Our aim is to provide a short and self contained synthesis which generalise and unify various related and unrelated works involving what we call $\Phi$-Sobolev functional inequalities. Such inequalities related to $\Phi$-entropies can be seen in particular as an inclusive interpolation between Poincaré and Gross logarithmic Sobolev inequalities. In addition to the known material, extensions are provided and improvements are given for some aspects. Stability by tensor products, convolution, and bounded perturbations are addressed. We show that under simple convexity assumptions on $\Phi$, such inequalities hold in a lot of situations, including hyper-contractive diffusions, uniformly strictly log-concave measures, Wiener measure (paths space of Brownian Motion on Riemannian Manifolds) and generic Poisson space (includes paths space of some pure jumps Lévy processes and related infinitely divisible laws). Proofs are simple and relies essentially on convexity. We end up by a short parallel inspired by the analogy with Boltzmann-Shannon entropy appearing in Kinetic Gases and Information Theories.

In this article we shall investigate the minimal and the extremal solutions of quasilinear elliptic equation with a positive nonlinear term in the right hand side. More precisely we shall study the boundary value problem \[ \left\{ \begin{array}{lcl} L_{p}(u) = \lambda f(u),&\text{in}&\Omega ,\\ u = 0,&\text{on}&\partial\Omega , \end{array} \right. \] where $\lambda$ is a nonnegative parameter, $\Omega$ is a domain of $R^{N}$ and $L_{p}(\cdot )(p > 1)$ is the $p$-Laplace operator defined by $L_{p}(\cdot ) =-\mathrm{div}(|\nabla\cdot |^{p-2}\nabla\cdot))$. We assume that $f(t)$ is increasing on $[0,\infty )$ and strictly convex with $f(0) > 0$. Under some additional conditions, we first establish the existence of the minimal solution $u_{\lambda}$ and the extremal solution $u^{*}$ to this equation and study their behaviors in connection with the linearized operator given by $L_{p}'(u)(\cdot ) =-\mathrm{div}\left( |\nabla u|^{p-2}\left(\nabla\cdot +(p-2)\frac{(\nabla u, \nabla\cdot )}{|\nabla u|^{2}}\nabla u\right)\right)$. The minimal solution $u_{\lambda}\in C^{1,\sigma }(\Bar{\Omega })$ is defined as the smallest solution among all possible classical solutions, and the extremal solution is defined as an increasing limit of $u_{\lambda}$ in $W_{0}^{1,p}(\Omega )$ as $\lambda \to \lambda^{*}$ (the extremal value). Though $L_{p}'(u_{\lambda})(\cdot )$ is, roughly speaking, a degenerate elliptic operator, it is shown that $L_{p}'(u_{\lambda})(\cdot )$ has a compact inverse from $L^{2}(\Omega )$ to itself if $u_{\lambda}$ is minimal. Moreover the self-adjoint operator $L_{p}'(u_{\lambda})(\cdot )-\lambda f'(u_{\lambda})$ on $L^{2}(\Omega )$ has a positive first eigenvalue if $\lambda$ is sufficiently small and a nonnegative first eigenvalue for any $\lambda\in (0,\lambda ^{*})$. Finally in Section 10 we give the characterizations of the extremal solution which are essentially depend upon the value of $p$ and the topology of $\Omega$ (see Theorem 10.1 and subsequent Propositions). When $\Omega$ is a ball, we investigate these problems rather precisely using the weighted Hardy type inequality with a sharp missing term.

Let $(X,\Sigma ,\mu )$ be a measure space. It is well known that $l^{p}(X) \subseteq l^{q}(X)$ whenever $0 < p \leq q \leq \infty$. Subramanian [12] characterized all positive measures $\mu$ on $(X,\Sigma )$ for which $L^{p}(\mu ) \subseteq L^{q}(\mu )$ whenever $0 < p \leq q \leq \infty$ and Romero [10] completed and improved some results of Subramanian [12]. Miamee [6] considered the more general inclusion $L^{p}(\mu ) \subseteq L^{q}(\nu )$ where $\mu$ and $\nu$ are two measures on $(X,\Sigma )$.

Let $L(p_{1}, q_{1})(X,\mu )$ and $L(p_{2},q_{2})(X,\nu )$ be two Lorentz spaces,where $0 < p_{1}, p_{2} < \infty$ and $0 < q_{1}, q_{2} \leq \infty$. In this work we generalized these results to the Lorentz spaces and investigated that under what conditions $L(p_{1}, q_{1})(X,\mu ) \subseteq L(p_{2},q_{2})(X,\nu )$ for two different measures $\mu$ and $\nu$ on $(X,\Sigma )$.

In [1] and [2], we introduced and investigated “unstable K-theory” $U_{n}(X) = [X,U(n)]$ and showed the relation with the Adams e-invariant. In this paper, we offer a theorem relating to $U_{n}(X)$ and show an application in connection with stable splittings of $G_{2}$.