We establish an algebraic version of the classical result that a $G$-map $f$ between $G$-complexes which restricts to a homotopy equivalence $f^H$ on $H$-fixed sets for all subgroups $H$ of $G$ is a $G$-homotopy equivalence. This is used to give an alternative proof of a theorem of Bouc. We also include a number of illustrations and applications.

The classical Helmholtz model for wakes behind an obstacle in perfect fluid mechanics boils down in complex analysis into a Riemann problem with free boundary, the free boundary being the wake and being such that the Riemann map parametrizes it as arc-length parametrization. We generalize a method due to Coifman and Meyer to solve this problem for a class of non-smooth obstacles.

We develop a general theory for asymptotically $\operatorname{CAT}(0)$ groups; these are groups acting geometrically on a geodesic space, all of whose asymptotic cones are $\operatorname{CAT}(0)$.

This paper proves the Corona Theorem to be affirmative for domains in the complex plane bounded by thick subsets of a Lipschitz graph. Specifically, the boundary of these domains $E_0$ has a Carleson lower density:

$$ \Lambda\left(B(z,r) \cap E_0\right) > \epsilon_0 r \quad\text{for all } z\in E_0, \quad \text{and all } r>0. $$

The goal of this paper is to study the Riesz transforms $\nabla A^{-1/2}$ where $A$ is the Schrödinger operator $-\Delta-V$, $V\ge 0$, under different conditions on the potential $V$. We prove that if $V$ is strongly subcritical, $\nabla A^{-1/2}$ is bounded on $L^p(\mathbb{R}^N)$, $N\ge3$, for all $p\in(p_0';2]$ where $p_0'$ is the dual exponent of $p_0$ where $2<\frac{2N}{N-2}<p_0<\infty$; and we give a counterexample to the boundedness on $L^p(\mathbb{R}^N)$ for $p\in(1;p'_0)\cup(p_{0*};\infty)$ where $p_{0*}:=\frac{p_0N}{N+p_0}$ is the reverse Sobolev exponent of $p_0$. If the potential is strongly subcritical in the Kato subclass $K_N^{\infty}$, then $\nabla A^{-1/2}$ is bounded on $L^p(\mathbb{R}^N)$ for all $p\in(1;2]$, moreover if it is in $L^{N/2}_w(\mathbb{R}^N)$ then $\nabla A^{-1/2}$ is bounded on $L^p(\mathbb{R}^N)$ for all $p\in(1;N)$. We prove also boundedness of $V^{1/2}A^{-1/2}$ with the same conditions on the same spaces. Finally we study these operators on manifolds. We prove that our results hold on a class of Riemannian manifolds.

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like

$$(-\Delta)^s v = \mu \quad \text{in } \mathbb{R}^N,$$

with vanishing conditions at infinity. Here $\mu$ is a bounded Radon measure whose support is compactly contained in $\mathbb{R}^N$, $N\geq2$, and $-(\Delta)^s$ is the fractional Laplace operator of order $s\in (1/2,1)$.

We prove that a generic second order differential equation in the projective plane has no algebraic solutions when the bidegree is big enough. We also prove an analogous result for webs on $\mathbb{P}^{2}$.

In this paper, we prove that if $\Psi $ is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation

\begin{equation} \label{fAbs} u_t -\Delta u= |u|^\alpha u, \tag{NLH} \end{equation}

in the unit ball of ${\mathbb R}^N $, $N= 3$, with Dirichlet boundary conditions, then the solution of \eqref{fAbs} with initial value $\lambda \Psi$ blows up in finite time if $|\lambda -1|>0$ is sufficiently small and if $\alpha >0$ is sufficiently small. The proof depends on showing that the inner product of $\Psi $ with the first eigenfunction of the linearized operator $L= -\Delta - (\alpha +1) | \Psi | ^\alpha $ is nonzero.

The aim of this paper is to establish a Meyer's type higher integrability result for weak solutions of possibly degenerate parabolic systems of the type

$$ \partial_t u - \operatorname{div} a(x,t,Du)= \operatorname{div} \bigl(|F|^{p(x,t)-2}F\bigr).$$

The vector-field $a$ is assumed to fulfill a non-standard $p(x,t)$-growth condition. In particular it is shown that there exists $\varepsilon >0$ depending only on the structural data such that there holds:

$$|Du|^{p(\cdot)(1+\varepsilon)}\in L^1_{\operatorname{loc}},$$

together with a local estimate for the $p(\cdot)(1+\varepsilon)$-energy.