For weak solutions to the three-dimensional Navier-Stokes equations the interior regularity problem for the renormalized velocity $u(1+|u|^2)^{-\alpha/2}$ and pressure $p(1+|u|^2)^{-\beta/2}$ is investigated. If a velocity component is locally semibounded and $\nabla u$ slightly more regular than suitable weak solutions the regularity estimates for the renormalized velocity are improved. Furthermore, estimates for the negative part of a renormalized pressure are presented.

We study formal power series solutions to the initial value problem for the Burgers type equation $\partial_t u-\Delta u = X\big(f(u)\big)$ with polynomial nonlinearity $f$ and a vector field $X$, and prove that they belong to the formal Gevrey class $G^2$. Next we give counterexamples showing that the solution, in general, is not analytic in time at $t=0$. We also prove the existence of non-constant globally analytic solutions.

The Lie algebra of the affine group is generated by two operators $A$ and $B$ satisfying the commutator rule $ [A,B] = B$. A version of the uncertainty principle is designed such that - in the time domain - the extremal functions are real valued. The uncertainty inequality naturally contains a parameter. In the application the wavelet transform based on the extremal functions gives a model for the first stage of the hearing perception in the inner ear (the cochlea). The parameter in the uncertainty inequality is associated to the position along the cochlea.

We adapt a technique developed by Bojarski and Iwaniec in their celebrated 1983 paper [2] to prove second order differentiability results for $p$-harmonic functions to the case of the Heisenberg group. We prove that for $2\le p<4$ we have $X_i ( |Xu|^{(p-2)/p}\, X_j u) \in L^2_{\rm loc} (\Omega )$ and $X_i (|Xu|^p ) \in L^2_{\rm loc} (\Omega)$, where $u$ is a $p$-harmonic function in the Heisenberg group $\mathbb{H}^n$.

Every nonvanishing univalent function $f(z)$ in the disk $\Delta^* = \widehat{\mathbb{C}} \setminus \overline{\Delta}, \Delta = \{|z| < 1\}$, for example, with hydrodynamical normalization, generates a complex isotopy $f_t (z) = t f(t^{-1} z): \Delta^* \times \Delta \to \widehat{\mathbb{C}}$, which is a special case of holomorphic motions and plays an important role in many topics. Let $q_f$ denote the minimal dilatation among quasiconformal extensions of $f$ to $\widehat{\mathbb{C}}$. In 1995, R. Kühnau raised the questions whether the dilatation function $q_f(r) = q_{f_r}$ is real analytic and whether the function $f$ can be reconstructed if $q_f(r)$ is given. The analyticity of $q_f$ was known only for ellipses and for the Cassini ovals. Our main theorem provides a wide class of maps with analytic dilatations and implies also a negative answer to the second question.

We study the degree of regularity of the Jacobian determinant of a mapping of finite distortion $K$, under suitable integrability assumptions on $K$.

We classify the second order, linear, two by two systems for which the two fundamental theorems for planar harmonic mappings, the Radó--Kneser--Choquet theorem and the H. Lewy theorem, hold. They are those which, up to a linear change of variable, can be written in diagonal form with \emph{the same} operator on both diagonal blocks. In particular, we prove that the aforementioned theorems cannot be extended to solutions of either the Lamé system of elasticity, or of elliptic systems in diagonal form, even with just slightly different operators for the two components.

The central theme running through this article is the embedding of Sobolev weighted spaces into Sobolev-Orlicz spaces under minimal assumptions on the weight. As application we prove the existence and regularity of the minimum of functionals of the Calculus of Variations with growth governed by the weight we consider.

We study local boundedness of the local minimizers of functionals with $(p,q)$-growth on metric measure spaces equipped with a doubling measure and supporting a weak Poincar\'e inequality. The metric space is not required to be complete.

Let $\Omega$ be an open subset of $\mathbb{R}^{n}$, we establish regularity results for solutions of some degenerate nonhomogeneous equations of the type $${\rm div}\langle {A}(x)Du,Du\rangle^{\frac{p-2}{2}}{ A}(x)Du={\rm div} F{\rm in} \Omega$$ where $p\ge 2$. The nonnegative function $\mathcal K(x)$, which measures the degree of degeneracy of ellipticity bounds, lies in the exponential class, i.e. $\mathrm {exp}(\lambda \mathcal K(x))$ is integrable for some $\lambda>0$. Under this assumption, the gradient of a finite energy solution of (1) lies in the Orlicz-Zygmund class $L^{p}\log^{-1}L(\Omega)$. Our results states that the gradient of such solution is more regular provided $\lambda$ is sufficiently large and the datum $F=F(x)$ belongs to a suitable Orlicz-Zygmund class.

We study the distribution of algebraic integers with prescribed factorization properties in short intervals and prove that for a large class of such numbers from a fixed algebraic number field $K$ with a non-trivial class group, every interval of the form $(x, x+x^{\theta})$ with a fixed $\theta >1/2$ contains absolute value of the norm of such algebraic integer provided $x\geq x_0$. The constant $x_0$ effectively depends on $K$ and $\theta$.