Holder continuity for vector-valued minimizers of quadratic functionals

In this article we give a sufficient condition for interior everywhere Holder continuity of weak minimizers of a class of quadratic functionals with coefficients A(ij)(alpha beta)(center dot, u) belonging to the VMO-class, uniformly with respect to u is an element of R-N, and continuous with respect to u. The condition is global. It is typical for the functionals belonging to the class that the continuity moduli of their coefficients become slowly growing sufficiently far from zero. Some features of the main result are illustrated by examples.Web of Scienceart. no. 6

On Morrey and BMO regularity for gradients of weak solutions to nonlinear elliptic systems with non-differentiable coefficients

We consider weak solutions to nonlinear elliptic systems with non-differentiable coefficients which principal parts are split into linear and nonlinear ones. Assuming that the nonlinear part $g(x,u,z)$ is equipped by sub-linear growth in $z$ only for big value of $|z|$ (but the growth is arbitrarily close to the linear one), we prove the Morrey and BMO regularity for gradient of weak solutions

A note on regular points for solutions of parabolic systems

summary:A vector valued function $u=u(x,t)$, solution of a quasilinear parabolic system cannot be too close to a straight line without being regular

Regularity on the interior for the gradient of weak solutions to nonlinear second-order elliptic systems

We consider weak solutions to the Dirichlet problem for nonlinear elliptic systems. Under suitable conditions on the coefficients of the systems we obtain everywhere H"older regularity on the interior for the gradients of weak solutions. Our sufficient condition for the regularity works even though an excess of the gradient of solution is not very small. More precise partial regularity on the interior can be deduced from our main result. The main result is illustrated through examples at the end of this article

On Hölder regularity for vector-valued minimizers of quasilinear functionals

summary:We discuss the interior Hölder everywhere regularity for minimizers of quasilinear functionals of the type $$\mathcal A(u;\Omega )=\int _{\Omega } A_{ij}^{\alpha \beta }(x,u) D_{\alpha }u^iD_{\beta }u^j\,{\rm d}x$$ whose gradients belong to the Morrey space $L^{2,n-2}(\Omega ,\mathbb R^{nN})$

On Liouville theorem and Hölder continuity of weak solutions to some quasilinear elliptic systems of higher order

summary:The aim of this paper is to show that the Liouville-type property is a sufficient and necessary condition for the regularity of weak solutions of quasilinear elliptic systems of higher orders

A note on regularity for nonlinear elliptic systems

summary:The $L^{2,\lambda }$ - regularity of the gradient of weak solutions to nonlinear elliptic systems is proved

A note on regular points for solutions of nonlinear elliptic systems

summary:It is shown in this paper that gradient of vector valued function $u(x),$ solution of a nonlinear elliptic system, cannot be too close to a straight line without $u(x)$ being regular