Let u be a solution to a second order elliptic equation with singular magnetic fields, vanishing continuously on an open subset Γ of the boundary of a Lipschitz domain. An elementary proof of the doubling property for u² over balls centered at some points near Γ is presented. Moreover, we get the unique continuation at the boundary of Dini domains for elliptic operators.

Let $K\subset Q$ be compact, convex sets in ${ℝ}^n$ with $\stackrel{\circ }{K}\ne \varnothing$ and let $P\left(D\right)$ be a linear, constant coefficient PDO. It is characterized in various ways when each zero solution of $P\left(D\right)$ in the space $ℰ\left(K\right)$ of all $C^{\infty }$-functions on $K$ extends to a zero solution in $ℰ\left(Q\right)$ resp. in $ℰ\left({ℝ}^n\right)$. The most relevant characterizations are in terms of Phragmén-Lindelöf conditions on the zero variety of $P$ in ${ℂ}^n$ and in terms of fundamental solutions for $P\left(D\right)$ with lacunas.

A recent result of Bahouri shows that continuation from an open set fails in general for solutions of $ℒu=Vu$ where $V\in C^{\infty }$ and $ℒ={\sum }_{j=1}^{N-1}{X}_j^2$ is a (nonelliptic) operator in ${\mathbf{R}}^N$ satisfying Hörmander’s condition for hypoellipticity. In this paper we study the model case when $ℒ$ is the subelliptic Laplacian on the Heisenberg group and $V$ is a zero order term which is allowed to be unbounded. We provide a sufficient condition, involving a first order differential inequality, for nontrivial solutions of $ℒu=Vu$ to have a finite order...

We study a general mathematical model linked with various physical models. Especially, we focus on those models established by King or Spencer-Davis-Voorhees related to thin films extending the lubrication model studied by Bernis-Friedman. According to the initial data, we prove that, either, blow up or global existence can be obtained.

We prove the existence of global and regular solutions to the Navier-Stokes equations in cylindrical type domains under boundary slip conditions, where coordinates are chosen so that the x₃-axis is parallel to the axis of the cylinder. Regular solutions have already been obtained on the interval [0,T], where T > 0 is large, on the assumption that the L₂-norms of the third component of the force field, of derivatives of the force field, and of the velocity field with respect to the direction of...

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy’s version of the uncertainty principle. We also obtain corresponding results for heat evolutions.

We prove an inequality of the type$$\parallel |x{|}^{r_f}{\parallel }_{L^p({\mathbf{R}}^n)}\le c(n,p,q,r)\parallel |x{|}^{\tau +\mu }\Delta f{\parallel }_{L^q({\mathbf{R}}^n)}.$$This is then used to derive the unique continuation property for the differential inequality $\left|\Delta f\right(x\left)\right|\le \left|v\right(x\left)\right|\left|f\right(x\left)\right|$ under suitable local integrability assumptions on the function $v$.

The purpose of this talk is to present some recent results about the Cauchy theory of the gravity water waves equations (without surface tension). In particular, we clarify the theory as well in terms of regularity indexes for the initial conditions as fin terms of smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). Our main result is that, following the approach developed in [1, 2], after suitable para-linearizations, the system can be arranged into...