Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to $x$.Comment: 24 pages, to be submitte

On the L_p-solvability of higher order parabolic and elliptic systems with BMO coefficients

We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.Comment: 44 pages, introduction revised, references expanded. To appear in Arch. Rational Mech. Ana

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

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

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

Maximum principle for a kind of elliptic systems with Morrey data

We consider nonlinear elliptic systems satisfying componentwise coercivity condition. The nonlinear terms have controlled growths with respect to the solution and its gradient, while the behaviour in the independent variable is governed by functions in Morrey spaces. We obtain maximum principle for such kind of systems