Oblique derivative problem for elliptic equations in non-divergence form with VMOVMO coefficients

summary:A priori estimates and strong solvability results in Sobolev space $W^{2,p}(\Omega)$, $1<p<\infty$ are proved for the regular oblique derivative problem $$\begin{cases} \sum_{i,j=1}^n a^{ij}(x)\frac{\partial^2u}{\partial x_i\partial x_j} =f(x) \text{ a.e. } \Omega \\ \frac{\partial u}{\partial \ell}+\sigma(x)u =\varphi(x) \text{ on } \partial \Omega \end{cases}$$ when the principal coefficients $a^{ij}$ are $V\kern -1.2pt MO\cap L^\infty$ functions

Nonstationary Venttsel problems with discontinuous data

The paper deals with Venttsel boundary problems for second-order linear and quasilinear parabolic operators with discontinuousprincipal coefficients. These are supposed to be functions of vanishing mean oscillationwith respect to the space variables, while only measurabilityis required in the time-variable. We derive aprioriestimates in composite Sobolev spaces for the strong solutions, and develop maximal regularity and strong solvability theory for such problems

Survey on gradient estimates for nonlinear elliptic equations in various function spaces

Very general nonvariational elliptic equations of $p$-Laplacian type are treated. An optimal Calderón-Zygmund theory is developed for such a nonlinear elliptic equation in divergence form in the setting of various function spaces including Lebesgue spaces, Orlicz spaces, weighted Orlicz spaces, and variable exponent Lebesgue spaces. The addressed arguments also apply to Morrey spaces, Lorentz spaces and generalized Orlicz spaces

Global Solvability of Dirichlet Problem for Fully Nonlinear Elliptic Systems

We show existence theorems of global strong solutions of Dirichlet problem for second order fully nonlinear systems that satisfy the Campanato’s condition of ellipticity. We use the Campanato’s near operators theory