The interior $C^{1,\gamma}$-regularity for weak solutions with BMO-gradient of nonlinear nonautonomous second order systems satisfying Legendre-Hadamard ellipticity condition is investigated.

It is shown that the reflection principle holds for $K$--quasiminimizers in $\mathbb{R}^n$, $n \geq 2$, provided that $K \in [1, 2)$. For $n = 1$ the principle holds for all $K \geq 1$ and an example shows that $K$ is not preserved in the reflection process. A local integrability result up to the boundary is proved for the derivative of a quasiminimizer in $\mathbb{R}^n$, $n \geq 1$; the result is needed for the reflection principle.

For a class of strongly elliptic, second order systems $L$ with rough coefficients on a Lipschitz domain $\Omega$, we show that if $Lu=0$ on $\Omega$ and $u$ vanishes on an open subset of the boundary, then weak a priori hypotheses on the nontangential maximal function of $u$ lead to strong estimates on $\nabla u$, in nontangential and Besov norms, near this subset.

In this study we deal with the nonlinear Riemann--Hilbert problem (in short (RHP)) for generalized analytic functions in multiply connected domains. Using a similarity principle for multiply connected domains (presented here for the first time), we reduce the nonlinear RHP for generalized analytic functions to a corresponding nonlinear RHP for holomorphic functions with Hölder continuous boundary data. Then the Newton--Kantorovič method combined with a continuation procedure as well as a new existence theorem for holomorphic solutions, which is based on topological degree arguments, leads to existence of at least two topologically different generalized analytic functions solving the nonlinear RHP.

We prove a long time existence of special regular solutions to the Navier-Stokes equations in an axially symmetric domain in $\mathbb{R}^3$, with boundary slip conditions and with inflow and outflow. We assume that an initial angular component of velocity and an angular component of the external force and angular derivatives of the cylindrical components of initial velocity and of the external force are sufficiently small in corresponding norms. We assume also that inflow and outflow is sufficiently close to homogeneous. Then there exists a solution such that velocity belongs to $W_{5/2}^{2,1}(\Omega^T)$ and gradient of pressure to $L_{5/2}(\Omega^T)$, and we do not have restrictions on~$T$.

The Schwarz problem for the Cauchy-Riemann equation and the Dirichlet and Neumann problems for the Poisson equation are explicitly solved in a half disc and a half ring of the complex plane. The respective Poisson kernels and the Green and Neumann functions are given.

We show that, for $\Omega$ a bounded convex domain of $\mathbb{R}^2$, any $2\times 2$ symmetric matrix $A(x)$ with $\det A(x)=1$ for a.e. $x\in\Omega$ satisfying the ellipticity bounds $$\frac{|\xi|^2}{H}\le \langle A(x)\xi,\xi\rangle \le H|\xi|^2$$ for a.e. $x\in\Omega$ and for all $\xi\in\mathbb{R}^2$ can be approximated, in the sense of $G$-convergence, by a sequence of matrices of the type $$\left(\begin{matrix}\gamma_j(x)& 0\\ 0&\frac{1}{\gamma_j(x)}\end{matrix}\right)$$ with $$H-\sqrt{H^2-1}\le \gamma_j(x)\le H+\sqrt{H^2-1}\,.$$

Aim of this paper is to introduce a generalization of Gelfand-Shilov classes by means of estimates based on a complete polyhedron (see for instance Gindikin-Volevich [10]). This class includes the standard Gelfand-Shilov functions and their anisotropic version.

In this paper we study the minimal periods on a holomorphic map which are preserved by any of its deformation considering separately the case of continuous and holomorphic homotopy. A complete description of the set of such minimal periods for holomorphic self-map of a compact Riemann surface is given. It shows that a nature of answer depends on the geometry of the surface distinguishing the parabolic case of the Riemann sphere, elliptic case of tori and the hyperbolic case of a surface of genus $\geq 2$.

We study a division problem for holomorphic functions that vanish to sufficiently high order near the singularity of a singular complex space.