Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions

We develop the concept and the calculus of anti-self dual (ASD) Lagrangians which seems inherent to many questions in mathematical physics, geometry, and differential equations. They are natural extensions of gradients of convex functions --hence of self-adjoint positive operators-- which usually drive dissipative systems, but also rich enough to provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of functionals of the form $I(u)=L(u, \Lambda u)$ (resp. $I(u)=\int_{0}^{T}L(t, u(t), \dot u (t)+\Lambda_{t}u(t))dt$) where $L$ is an anti-self dual Lagrangian and where $\Lambda_{t}$ are essentially skew-adjoint operators. However, and just like the self (and antiself) dual equations of quantum field theory (e.g. Yang-Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional $I$, but because they are also zeroes of the Lagrangian $L$ itself.Comment: 50 pages. For the most updated version of this paper, please visit http://www.pims.math.ca/~nassif/pims_papers.htm

Sobolev inequalities for the Hardy-Schr\"odinger operator: Extremals and critical dimensions

In this expository paper, we consider the Hardy-Schr\"odinger operator $-\Delta -\gamma/|x|^2$ on a smooth domain \Omega of R^n with 0\in\bar{\Omega}, and describe how the location of the singularity 0, be it in the interior of \Omega or on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter rewrites: $C(\int_{\Omega}\frac{u^{p}}{|x|^s}dx)^{\frac{2}{p}}\leq \int_{\Omega} |\nabla u|^2dx-\gamma \int_{\Omega}\frac{u^2}{|x|^2}dx$ for all $u\in H^1_0(\Omega)$, where \gamma <n^2/4, s\in [0,2) and p:=2(n-s)/(n-2). We address questions regarding the explicit values of the optimal constant C, as well as the existence of non-trivial extremals attached to these inequalities. Scale invariance properties lead to situations where the best constants do not depend on the domain and are not attainable. We consider two different approaches to "break the homogeneity" of the problem: One approach was initiated by Brezis-Nirenberg and by Janelli. It is suitable for the case where 0 is in the interior of \Omega, and consists of considering lower order perturbations of the critical nonlinearity. The other approach was initiated by Ghoussoub-Kang , C.S. Lin et al. and Ghoussoub-Robert. It consists of considering domains where the singularity is on the boundary. Both of these approaches are rich in structure and in challenging problems. If 0\in \Omega, a negative linear perturbation suffices for higher dimensions, while a positive "Hardy-singular interior mass" is required in lower dimensions. If the singularity is on the boundary, then the local geometry around 0 plays a crucial role in high dimensions, while a positive "Hardy-singular boundary mass" is needed for the lower dimensions.Comment: Expository paper. 48 page