Interior \(W^{2, p(x)}\)-regularity for elliptic equations of nondivergence form
The paper deals with second order uniformly elliptic equations of the form \[ \sum_{i,j=1}^n a^{ij}(x) \dfrac{\partial^2 u}{\partial x_i\partial x_j}=f\quad \text{a.e. in} \;\Omega\subset \mathbb{R}^n, \] where the principal coefficients belong to \(L^\infty_{\text{loc}}(\Omega)\cap VMO_{\text{loc}}(\Omega).\) Assuming log-Hölder continuity of \(p(\cdot),\) the author proves that \(f\in L^{p(x)}(\Omega)\) implies \(u\in W^{2,p(x)}_{\text{loc}}(\Omega)\) for the strong solutions.
scientific article; zbMATH DE number 6511580
scientific article