Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity

Let $\Omega ?{\mathbb{R}}^N$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1)$\mathrm{?}\zeta =G\zeta ,\zeta {|}_{\Gamma }=0,$ vanishes if $G\in {\mathsf{L}}^1(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ and $\zeta \in {\mathsf{W}}^{1,1}(\Omega ;{\mathbb{R}}^N)$. In particular, square-integrable solutions ζ of (1) with $G\in {\mathsf{L}}^1\cap {\mathsf{L}}^2(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ vanish. As a consequence, we prove that:$???:{\mathsf{C}}_{°}^{\infty }(\Omega ,\Gamma ;{\mathbb{R}}^3)\to [0,\infty ),u?6\mathrm{sym}\left(\mathrm{?}u{P}^{?1}\right)6_{{\mathsf{L}}^2(\Omega )}$ is a norm if $P\in {\mathsf{L}}^{\infty }(\Omega ;{\mathbb{R}}^{3\times 3})$ with $\mathrm{Curl}P\in {\mathsf{L}}^p(\Omega ;{\mathbb{R}}^{3\times 3})$, $\mathrm{Curl}{P}^{?1}\in {\mathsf{L}}^q(\Omega ;{\mathbb{R}}^{3\times 3})$ for some $p,q>1$ with $1/p+1/q=1$ as well as $\det P?c^{+}>0$. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let $\Phi \in {\mathsf{H}}^1(\Omega ;{\mathbb{R}}^3)$, $\Omega ?{\mathbb{R}}^3$, satisfy $\mathrm{sym}(\mathrm{?}{\Phi }^{?}\mathrm{?}\Psi )=0$ for some $\Psi \in {\mathsf{W}}^{1,\infty }(\Omega ;{\mathbb{R}}^3)\cap {\mathsf{H}}^2(\Omega ;{\mathbb{R}}^3)$ with $\det \mathrm{?}\Psi ?c^{+}>0$. Then there exists a constant translation vector $a\in {\mathbb{R}}^3$ and a constant skew-symmetric matrix $A\in \mathfrak{so}(3)$, such that $\Phi =A\Psi +a$.     

Jumps of computably enumerable equivalence relations

We study computably enumerable equivalence relations (or, ceers), under computable reducibility ≤, and the halting jump operation on ceers. We show that every jump is uniform join-irreducible, and thus join-irreducible. Therefore, the uniform join of two incomparable ceers is not equivalent to any jump. On the other hand there exist ceers that are not equivalent to jumps, but are uniform join-irreducible: in fact above any non-universal ceer there is a ceer which is not equivalent to a jump, and is uniform join-irreducible. We also study transfinite iterations of the jump operation. If a is an ordinal notation, and E is a ceer, then let $E^{(a)}$ denote the ceer obtained by transfinitely iterating the jump on E along the path of ordinal notations up to a. In contrast with what happens for the Turing jump and Turing reducibility, where if a set X is an upper bound for the A-arithmetical sets then $X^{(2)}$ computes $A^{(\omega )}$, we show that there is a ceer R such that $R\ge {\mathrm{Id}}^{(n)}$, for every finite ordinal n, but, for all k, $R^{(k)}?{\mathrm{Id}}^{(\omega )}$ (here Id is the identity equivalence relation). We show that if $a,b$ are notations of the same ordinal less than ${\omega }^2$, then $E^{(a)}\equiv E^{(b)}$, but there are notations $a,b$ of ${\omega }^2$ such that ${\mathrm{Id}}^{(a)}$ and ${\mathrm{Id}}^{(b)}$ are incomparable. Moreover, there is no non-universal ceer which is an upper bound for all the ceers of the form ${\mathrm{Id}}^{(a)}$ where a is a notation for ${\omega }^2$.     

On a p-Laplace equation with multiple critical nonlinearities

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{u}^{p?1}={|x|}^{?s}{u}^{p^{?}(s)?1}+u^{p^{?}?1}$ admits a positive weak solution in ${\mathbb{R}}^n$ of class $D_1^p({\mathbb{R}}^n)\cap C^1({\mathbb{R}}^n?\{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1={[(n?p)/p]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in D_1^p({\mathbb{R}}^n)$ is a weak solution in ${\mathbb{R}}^n$ of $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{|u|}^{p?2}u={|x|}^{?s}{|u|}^{p^{?}(s)?2}u+{|u|}^{q?2}u$, then $u\equiv 0$ when either $1<q<p^{?}$, or $q>p^{?}$ and u is also of class $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^n?\{0\})$.     

Commutators and generalized local Morrey spaces

In this paper we study the behavior of Hardy–Littlewood maximal operator and the action of commutators in generalized local Morrey spaces $L{M}_{\{x_0\}}^{p,\phi }({\mathbb{R}}^n)$ and generalized Morrey spaces $M^{p,\phi }({\mathbb{R}}^n)$.     

On the norms of quaternionic harmonic projection operators

As a consequence of integral bounds for three classes of quaternionic spherical harmonics, we prove some bounds from below for the $(L^p,L^2)$ norm of quaternionic harmonic projectors, for $p\in [1,2]$.     

Bilinear forms on homogeneous Sobolev spaces

In this paper we characterize the boundedness of the bilinear form defined by

$(f,g)\in {\dot{H}}^s(\mathbb{R})\times {\dot{H}}^s(\mathbb{R})\to \underset{\mathbb{R}}{\int }{(?\mathrm{\Delta })}^{s/2}(fg)(x){(?\mathrm{\Delta })}^{s/2}(b)(x)dx,$

in the product of homogeneous Sobolev spaces ${\dot{H}}^s(\mathbb{R})\times {\dot{H}}^s(\mathbb{R})$, $0<s<1/2$. We deduce a characterization of the space of pointwise multipliers from ${\dot{H}}^s(\mathbb{R})$ to its dual ${\dot{H}}^{?s}(\mathbb{R})$ in terms of trace measures.     

Regularity in Lp Sobolev spaces of solutions to fractional heat equations

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations with a nonlocal operator P:

(*)$Pu+{?}_{t}u=f(x,t),\text{ for }x\in \mathrm{\Omega }?{\mathbb{R}}^n,t\in I?\mathbb{R}.$

1) For strongly elliptic pseudodifferential operators (ψdo's) P on ${\mathbb{R}}^n$ of order $d\in {\mathbb{R}}_{+}$, a symbol calculus on ${\mathbb{R}}^{n+1}$ is introduced that allows showing optimal regularity results, globally over ${\mathbb{R}}^{n+1}$ and locally over $\mathrm{\Omega }\times I$:

$f\in H_{p,\mathrm{loc}}^{(s,s/d)}(\mathrm{\Omega }\times I)?u\in H_{p,\mathrm{loc}}^{(s+d,s/d+1)}(\mathrm{\Omega }\times I),$

for $s\in \mathbb{R}$, $1<p<\infty$. The $H_p^{(s,s/d)}$ are anisotropic Sobolev spaces of Bessel-potential type, and there is a similar result for Besov spaces.2) Let Ω be smooth bounded, and let P equal ${(?\mathrm{\Delta })}^a$ ($0<a<1$), or its generalizations to singular integral operators with regular kernels, generating stable Lévy processes. With the Dirichlet condition $\mathrm{supp}u?\stackrel{￣}{\mathrm{\Omega }}$, the initial condition $u{|}_{t=0}=0$, and $f\in L_p(\mathrm{\Omega }\times I)$, (*) has a unique solution $u\in L_p(I;H_p^{a(2a)}(\stackrel{￣}{\mathrm{\Omega }}))$ with ${?}_{t}u\in L_p(\mathrm{\Omega }\times I)$. Here $H_p^{a(2a)}(\stackrel{￣}{\mathrm{\Omega }})={\dot{H}}_p^{2a}(\stackrel{￣}{\mathrm{\Omega }})$ if $a<1/p$, and is contained in ${\dot{H}}_p^{2a?\epsilon }(\stackrel{￣}{\mathrm{\Omega }})$ if $a=1/p$, but contains nontrivial elements from $d^a{\stackrel{￣}{H}}_p^a(\mathrm{\Omega })$ if $a>1/p$ (where $d(x)=\mathrm{dist}(x,?\mathrm{\Omega })$). The interior regularity of u is lifted when f is more smooth.     

Improved Cotlar's inequality in the context of local Tb theorems

In the context of local Tb theorems with $L^p$ testing conditions we prove an enhanced Cotlar's inequality. This is related to the problem of removing the so called buffer assumption of Hytönen–Nazarov, which is the final barrier for the full solution of S. Hofmann's problem. We also investigate the problem of extending the Hytönen–Nazarov result to non-homogeneous measures. We work not just with the Lebesgue measure but with measures μ in ${\mathbb{R}}^d$ satisfying $\mu (B(x,r))\le C{r}^n$, $n\in (0,d]$. The range of exponents in the Cotlar type inequality depend on n. Without assuming buffer we get the full range of exponents $p,q\in (1,2]$ for measures with $n\le 1$, and in general we get $p,q\in [2??(n),2]$, $?(n)>0$. Consequences for (non-homogeneous) local Tb theorems are discussed.     

Two remarks on liftings of maps with values into S1

Given a map $u\in L_{\mathrm{loc}}^1(\Omega ,S^1)$ with some regularity: ${|u|}_X=R<\infty$, we consider the problem of finding a lifting φ of u (i.e. a measurable function satisfying $u={\mathrm{e}}^{\mathrm{i}\phi }$) with the same regularity and with an optimal control ${|\phi |}_X?g(R)$. Two cases are treated here:(i) ${|?|}_X$ is a $W^{s,p}(0,1)$-seminorm, with $0<s<1<p$ and $sp>1$. We find a lifting φ such that ${|\phi |}_{W^{s,p}(I)}?C(R+R^{1/s})$ and we show that the exponent $1/s$ cannot be improved.(ii) ${|?|}_X$ is the $\mathit{BV}(\Omega )$-seminorm where $\Omega ?{\mathbf{R}}^d$ is a smooth open set. We give a simplified proof of a previous result [J. Dàvila, R. Ignat, Lifting of BV functions with values in $S^1$, C. R. Acad. Sci. Paris, Ser. I 337 (3) (2003) 159–164]: there exists $\phi \in \mathit{BV}(\Omega )$ satisfying ${|\phi |}_{\mathit{BV}}?2R$. To cite this article: B. Merlet, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A new characterization of Sobolev spaces

Our main result is the following: Let $g\in L^p({\mathbb{R}}^N)$, $1<p<+\infty$, be such that

$\underset{n\in \mathbb{N}}{\sup }\underset{|g(x)?g(y)|>{\delta }_n}{\underset{{\mathbb{R}}^N}{\int }\underset{{\mathbb{R}}^N}{\int }}\frac{{\delta }_n^p}{{|x?y|}^{N+p}}\mathrm{d}x\mathrm{d}y<+\infty ,$

for some arbitrary sequence of positive numbers ${({\delta }_n)}_{n\in \mathbb{N}}$ with ${\lim }_{n\to \infty }{\delta }_n=0$. Then $g\in W^{1,p}({\mathbb{R}}^N)$.This extends a result from H.-M. Nguyen (2006). To cite this article: J. Bourgain, H.-M. Nguyen, C. R. Acad. Sci. Paris, Ser. I 343 (2006).