Searcharxiv⌕ Search

arXiv subjects

Patrick J. Rabier

Publications and source records attributed to Patrick J. Rabier.

9 recordsLinked to original sources
arXiv2016

$L^{p}$ measure of growth and higher order Hardy-Sobolev-Morrey inequalities

When the growth at infinity of a function $u$ on $\Bbb{R}^{N}$ is compared with the growth of $|x|^{s}$ for some $s\in \Bbb{R},$ this comparison is invariably made pointwise. This paper argues that the comparison can also be made in a suitably defined $L^{p}$ sense for every $1\leq p<\infty $ and that, in this perspective, inequalities of Hardy, Sobolev or Morrey type account for the fact that sub $|x|^{-N/p}$ growth of $\nabla u$ in the $% L^{p} $ sense implies sub $|x|^{1-N/p}$ growth of $u$ in the $L^{q}$ sense for well chosen values of $q.$ By investigating how sub $|x|^{s}$ growth of $\nabla ^{k}u$ in the $L^{p}$ sense implies sub $|x|^{s+j}$ growth of $\nabla ^{k-j}u$ in the $L^{q}$ sense for (almost) arbitrary $s\in \Bbb{R}$ and for $q$ in a $p$-dependent range of values, a family of higher order Hardy/Sobolev/Morrey type inequalities is obtained, under optimal integrability assumptions. These optimal inequalities take the form of estimates for $\nabla ^{k-j}(u-π_{u}),1\leq j\leq k,$ where $π_{u}$ is a suitable polynomial of degree at most $k-1,$ which is unique if and only if $s<-k.$ More generally, it can be chosen independent of $(s,p)$ when $s$ remains in the same connected component of $\Bbb{R}\backslash \{-k,...,-1\}.$

math.AP↗
arXiv2016

L^p regularity of homogeneous elliptic differential operators with constant coefficients on R^N

Let $A$ be a homogeneous elliptic differential operator of order $m$ on $% \Bbb{R}^{N}$ with constant complex coefficients. A partial version of the main result is as follows: Suppose that $u\in L_{loc}^{1}$ and that $Au\in L^{p}$ for some $1<p<\infty .$ Then, all the partial derivatives of order $m$ of $u$ are in $L^{p}$ if and only if $|u|$ grows slower than $|x|^{m}$ at infinity, provided that growth is measured in an $L^{1}$-averaged sense over balls with increasing radii. The necessity provides an alternative answer to the pointwise growth question investigated with mixed success in the literature. Only a few special cases of the sufficiency are already known, mostly when $A=Δ.$ The full result gives a similar necessary and sufficient growth condition for the derivatives of $u$ of any order $k\geq 0$ to be in $L^{p}$ when $Au$ satisfies a suitable (necessary) condition. This is generalized to exterior domains under mandatory restrictions on $N$ and $p$ and to Douglis-Nirenberg elliptic systems whose entries are homogeneous operators with constant coefficients and possibly different orders.

math.AP↗
arXiv2015

Integral inequalities for infimal convolution and Hamilton-Jacobi equations

Let $f,g:\Bbb{R}^{N}\rightarrow (-\infty ,\infty ]$ be Borel measurable, bounded below and such that $\inf f+\inf g\geq 0.$ We prove that with $ m_{f,g}:=(\inf f-\inf g)/2,$ the inequality $||(f-m_{f,g})^{-1}||_{ϕ}+||(g+m_{f,g})^{-1}||_{ϕ}\leq 4||(f\Box g)^{-1}||_{ϕ}$ holds in every Orlicz space $L_{ϕ},$ where $f\Box g$ denotes the infimal convolution of $f$ and $g$ and where $||\cdot ||_{ϕ}$ is the Luxemburg norm (i.e., the $L^{p}$ norm when $L_{ϕ}=L^{p}$). Although no genuine reverse inequality can hold in any generality, we also prove that such reverse inequalities do exist in the form $||(f\Box g)^{-1}||_{ϕ}\leq 2^{N-1}(||(\check{f}-m_{f,g})^{-1}||_{ϕ}+||(\check{ g}+m_{f,g})^{-1}||_{ϕ}),$ where $\check{f}$ and $\check{g}$ are suitable transforms of $f$ and $g$ introduced in the paper and reminiscent of, yet very different from, nondecreasing rearrangement. Similar inequalities are proved for other extremal operations and applications are given to the long-time behavior of the solutions of the Hamilton-Jacobi and related equations.

math.FA↗
arXiv2015

Some measure-theoretic properties of generalized means

If $Λ$ is a measure space, $u:Λ^{m}\rightarrow \Bbb{R}$ is a given function and $N\geq m,$ the function $U(x_{1},...,x_{N})=\left( \begin{array}{l} N \\ m \end{array} \right) ^{-1}\sum_{1\leq i_{1}<\cdots <i_{m}\leq N}u(x_{i_{1}},...,x_{i_{m}}) $ is called the generalized $N$-mean with kernel $u,$ a terminology borrowed from $U$-statistics. Physical potentials for systems of particles are also defined by generalized means. This paper investigates whether various measure-theoretic concepts for generalized $N$-means are equivalent to the analogous concepts for their kernels: a.e. convergence of sequences, measurability, essential boundedness and integrability with respect to absolutely continuous probability measures. The answer is often, but not always, positive. This information is crucial in some problems addressing the existence of generalized means satisfying given conditions, such as the classical Inverse Problem of statistical physics (in the canonical ensemble).

math.FA↗
arXiv2013

Differentiability of quasiconvex functions on separable Banach spaces

We investigate the differentiability properties of real-valued quasiconvex functions f defined on a separable Banach space X. Continuity is only assumed to hold at the points of a dense subset. If so, this subset is automatically residual. Sample results that can be quoted without involving any new concept or nomenclature are as follows: (i) If f is usc or strictly quasiconvex, then f is Hadamard differentiable at the points of a dense subset of X (ii) If f is even, then f is continuous and Gateaux differentiable at the points of a dense subset of X. In (i) or (ii), the dense subset need not be residual but, if X is also reflexive, it contains the complement of a Haar null set. Furthermore, (ii) remains true without the evenness requirement if the definition of Gateaux differentiability is generalized in an unusual, but ultimately natural, way. The full results are much more general and substantially stronger. In particular, they incorporate the well known theorem of Crouzeix, to the effect that every real-valued quasiconvex function on R^N is Frechet differentiable a.e.

math.FA↗
arXiv2012

Points of continuity of quasiconvex functions on topological vector spaces

We give necessary and sufficient conditions for a real-valued quasiconvex function f on a Baire topological vector space X (in particular, Banach or Frechet space) to be continuous at the points of a residual subset of X. These conditions involve only simple topological properties of the lower level sets of f. A main ingredient consists in taking advantage of a remarkable property of quasiconvex functions relative to a topological variant of essential extrema on the open subsets of X. One application is that if f is quasiconvex and continuous at the points of a residual subset of X, then with a single possible exception, f^{-1}(a) is nowhere dense or has nonempty interior, as is the case for everywhere continuous functions. As a barely off-key complement, we also prove that every usc quasiconvex function is quasicontinuous in the (classical) sense of Kempisty since this interesting property does not seem to have been noticed before.

math.OC↗
arXiv2012

Quasiconvexity and density topology

We prove that if f : R^N --> R is quasiconvex and U is open in the density topology of R^N, then sup_U f = ess sup_U f, while inf_U f = ess inf_U f if and only if the equality holds when U = R^N. The first (second) property is typical of lsc (usc) functions and, even when U is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions. This property ensures that the pointwise extrema of f on any nonempty density open subset can be arbitrarily closely approximated by values of f achieved on "large" subsets, which may be of relevance in a variety of issues. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero.

math.MG↗
arXiv2011

Special embeddings of weighted Sobolev spaces with nontrivial power weights

In prior work, the author has characterized the real numbers $a,b,c$ and $1\leq p,q,r<\infty $ such that the weighted Sobolev space $W_{\{a,b\}}^{(q,p)}(R^{N}\backslash \{0}):=\{u\in L_{loc}^{1}(R^{N}\backslash \{0}):|x|^{\frac{a}{q}}u\in L^{q}(R^{N}),|x|^{\frac{b}{p}}\nabla u\in (L^{p}(R^{N}))^{N}\}$ is continuously embedded into $L^{r}(R^{N};|x|^{c}dx) :=\{u\in L_{loc}^{1}(R^{N}\backslash \{0}):|x|^{\frac{c}{r}}u\in L^{r}(R^{N})\}$. This paper discusses the embedding question for $W_{\{a,b\}}^{(\infty, p)}(R^{N}\backslash \{0}):=\{u\in L_{loc}^{1}(R^{N}\backslash \{0}):|x|^{a}u\in L^{\infty}(R^{N}),|x|^{\frac{b}{p}}\nabla u\in (L^{p}(R^{N}))^{N}\}$, which is not the space obtained by the formal substitution $q=\infty$ in the previous definition of $W_{\{a,b\}}^{(q,p)}(R^{N}\backslash \{0}),$ unless $a=0$. The corresponding embedding theorem identifies all the real numbers $a,b,c$ and $1\leq p,r<\infty $ such that $W_{\{a,b\}}^{(\infty, p)}(R^{N} \backslash \{0})$ is continuously embedded in $L^{r}(R^{N};|x|^{c}dx)$. A notable feature is that such embeddings exist only when $a\neq 0$ and, in particular, have no analog in the unweighted setting. It is also shown that the embeddings are always accounted for by multiplicative rather than just additive norm inequalities. These inequalities are natural extensions of the Caffarelli-Kohn-Nirenberg inequalities which, in their known form, are restricted to functions of $C_{0}^{\infty}(R^{N})$ and do not incorporate supremum norms.

math.AP↗