}(x) u_{x_i} +c(x)u=f(x)\text{ in }\Omega\subset\mathbb R^n, \] assuming that the equation is uniformly elliptic and \(b_i\), \(c\in L^\infty(\Omega)\), \(f\in L^n(\Omega)\). Let \(u\in W^{2,n}(\Omega)\) be a strong solution. It is proved that \(u\) is analytic in the variable \(x_n\) if the coefficients \(a_{ij}\) are continuous and \(a_{ij}\), \(b_i\), \(c\), \(f\) are analytic in the variable \(x_n\). In dimension two, analyticity is obtained without continuity assumption on \(a_{ij}\). Another result of the paper under review gives the Hölder continuity of the second-order derivatives of \(u\) in a direction if the coefficients \(a_{ij}\), \(b_i\), \(c\) and the inhomogeneous term \(f\) are Hölder-continuous in this direction. A brief discussion on equations of divergence form is included.