Searcharxiv⌕ Search

arXiv subjects

Joan Orobitg

Publications and source records attributed to Joan Orobitg.

16 recordsLinked to original sources
arXiv2026

The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations

We prove the persistence of boundary smoothness of vortex patches for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations generalize the Euler equations by including an additional parameter, the Rossby radius $\varepsilon^{-1}$, which modifies the relationship between the streamfunction and the (potential) vorticity. In addition, we prove that solutions of the QGSW equations converge locally in time to the corresponding Euler solutions as $\varepsilon \to 0$ in little Hölder spaces.

math.AP↗
arXiv2022

Skeleton for the one-dimensional aggregation equation

For the aggregation equation in $\mathbb{R}$, we consider the evolution of an initial density corresponding to the characteristic function of some set $Ω_0$. We study the limit measure at the blow up time 1 for $Ω_0$ open or compact and we inspect the limit set (skeleton) where this measure is supported.

math.AP↗
arXiv2016

Fractional differentiability for solutions of nonlinear elliptic equations

We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,ξ)$ enjoys $B^α_{\frac{n}α, q}$ smoothness, local well-posedness is found in $B^α_{p,q}$ for certain values of $p\in[2,\frac{n}α)$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,ξ)=A(x)ξ$, $G=0$ and $A\in B^α_{\frac{n}α,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^α_{p,q}$ for each $p<\frac{n}α$. Our main tool in the proof is a more general result, that holds also if ${\mathcal A}$ has growth $s-1$ in $Du$, $2\leq s\leq n$, and asserts local well-posedness in $L^q$ for each $q>s$, provided that $x\mapsto{\mathcal A}(x,ξ)$ satisfies a locally uniform $VMO$ condition.

math.AP↗
arXiv2016

Flows for non-smooth vector fields with subexponentially integrable divergence

In this paper, we study flows associated to Sobolev vector fields with subexponentially integrable divergence. Our approach is based on the transport equation following DiPerna-Lions [DPL89]. A key ingredient is to use a quantitative estimate of solutions to the Cauchy problem of transport equation to obtain the regularity of density functions.

math.CA↗
arXiv2015

Beltrami equations with coefficient in the fractional Sobolev space $W^{θ, \frac2θ}$

In this paper, we look at quasiconformal solutions $ϕ:\mathbb{C}\to\mathbb{C}$ of Beltrami equations $$ \partial_{\overline{z}} ϕ(z)=μ(z)\,\partial_z ϕ(z). $$ where $μ\in L^\infty(\mathbb{C})$ is compactly supported on $\mathbb{D}$, $\|μ\|_\infty<1$ and belongs to the fractional Sobolev space $W^{α, \frac2α}(\mathbb{C})$. Our main result states that $$\log\partial_zϕ\in W^{α, \frac2α}(\mathbb{C})$$ whenever $α>\frac12$. Our method relies on an $n$-dimensional result, which asserts the compactness of the commutator $$[b,(-Δ)^\fracβ{2}]:L^\frac{np}{n-βp}(\mathbb{R}^n)\to L^p(\mathbb{R}^n)$$ between the fractional laplacian $(-Δ)^\frac\beta2$ and any symbol $b\in W^{β,\frac{n}β}(\mathbb{R}^n)$, provided that $1<p<\frac{n}β$.

math.CV↗
arXiv2015

Linear transport equations for vector fields with subexponentially integrable divergence

We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline integrability assumptions on the divergence of the velocity field $b$. For $W^{1,1}_{loc}$ vector fields $b$ satisfying $\frac{|b(x,t)|}{1+|x|}\in L^1(0,T; L^1)+L^1(0,T; L^\infty)$ and $$\operatorname{div} b\in L^1(0,T;L^\infty) + L^1\left(0,T; \operatorname{Exp}\left(\frac{L}{\log L}\right)\right),$$ we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every $γ>1$, we construct an example of a bounded autonomous velocity field $b$ with $$\operatorname{div} b\in \operatorname{Exp}\left(\frac{L}{\log^γL}\right) ,$$ for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the $BV$ setting are also addressed.

math.AP↗
arXiv2014

The maximal Beurling transform associated with squares

It is known that the improved Cotlar's inequality $B^{*}f(z) \le C M(Bf)(z)$, $z\in\mathbb C$, holds for the Beurling transform $B$, the maximal Beurling transform $B^{*}f(z)=$ $\displaystyle\sup_{\varepsilon >0}\left|\int_{|w|>\varepsilon}f(z-w) \frac{1}{w^2} \,dw\right|$, $z\in\mathbb C$, and the Hardy--Littlewood maximal operator $M$. In this note we consider the maximal Beurling transform associated with squares, namely, $B^{*}_Sf(z)=\displaystyle\sup_{\varepsilon >0}\left|\int_{w\notin Q(0,\varepsilon)}f(z-w) \frac{1}{w^2} \,dw \right|$, $z\in\mathbb C$, $Q(0,\varepsilon)$ being the square with sides parallel to the coordinate axis of side length $\varepsilon$. We prove that $B_{S}^{*}f(z) \le C M^2(Bf)(z)$, $z\in\mathbb C$, where $M^2=M \circ M$ is the iteration of the Hardy--Littlewood maximal operator, and $M^2$ cannot be replaced by $M$.

math.CA↗
arXiv2013

$L^p$ estimates for the maximal singular integral in terms of the singular integral

This paper continues the study, initiated in the works {MOV} and {MOPV}, of the problem of controlling the maximal singular integral $T^{*}f$ by the singular integral $Tf$. Here $T$ is a smooth homogeneous Calderón-Zygmund singular integral operator of convolution type. We consider two forms of control, namely, in the weighted $L^p(ω)$ norm and via pointwise estimates of $T^{*}f$ by $M(Tf)$ or $M^2(Tf)$\,, where $M$ is the Hardy-Littlewood maximal operator and $M^2=M \circ M$ its iteration. The novelty with respect to the aforementioned works, lies in the fact that here $p$ is different from 2 and the $L^p$ space is weighted.

math.AP↗
arXiv2012

Beltrami equation with coefficient in Sobolev and Besov spaces

Our goal in this work is to present some function spaces on the complex plane $\C$, $X(\C)$, for which the quasiregular solutions of the Beltrami equation, $\bar\partial f (z) = μ(z) \partial f (z)$, have first derivatives locally in $X(\C)$, provided that the Beltrami coefficient $μ$ belongs to $X(\C)$.

math.AP↗
arXiv2010

Estimates for the maximal singular integral in terms of the singular integral:the case of even kernels

The purpose of this paper is to describe the smooth homogeneous Calderon-Zygmund operators for which the maximal singular integral T*f may be controlled by the singular integral Tf. We consider two types of control. The first is the L2 estimate of T*f by Tf, namely the estimate of the L2 norm of T*f by a constant times the L2 norm of Tf. The second is the pointwise estimate of T*f(x) by a constant times M(Tf)(x), where M denotes the Hardy-Littlewood maximal operator. Notice that this is an improved variant of Cotlar's inequality, because the term Mf(x) is missing on the right hand side. Our main result states that, for even operators, both are equivalent to a purely algebraic condition formulated in terms of the expansion of the kernel in spherical harmonics. The condition holds by higher order Riesz transforms, which then satisfy an improved version of Cotlar's inequality

math.CA↗
arXiv2010

New estimates for the maximal singular integral

In this paper we pursue the study of the problem of controlling the maximal singular integral $T^{*}f$ by the singular integral $Tf$. Here $T$ is a smooth homogeneous Calderón-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the $L^2(\Rn)$ norm and via pointwise estimates of $T^{*}f$ by $M(Tf)$ or $M^2(Tf)$, where $M$ is the Hardy-Littlewood maximal operator and $M^2=M \circ M$ its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type $T^\star \circ T$ arises. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak $(1,1)$ estimates, which are no longer true for odd kernels. For odd kernels we obtain sharp weaker inequalities involving a weak $L^1$ estimate for functions in $L LogL$.

math.CA↗
arXiv2008

Extra cancellation of even Calderon-Zygmund operators and quasiconformal mappings

We discuss a special class of Beltrami coefficients whose associated quasiconformal mapping is bilipschitz. These are of the form the characteristic function of a planar bounded domain with smooth boundary of class C 1+epsilon times a density of class Lip epsilon on the domain. The crucial fact in the argument is the special extracancellation property of even Calderon-Zygmund kernels, namely that they have zero integral on half the unit ball. This property is expressed in a particularly suggestive way and is shown to have far-reaching consequences. The main result may also be viewed as a Lipschitz regularity result for the Beltrami equation, and so for certain planar second order elliptic equations in divergence form.

math.CA↗
arXiv2007

Beltrami equations with coefficient in the Sobolev space $W^{1,p}$

We study the removable singularities for solutions to the Beltrami equation $\bar\partial f=μ\partial f$, assuming that the coefficient $μ$ lies on some Sobolev space $W^{1,p}$, $p\leq 2$. Our results are based on an extended version of the well known Weyl's lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported Beltrami coefficient $μ\in W^{1,2}$ preserve compact sets of $σ$-finite length and vanishing analytic capacity, even though they need not be bilipschitz.

math.AP↗
arXiv2006

Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

The classical Painlevé theorem tells that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains the corresponding critical dimension is $\frac{2}{K+1}$. We show that when $K>1$, unexpectedly one has improved removability. More precisely, we prove that sets $E$ of $σ$-finite Hausdorff $\frac{2}{K+1}$-measure are removable for bounded $K$-quasiregular mappings. On the other hand, $\dim(E) = \frac{2}{K+1}$ is not enough to guarantee this property. We also study absolute continuity properties of pull-backs of Hausdorff measures under $K$-quasiconformal mappings, in particular at the relevant dimensions 1 and $\frac{2}{K+1}$. For general Hausdorff measures ${\cal H}^t$, $0 < t < 2$, we reduce the absolute continuity properties to an open question on conformal mappings.

math.CV↗