arXiv subjects
Publications and source records attributed to Michele Caselli.
We give a short proof of global regularity for the critical dissipative SQG equation in two dimensions. The argument is based on a pointwise representation of the Leibniz commutator for the half-Laplacian, recently obtained by the author with Luca Gennaioli, which simplifies and unifies the nonlinear estimates used in previous maximum principle proofs.
We study a notion of fractional $s$-mass for codimension-two currents on closed Riemannian manifolds, defined via energy minimization with a prescribed Jacobian constraint. We prove equi-coercivity and $\Gamma$-convergence, with respect to the flat topology, of the $s$-mass on general codimension-two currents. We also prove several additional results for fixed $s$. We establish improved regularity for $s$-harmonic maps that are minimizing among competitors with vanishing Jacobian and show that their singular set has Minkowski dimension at most $n-3$. Moreover, we show that the $s$-mass defined via weak linking, as recently introduced by the authors, agrees with the prescribed Jacobian formulation used here, clarifying the extent to which the $s$-mass depends, or ultimately does not depend, on the way singularities are prescribed.
We introduce a simple new method, based on the Caffarelli-Silvestre extension and a Duhamel-type formula, to derive exact pointwise identities for fractional commutators and nonlinear compositions associated with the fractional Laplacian on general Riemannian manifolds. As applications, we obtain a pointwise fractional Leibniz rule, a fractional Bochner's formula with an explicit Ricci curvature term, apparently the first of this kind, and exact remainders in the C\'ordoba-C\'ordoba and Kato inequalities for the fractional Laplacian. All these formulas are new even in the Euclidean space.
We construct an entire saddle solution to the non-magnetic complex Ginzburg-Landau system in three dimensions, whose zero set is the union of two perpendicular, intersecting lines, and whose blow-downs concentrate on these lines with multiplicity one.
We consider sequences of maps from an $(n+m)$-dimensional domain into the $(n-1)$-sphere, which satisfy a natural $p$-energy growth, as $p$ approaches $n$ from below. We prove that, up to subsequences, the Jacobians of such maps converge in the flat topology to an integral $m$-current, and that the $p$-energy Gamma-converges to the mass of the limit current. As a corollary, we deduce that the Jacobians of $p$-energy minimizing maps converge to an integral $m$-current that is area-minimizing in a suitable cobordism class, depending on the boundary datum. Moreover, we obtain new estimates for the minimal $p$-energy of maps with prescribed singularities.
For $s \in (0,1)$ small, we show that the only cones in $\mathbb{R}^2$ stationary for the $s$-perimeter and stable in $\mathbb{R}^2 \setminus \{0\}$ are half-planes. This is in direct contrast with the case of the classical perimeter or the regime $s$ close to $1$, where nontrivial cones as $\{xy>0\} \subset \mathbb{R}^2$ are stable for inner variations.
For $s\in (0,1)$ we introduce a notion of fractional $s$-mass on $(n-2)$-dimensional closed, orientable surfaces in $\R^n$. Moreover, we prove its $\Gamma$-convergence, with respect to the flat topology, and pointwise convergence to the $(n-2)$-dimensional area.
This article studies the canonical Hilbert energy $H^{s/2}(M)$ on a Riemannian manifold for $s\in(0,2)$, with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type $$ \mathcal E(v)=[v]^2_{H^{s/2}(M)}+\int_M F(v) \, dV \,, \,\,\, F \ge 0 \,, $$ is given, which includes, in particular, the case of nonlocal $s$-minimal surfaces. Finally, we prove some estimates for the Caffarelli-Silvestre extension problem, which are of general interest. This work is motivated by a recent article by the authors, which proves the nonlocal version of a conjecture of Yau.
In this work we study the asymptotics of the fractional Laplacian as $s\to 0^+$ on any complete Riemannian manifold $(M,g)$, both of finite and infinite volume. Surprisingly enough, when $M$ is not stochastically complete this asymptotics is related to the existence of bounded harmonic functions on $M$. As a corollary, we can find the asymptotics of the fractional $s$-perimeter on (essentially) every complete manifold, generalising both the existing results for $\mathbb{R}^n$ and for the Gaussian space. In doing so, from many sets $E\subset M$ we are able to produce a bounded harmonic function associated to $E$, which in general can be non-constant.
We introduce nonlocal minimal surfaces on closed manifolds and establish a far-reaching Yau-type result: in every closed, $n$-dimensional Riemannian manifold we construct infinitely many nonlocal $s$-minimal surfaces. We prove that, when $s\in (0,1)$ is sufficiently close to $1$, the constructed surfaces are smooth for $n=3$ and $n=4$, while for $n\ge 5$ they are smooth outside of a closed set of dimension $n-5$. Moreover, we prove surprisingly strong regularity and rigidity properties of finite Morse index $s$-minimal surfaces such as a "finite Morse index Bernstein-type result". These properties make nonlocal minimal surfaces ideal objects on which to apply min-max variational methods.
We prove Lipschitz continuity results for solutions to a class of obstacle problems under standard growth conditions of $p$-type, $p \geq 2$. The main novelty is the use of a linearization technique going back to [28] in order to interpret our constrained minimizer as a solution to a nonlinear elliptic equation, with a bounded right-hand side. This leads us to start a Moser iteration scheme which provides the $L^\infty$ bound for the gradient. The application of a recent higher differentiability result [24] allows us to simplify the procedure of the identification of the Radon measure in the linearization technique employed in [32]. To our knowledge, this is the first result for non-autonomous functionals with standard growth conditions in the direction of the Lipschitz regularity.
We establish the higher differentiability of solutions to a class of obstacle problems for integral functionals where the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher differentiability property of the weak solution v is related to the regularity of the assigned , under a suitable Sobolev assumption on the partial map that measures the oscillation of f with respect to the x variable. The main novelty is that such assumption is independent of the dimension n and that, in the case p<=n-2, improves previous known results.
In this paper we prove the local boundedness as well as the local Lipschitz continuity for solutions to a class of obstacle problems of the type $$\min\left\{\int_\Omega {F(x, Dz)}: z\in \mathcal{K}_{\psi}(\Omega)\right\}.$$ Here $\mathcal{K}_{\psi}(\Omega)$ is set of admissible functions $z \in W^{1,p}(\Omega)$ such that $z \ge \psi$ a.e. in $\Omega$, $\psi$ being the obstacle and $\Omega$ being an open bounded set of $\mathbb{R}^n$, $n \ge 2$. The main novelty here is that we are assuming $ F(x, Dz)$ satisfying $(p,q)$-growth conditions {and less restrictive assumptions on the obstacle with respect to the existing regularity results}.