We consider the Hardy-Hénon system $-\Delta u =|x|^a v^p$, $-\Delta v =|x|^b u^q$ with $p,q>0$ and $a,b\in {\mathbb R}$ and we are concerned in particular with the Liouville property, i.e., the nonexistence of positive solutions in the whole space ${\mathbb R}^N$. In view of known results, it is a natural conjecture that this property should be true if and only if $(N+a)/(p+1)+(N+b)/(q+1)>(N-2)$. In this paper, we prove the conjecture for dimension $N=3$ in the case of bounded solutions and in dimensions $N\le 4$ when $a,b\le 0$, among other partial nonexistence results. As far as we know, this is the first optimal Liouville-type result for the Hardy-H\'enon system. Next, as applications, we give results on singularity and decay estimates as well as a priori bounds of positive solutions.

This paper is concerned with the study of the Strong Maximum Principle for semicontinuous viscosity solutions of fully nonlinear, second-order parabolic integro-differential equations. We study separately the propagation of maxima in the horizontal component of the domain and the local vertical propagation in simply connected sets of the domain. We give two types of results for horizontal propagation of maxima: one is the natural extension of the classical results of local propagation of maxima and the other comes from the structure of the nonlocal operator. As an application, we use the Strong Maximum Principle to prove a Strong Comparison Result of viscosity sub- and supersolution for integro-differential equations.

In the \textit{two-well problem} we look for a map $u$ which satisfies Dirichlet boundary conditions and whose gradient $Du$ assumes values in $SO\left( 2\right) A\cup SO\left( 2\right) B=\mathbb{S}_{A}\cup \mathbb{S}_{B},$ for two given invertible matrices $A,B$ (an element of $SO\left( 2\right) A$ is of the form $RA$ where $R$ is a rotation). In the original approach by Ball and James [1], [2] $A$, $B$ are two matrices such that $\det B>\det A>0$ and $\operatorname*{rank}\left\{ A-B\right\} =1.$ It was proved in the 1990's (see [4], [5], [6], [7], [17]) that a map $u$ satisfying given boundary conditions and such that $Du\in\mathbb{S} _{A}\cup\mathbb{S}_{B}$ exists in the Sobolev class $W^{1,\infty} (\Omega;\mathbb{R}^{2})$ of Lipschitz continuous maps. However, for orthogonal matrices it was also proved (see [3], [8], [9], [10], [11], [12], [16]) that solutions exist in the class of piecewise-$C^{1}$ maps, in particular in the class of piecewise-affine maps. We prove here that this possibility does not exist for other nonsingular matrices $A$, $B$: precisely, the two-well problem can be solved by means of piecewise-affine maps only for orthogonal matrices.

The Lagrangian Averaged Navier-Stokes (LANS) equations are a recently derived approximation to the Navier-Stokes equations. Existence of global solutions for the LANS equation has been proven for initial data in the Sobolev space $H^{3/4,2}(\mathbb{R}^3)$ and in the Besov space $B^{n/2}_{2,q}(\mathbb{R}^n)$. In this paper, we use an interpolation-based method to prove the existence of global solutions to the LANS equation with initial data in $B^{3/p}_{p,q}(\mathbb{R}^3)$ for any $p>n$.

Is ignition or extinction the fate of an exothermic chemical reaction occurring in a bounded region within a heat conductive solid consisting of a porous medium? In the spherical case, the reactor is modeled by a system of reaction-diffusion equations that reduces to a linear heat equation in a shell, coupled at the internal boundary to a nonlinear ODE modeling the reaction region. This ODE can be regarded as a boundary condition. This model allows the complete analysis of the time evolution of the system: there is always a global attractor. We show that, depending on physical parameters, the attractor contains one or three equilibria. The latter case has special physical interest: the two equilibria represent attractors ("extinction" or "ignition") and the third equilibrium is a saddle. The whole system is well approximated by a single ODE, a "reduced" model, justifying the "heat transfer coefficient" approach of Chemical Engineering.

We study a second-order parabolic equation in divergence form in a spatial domain separated in two parts by a hyperplane. The equation is uniformly parabolic in one of the parts and degenerates with respect to a small parameter $\varepsilon$ on the other part. We show that weak solutions to this equation are Hölder continuous with the Hölder exponent independent of ${\varepsilon}$.

In this note an ${\mathcal{R}}$-bounded ${\mathcal{H}}^\infty$-calculus for linear operators associated to cylindrical boundary value problems is proved. The obtained results are based on an abstract result on operator-valued functional calculus by N. Kalton and L. Weis; cf. [28]. Cylindrical in this context means that both domain and differential operator possess a certain cylindrical structure. In comparison to standard methods (e.g. localization procedures), our approach appears less technical and provides short proofs. Besides, we are even able to deal with some classes of equations on rough domains. For instance, we can extend the well-known (and in general sharp) range for $p$ such that the (weak) Dirichlet Laplacian admits an ${\mathcal{H}}^\infty$-calculus on $L^p(\Omega)$, from $(3+\varepsilon)'<p<3+\varepsilon$ to $(4+\varepsilon)'<p<4+\varepsilon$ for three-dimensional bounded or unbounded Lipschitz cylinders $\Omega$. Our approach even admits mixed Dirichlet Neumann boundary conditions in this situation.