Explicit minimisers for anisotropic Riesz energies
In this paper we describe explicitly the energy minimisers of a class of nonlocal interaction energies where the attraction is quadratic, and the repulsion is Riesz-like and anisotropic.
math.AP↗
arXiv subjects
Joan Verdera
Publications and source records attributed to Joan Verdera.
About the Jones-Wolff Theorem on the Hausdorff dimension of harmonic measure
These are the lecture notes of a seminar held at the Universitat Autònoma de Barcelona where the Jones-Wolff theorem about the dimension of harmonic measure in the plane is explained in full detail, for non-expert readers.
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Explicit minimisers for anisotropic Coulomb energies in 3D
In this paper we consider a general class of anisotropic energies in three dimensions and give a complete characterisation of their minimisers. We show that, depending on the Fourier transform of the interaction potential, the minimiser is either the normalised characteristic function of an ellipsoid or a measure supported on a two-dimensional ellipse. In particular, it is always an ellipsoid if the transform is strictly positive, while when the Fourier transform is degenerate both cases can occur. Finally, we show an explicit example where loss of dimensionality of the minimiser does occur.
math.AP↗
The precise representative for the gradient of the Riesz potential of a finite measure
Given a finite nonnegative Borel measure $m$ in $\mathbb{R}^{d}$, we identify the Lebesgue set $\mathcal{L}(V_{s}) \subset \mathbb{R}^{d}$ of the vector-valued function $$V_{s}(x) = \int_{\mathbb{R}^{d}}\frac{x - y}{|x - y|^{s + 1}} \mathrm{d}m(y), $$ for any order $0 < s < d$. We prove that $a \in \mathcal{L}(V_{s})$ if and only if the integral above has a principal value at $a$ and $$\lim_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}} = 0.$$ In that case, the precise representative of $V_{s}$ at $a$ coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions.
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Stability of ellipsoids as the energy minimisers of perturbed Coulomb energies
In this paper we characterise the minimiser for a class of nonlocal perturbations of the Coulomb energy. We show that the minimiser is the normalised characteristic function of an ellipsoid, under the assumption that the perturbation kernel has the same homogeneity as the Coulomb potential, is even, smooth off the origin and sufficiently small. This result can be seen as the stability of ellipsoids as energy minimisers, since the minimiser of the Coulomb energy is the normalised characteristic function of a ball.
math.AP↗
The global regularity of vortex patches revisited
We prove persistence of the regularity of the boundary of vortex patches for a large class of transport equations in the plane. The velocity field is given by convolution of the vorticity with an odd kernel, homogeneous of degree $-1$ and of class $C^2$ off the origin.
math.AP↗
Birth and life of the $L^{2}$ boundedness of the Cauchy Integral on Lipschitz graphs
We review various motives for considering the problem of estimating the Cauchy Singular Integral on Lipschitz graphs in the $L^{2}$ norm. We follow the thread that led to the solution and then describe a few of the innumerable applications and ramifications of this fundamental result. We concentrate on its influence in complex analysis, geometric measure theory and harmonic measure.
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Capacitary differentiability of potentials of finite Radon measures
We study differentiability properties of a potential of the type $K\star μ$, where $μ$ is a finite Radon measure in $\mathbb{R}^N$ and the kernel $K$ satisfies $|\nabla^j K(x)| \le C\, |x|^{-(N-1+j)}, \quad j=0,1,2.$ We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vallée Poussin sense associated with the kernel $|x|^{-(N-1)}.$ We require that the first order remainder at a point is small when measured by means of a normalized weak capacity "norm" in balls of small radii centered at the point. This implies weak $L^{N/(N-1)}$ differentiability and thus $L^{p}$ differentiability in the Calderón--Zygmund sense for $1\le p < N/(N-1)$. We show that $K\star μ$ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for $K\star μ.$ As an application, we study level sets of newtonian potentials of finite Radon measures.
math.CA↗
On square functions with independent increments and Sobolev spaces on the line
We prove a characterization of some $L^p$-Sobolev spaces involving the quadratic symmetrization of the Calderón commutator kernel, which is related to a square function with differences of difference quotients. An endpoint weak type estimate is established for functions in homogeneous Hardy-Sobolev spaces $\dot H^1_α$. We also use a local version of this square function to characterize pointwise differentiability for functions in the Zygmund class.
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Differentiability properties of Riesz potentials of finite measures and non-doubling Calderón-Zygmund theory
We study differentiability properties of Riesz potentials of finite Borel measures in dimension d larger than 2. The Riesz kernel has homogeneity 2-d. In dimension 2 we consider logarithmic potentials. We introduce a notion of differentiability in the capacity sense, capacity being Newtonian capacity in dimension larger than 2 and Wiener capacity in the plane. It turns out that differentiability in the capacity sense at a point is related to the existence of principal values of the measure with respect to the vector valued Riesz potential x/|x|^d of homogeneity 1-d. This leads to Calderón-Zygmund theory for non-doubling measures. We prove that the Riesz potential of a finite Borel measure is differentiable in the capacity sense except for a set of zero C^1-harmonic capacity. This result is sharp. Surprisingly in the plane there are two distinct notions of differentiability in the capacity sense. For each of them we obtain the best possible result on the size of the exceptional set in terms of Hausdorff measures. We obtain for dimension larger than 2 results on Peano second order differentiability in the capacity sense with exceptional sets of zero Lebesgue measure. Finally, we apply our results to find a new proof of the fact that equilibrium measure is singular with respect to Lebesgue measure.
math.CA↗
The regularity of the boundary of a multidimensional aggregation patch
Let $d \geq 2$ and let $N(y)$ be the fundamental solution of the Laplace equation in $R^d$ We consider the aggregation equation $$ \frac{\partial ρ}{\partial t} + \operatorname{div}(ρv) =0, v = -\nabla N * ρ$$ with initial data $ρ(x,0) = χ_{D_0}$, where $χ_{D_0}$ is the indicator function of a bounded domain $D_0 \subset R^d.$ We now fix $0 < γ< 1$ and take $D_0$ to be a bounded $C^{1+γ}$ domain (a domain with smooth boundary of class $C^{1+γ}$). Then we have Theorem: If $D_0$ is a $C^{1 + γ}$ domain, then the initial value problem above has a solution given by $$ρ(x,t) = \frac{1}{1 -t} χ_{D_t}(x), \quad x \in R^d, \quad 0 \le t < 1$$ where $D_t$ is a $C^{1 + γ}$ domain for all $0 \leq t < 1$.
math.AP↗
Boundary regularity of rotating vortex patches
We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is of class C^infinity provided the patch is close enough to the bifurcation circle in the Lipschitz norm. The rotating patch is convex if it is close enough to the bifurcation circle in the C^2 norm. Our proof is based on Burbea's approach to V-states. Thus conformal mapping plays a relevant role as well as estimating, on Hölder spaces, certain non-convolution singular integral operators of Calderón-Zygmund type.
math.CA↗
Doubly connected V-states for the planar Euler equations
We prove existence of doubly connected V-states for the planar Euler equations which are not annuli. The proof proceeds by bifurcation from annuli at simple "eigenvalues". The bifurcated $V$-states we obtain enjoy a $m$-fold symmetry for some $m\ge 3.$ The existence of doubly connected $V$-states of strict $2$-fold symmetry remains open.
math.AP↗
Capacities associated with scalar signed Riesz kernels, and analytic capacity
The real and imaginari parts of the Cauchy kernel in the plane are scalar Riesz kernels of homogeneity -1. One can associate with each of them a natural notion of capacity related to bounded potentials. The main result of the paper asserts that these capacities are comparable to classical analytic capacity, thus stressing the real variables nature of analytic capacity. Higher dimensional versions of this result are also considered.
math.CA↗
On rotating doubly connected vortices
In this paper we consider rotating doubly connected vortex patches for the Euler equations in the plane. When the inner interface is an ellipse we show that the exterior interface must be a confocal ellipse. We then discuss some relations, first found by Flierl and Polvani, between the parameters of the ellipses, the velocity of rotation and the magnitude of the vorticity in the domain enclosed by the inner ellipse.
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A general form of Green Formula and Cauchy Integral Theorem
We prove a general form of Green Formula and Cauchy Integral Theorem for arbitrary closed rectifiable curves in the plane.
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The maximal singular integral : estimates in terms of the singular integral
This is an expository paper on the characterization of the even (or odd) smooth homogeneous convolution Calderón-Zygmund operators in R^n such that the maximal singular integral can be controlled in the L^2 norm by the singular integral. We focuss attention on special cases of the general statements to convey the main ideas of the proofs in a transparent way, as free as possible of the technical complications inherent to the general case. Particular attention is devoted to higher Riesz transforms. The exposition is based on two papers by J.Mateu, J.Orobitg, C.Pérez and J.Verdera published in 2010 and 2011.
math.CA↗
Duality in spaces of finite linear combinations of atoms
In this note we describe the dual and the completion of the space of finite linear combinations of $(p,\infty)$-atoms, $0<p\leq 1$ on ${\mathbb R}^n$. As an application, we show an extension result for operators uniformly bounded on $(p,\infty)$-atoms, $0<p < 1$, whose analogue for $p=1$ is known to be false. Let $0 < p <1$ and let $T$ be a linear operator defined on the space of finite linear combinations of $(p,\infty)$-atoms, $0<p < 1 $, which takes values in a Banach space $B$. If $T$ is uniformly bounded on $(p,\infty)$-atoms, then $T$ extends to a bounded operator from $H^p({\mathbb R}^n)$ into $B$.
math.FA↗