We prove existence/nonexistence and uniqueness of positive entire solutions for some semilinear elliptic equations on the Hyperbolic space.

We prove an existence theorem of Cauchy-Kovalevskaya type for the equation $D_{t}u(t,z)=f(t,z,u\left({\alpha }^{\left(0\right)}(t,z)\right),D_{z}u\left({\alpha }^{\left(1\right)}(t,z)\right),...,D_z^{k}u\left({\alpha }^{\left(k\right)}(t,z)\right))$ where f is a polynomial with respect to the last k variables.

We obtain new variants of weighted Gagliardo-Nirenberg interpolation inequalities in Orlicz spaces, as a consequence of weighted Hardy-type inequalities. The weights we consider need not be doubling.

We consider the problem of minimizing the energy$$E\left(u\right):={\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x+{\int }_{S_u\cap \Omega }\left(1+\left|\right[u\left]\right(x\left)\right|\right)\mathrm{d}{H}^{N-1}\left(x\right)$$among all functions $u\in SB{V}^2\left(\Omega \right)$ for which two level sets $\{u=l_i\}$ have prescribed Lebesgue measure ${\alpha }_i$. Subject to this volume constraint the existence of minimizers for $E(·)$ is proved and the asymptotic behaviour of the solutions is investigated.

We consider the problem of minimizing the energy $$E\left(u\right):={\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x+{\int }_{S_u\cap \Omega }\left(1+\left|\right[u\left]\right(x\left)\right|\right)\mathrm{d}{H}^{N-1}\left(x\right)$$ among all functions u ∈ SBV²(Ω) for which two level sets $\{u=l_i\}$ have prescribed Lebesgue measure ${\alpha }_i$. Subject to this volume constraint the existence of minimizers for E(.) is proved and the asymptotic behaviour of the solutions is investigated.

We consider a random, uniformly elliptic coefficient field $a$ on the lattice ${\mathbb{Z}}^d$. The distribution $\langle ·\rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\langle |{\nabla }_{x}G(t,x,y){{|}^2\rangle }^{1/2}$ and $\langle \left|{\nabla }_x{\nabla }_{y}G(t,x,y)\right|\rangle$. In particular, the elliptic Green’s function $G(x,y)$ satisfies optimal annealed bounds. In their recent work, the authors...

I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “bilinear estimates”. In addition to the $L^2$ theory, which is now quite well developed, I plan to discuss a more general point of view concerning the $L^p$ theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also plan to discuss...

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation...

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation...