The authors construct a periodic quasiregular function of any finite order $\rho$, $1\leq\rho < \infty$. This completes earlier work of O. Martio and U. Srebro.

Let $M$ be a random measure and $L$ be an elliptic pseudo-differential operator on $\mathbb{R}^d$. We study the solution of the stochastic problem $LX=M$, $X(0)=0$ when some homogeneity and integrability conditions are assumed. If $M$ is a Gaussian measure the process $X$ belongs to the class of Elliptic Gaussian Processes which has already been studied. Here the law of $M$ is not necessarily Gaussian. We characterize the solutions $X$ which are self-similar and with stationary increments in terms of the driving measure $M$. Then we use appropriate wavelet bases to expand these solutions and we give regularity results. In the last section it is shown how a percolation forest can help with constructing a self-similar Elliptic Process with non stable law.

Variational inequalities (free boundaries), governed by the $p$-parabolic equation ($p\geq 2$), are the objects of investigation in this paper. Using intrinsic scaling we establish the behavior of solutions near the free boundary. A consequence of this is that the time levels of the free boundary are porous (in $N$-dimension) and therefore its Hausdorff dimension is less than $N$. In particular the $N$-Lebesgue measure of the free boundary is zero for each $t$-level.

The random multiplicative measures on $\mathbb{R}$ introduced in Mandelbrot ([Mandelbrot 1996]) are a fundamental particular case of a larger class we deal with in this paper. An element $\mu$ of this class is the vague limit of a continuous time measure-valued martingale $\mu _{t}$, generated by multiplying i.i.d. non-negative random weights, the $(W_M)_{M\in S}$, attached to the points $M$ of a Poisson point process $S$, in the strip $H=\{(x,y)\in \mathbb{R}\times\mathbb{R}_+ ; 0 < y\leq 1\}$ of the upper half-plane. We are interested in giving estimates for the dimension of such a measure. Our results give these estimates almost surely for uncountable families $(\mu ^{\lambda})_{\lambda \in U}$ of such measures constructed simultaneously, when every measure $\mu^{\lambda}$ is obtained from a family of random weights $(W_M(\lambda))_{M\in S}$ and $W_M(\lambda)$ depends smoothly upon the parameter $\lambda\in U\subset\mathbb{R}$. This problem leads to study in several sense the convergence, for every $s\geq 0$, of the functions valued martingale $Z^{(s)}_t: \lambda \mapsto \mu_{t}^{\lambda }([0,s])$. The study includes the case of analytic versions of $Z^{(s)}_t(\lambda)$ where $\lambda\in\mathbb{C}^n$. The results make it possible to show in certain cases that the dimension of $\mu^{\lambda}$ depends smoothly upon the parameter. When the Poisson point process is statistically invariant by horizontal translations, this construction provides the new non-decreasing multifractal processes with stationary increments $s\mapsto \mu ([0,s])$ for which we derive limit theorems, with uniform versions when $\mu$ depends on $\lambda$.

We establish continuity, openness and discreteness, and the condition $(N)$ for mappings of finite distortion under minimal integrability assumptions on the distortion.

We consider the following degenerate parabolic equation $$ v_{t}=v\Delta v-\gamma|\nabla v|^{2}\quad\mbox{in $\mathbb{R}^{N} \times(0,\infty)$,} $$ whose behaviour depends strongly on the parameter $\gamma$. While the range $\gamma < 0$ is well understood, qualitative and analytical novelties appear for $\gamma>0$. Thus, the standard concepts of weak or viscosity solution do not produce uniqueness. Here we show that for $\gamma>\max\{N/2,1\}$ the initial value problem is well posed in a precisely defined setting: the solutions are chosen in a class $\mathcal{W}_s$ of local weak solutions with constant support; initial data can be any nonnegative measurable function $v_{0}$ (infinite values also accepted); uniqueness is only obtained using a special concept of initial trace, the $p$-trace with $p=-\gamma < 0$, since the standard concepts of initial trace do not produce uniqueness. Here are some additional properties: the solutions turn out to be classical for $t>0$, the support is constant in time, and not all of them can be obtained by the vanishing viscosity method. We also show that singular measures are not admissible as initial data, and study the asymptotic behaviour as $t\to \infty$.

We modify Hörmander's well-known weak type (1,1) condition for integral operators (in a weakened version due to Duong and McIntosh) and present a weak type $(p,p)$ condition for arbitrary operators. Given an operator $A$ on $L_2$ with a bounded $H^\infty$ calculus, we show as an application the $L_r$-boundedness of the $H^\infty$ calculus for all $r\in(p,q)$, provided the semigroup $(e^{-tA})$ satisfies suitable weighted $L_p\to L_q$-norm estimates with $2\in(p,q)$. This generalizes results due to Duong, McIntosh and Robinson for the special case $(p,q)=(1,\infty)$ where these weighted norm estimates are equivalent to Poisson-type heat kernel bounds for the semigroup $(e^{-tA})$. Their results fail to apply in many situations where our improvement is still applicable, e.g. if $A$ is a Schrödinger operator with a singular potential, an elliptic higher order operator with bounded measurable coefficients or an elliptic second order operator with singular lower order terms.

A lot is known about the forward iterates of an analytic function which is bounded by $1$ in modulus on the unit disk $\mathbb{D}$. The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880's to the 1980's, have provided conjugations which yield very precise descriptions of the dynamics. Backward-iteration sequences are of a different nature because a point could have infinitely many preimages as well as none. However, if we insist in choosing preimages that are at a finite hyperbolic distance each time, we obtain sequences which have many similarities with the forward-iteration sequences, and which also reveal more information about the map itself. In this note we try to present a complete study of backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk.

The long-time asymptotics of certain nonlinear, nonlocal, diffusive equations with a gradient flow structure are analyzed. In particular, a result of Benedetto, Caglioti, Carrillo and Pulvirenti [BCCP98] guaranteeing eventual relaxation to equilibrium velocities in a spatially homogeneous model of granular flow is extended and quantified by computing explicit relaxation rates. Our arguments rely on establishing generalizations of logarithmic Sobolev inequalities and mass transportation inequalities, via either the Bakry-Emery method or the abstract approach of Otto and Villani [OV00].

We prove variable coefficient analogues of results in [CCCDRVWW86] on Hilbert transforms and maximal functions along convex curves in the plane.