Functional Inequalities: New Perspectives and New Applications

This book is not meant to be another compendium of select inequalities, nor does it claim to contain the latest or the slickest ways of proving them. This project is rather an attempt at describing how most functional inequalities are not merely the byproduct of ingenious guess work by a few wizards among us, but are often manifestations of certain natural mathematical structures and physical phenomena. Our main goal here is to show how this point of view leads to "systematic" approaches for not just proving the most basic functional inequalities, but also for understanding and improving them, and for devising new ones - sometimes at will, and often on demand.Comment: 17 pages; contact Nassif Ghoussoub (nassif @ math.ubc.ca) for a pre-publication pdf cop

Well-posedness for a class of nonlinear degenerate parabolic equations

In this paper we obtain well-posedness for a class of semilinear weakly degenerate reaction-diffusion systems with Robin boundary conditions. This result is obtained through a Gagliardo-Nirenberg interpolation inequality and some embedding results for weighted Sobolev spaces

Parabolic equations on uniformly regular Riemannian manifolds and degenerate initial boundary value problems

In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying approach to problems with strong degeneracies in the interior or at the boundary.Comment: To appear in "Recent Developments of Mathematical Fluid Mechanics", Series: Advances in Mathematical Fluid Mechanics, Birkhaeuser-Verlag, Editors: G. P. Galdi, J. G. Heywood and R. Rannacher. Some misprints of the earlier version have been correcte

Results on entire solutions for a degenerate critical elliptic equation with anisotropic coefficients

In this paper, we study the following degenerate critical elliptic equations with anisotropic coefficients $$-div(|x_{N}|^{2\alpha}\nabla u)=K(x)|x_{N}|^{\alpha\cdot 2^{*}(s)-s}|u|^{2^{*}(s)-2}u {in} \mathbb{R}^{N}$$ where $x=(x_{1},...,x_{N})\in\mathbb{R}^{N},$ $N\geq 3,$ $\alpha>1/2,$ $0\leq s\leq 2$ and $2^{*}(s)=2(N-s)/(N-2).$ Some basic properties of the degenerate elliptic operator $-div(|x_{N}|^{2\alpha}\nabla u)$ are investigated and some regularity, symmetry and uniqueness results for entire solutions of this equation are obtained. We also get some variational identities for solutions of this equation. As a consequence, we obtain some nonexistence results for solutions of this equation.Comment: 29 page

Boundary-layers for a Neumann problem at higher critical exponents

We consider the Neumann problem $$(P)\qquad - \Delta v + v= v^{q-1} \ \text{in }\ \mathcal{D}, \ v > 0 \ \text{in } \ \mathcal{D},\ \partial_\nu v = 0 \ \text{on } \partial\mathcal{D} ,$$ where $\mathcal{D}$ is an open bounded domain in $\mathbb{R}^N,$ $\nu$ is the unit inner normal at the boundary and $q>2.$ For any integer, $1\le h\le N-3,$ we show that, in some suitable domains $\mathcal D,$ problem $(P)$ has a solution which blows-up along a $h-$dimensional minimal submanifold of the boundary $\partial\mathcal D$ as $q$ approaches from either below or above the higher critical Sobolev exponent ${2(N-h)\over N-h-2}.$Comment: 13 page

Infinitely many solutions for semilinear nonlocal elliptic equations under noncompact settings

In this paper, we study a class of semilinear nonlocal elliptic equations posed on settings without compact Sobolev embedding. More precisely, we prove the existence of infinitely many solutions to the fractional Brezis-Nirenberg problems on bounded domain.Comment: 29 pages, Typos are fixed, Intro and refereces are extende

The Strauss conjecture on asymptotically flat space-times

By assuming a certain localized energy estimate, we prove the existence portion of the Strauss conjecture on asymptotically flat manifolds, possibly exterior to a compact domain, when the spatial dimension is 3 or 4. In particular, this result applies to the 3 and 4-dimensional Schwarzschild and Kerr (with small angular momentum) black hole backgrounds, long range asymptotically Euclidean spaces, and small time-dependent asymptotically flat perturbations of Minkowski space-time. We also permit lower order perturbations of the wave operator. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and the assumed localized energy estimate, which is used in the remaining compact set.Comment: Final version, to appear in SIAM Journal on Mathematical Analysis. 17 page