arXiv subjects

Hardy Chan

Publications and source records attributed to Hardy Chan.

At least 19 recordsLinked to original sources

arXiv2025

$\Gamma$-convergence for higher order nonlocal phase transitions

For every $0 < s <3/4$, we study the asymptotic behavior of the $\varepsilon$-rescaled sum of the $s$-fractional Allen-Cahn energy and the squared $L^2$-norm of its first variation. We prove that the contribution of the first variation vanishes as $\varepsilon \to 0$. This implies the Gamma-convergence of the initial sum to either the classical perimeter or to the $2s$-fractional perimeter, depending on whether $s \ge 1/2$ or not. This contradicts the expectation of finding curvature-dependent terms in the limit, as suggested by the regime $3/4 \le s < 1$, and as known to hold in low dimensions in the local case.

math.AP↗

arXiv2025

Convergence of the fractional Yamabe flow for arbitrary initial energy

Since the seminal paper of Graham and Zworski (Invent. Math. 2003), conformal geometric problems are studied in the fractional setting. We consider the convergence of fractional Yamabe flow, which is previously known under small initial energy assumption. Inspired by the deep work of Brendle (J. Diff. Geom. 2005), we obtain the full convergence result for arbitrary initial energy, whenever the (fractional) positive mass conjecture is valid.

math.AP↗

arXiv2025

Global Stable Solutions to the Free Boundary Allen--Cahn and Bernoulli Problems in 3D are One-Dimensional

A long-standing conjecture of De Giorgi asserts that every monotone solution of the Allen--Cahn equation in $\mathbb{R}^{n+1}$ is one-dimensional if $n \leq 7$. A stronger version of the conjecture, also widely studied and often called ``the stable De Giorgi conjecture'', proposes that every stable solution in $\mathbb{R}^n$ must be one-dimensional for $n \leq 7$. To this date, both conjectures remain open for $3 \leq n \leq 7$. An elegant variant of this problem, advocated by Caffarelli, C\'ordoba, and Jerison since the 1990s, considers a free boundary version of the Allen--Cahn equation. This variant features a step-like double-well potential, leading to multiple free boundaries. Locally, near each free boundary, the solution satisfies the Bernoulli free boundary problem. However, the interaction of the free boundaries causes the global behavior of the solution to resemble that of the Allen--Cahn equation. In this paper, we establish the validity of the stable De Giorgi conjecture in dimension 3 for the free boundary Allen--Cahn equation and, as a corollary, we prove the corresponding De Giorgi conjecture for monotone solutions in dimension 4. To obtain these results, a key aspect of our work is to address a classical open problem in free boundary theory of independent interest: the classification of global stable solutions to the one-phase Bernoulli problem in three dimensions. This result, which is the core of our paper, implies universal curvature estimates for local stable solutions to Bernoulli, and serves as a foundation for adapting some classical ideas from minimal surface theory -- after significant refinements -- to the free boundary Allen--Cahn equation.

math.AP↗

arXiv2024

Spectral properties of L\'evy Fokker--Planck equations

Hermite polynomials, which are associated to a Gaussian weight and solve the Laplace equation with a drift term of linear growth, are classical in analysis and well-understood via ODE techniques. Our main contribution is to give explicit Euclidean formulae of the fractional analogue of Hermite polynomials, which appear as eigenfunctions of a L\'evy Fokker-Planck equation. We will restrict, without loss of generality, to radially symmetric functions. A crucial tool in our analysis is the Mellin transform, which is essentially the Fourier transform in logarithmic variable and which turns weighted derivatives into multipliers. This allows to write the weighted space in the fractional case that replaces the usual $L_r^2(\mathbb R^n, e^{|x|^2/4})$. After proving compactness, we obtain a exhaustive description of the spectrum of the L\'evy Fokker--Planck equation and its dual, the fractional Ornstein--Uhlenbeck problem, which forms a basis thanks to the spectral theorem for self-adjoint operators. As a corollary, we obtain a full asymptotic expansion for solutions of the fractional heat equation.

math.AP↗

arXiv2024

$\Gamma$-Limsup estimate for a nonlocal approximation of the Willmore functional

We propose a possible nonlocal approximation of the Willmore functional, in the sense of Gamma-convergence, based on the first variation ot the fractional Allen-Cahn energies, and we prove the corresponding $\Gamma$-limsup estimate. Our analysis is based on the expansion of the fractional Laplacian in Fermi coordinates and fine estimates on the decay of higher order derivatives of the one-dimensional nonlocal optimal profile. This result is the nonlocal counterpart of that obtained by Bellettini and Paolini, where they proposed a phase-field approximation of the Willmore functional based on the first variation of the (local) Allen-Cahn energies.

math.AP↗

arXiv2023

Nonlocal approximation of minimal surfaces: optimal estimates from stability

Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture. The primary goal of this paper is to set the ground for a new approximation based on nonlocal minimal surfaces. More precisely, we prove that if $\partial E$ is a stable $s$-minimal surface in $B_1\subset \mathbb R^3$ then: - $\partial E\cap B_{1/2}$ enjoys a $C^{2,\alpha}$ estimate that is robust as $s\uparrow 1$ (i.e. uniform in $s$); - the distance between different connected components of~$\partial E\cap B_{1/2}$ must be at least of order~$(1-s)^{\frac 1 2}$ (optimal sheet separation estimate); - interactions between multiple sheets at distances of order $(1-s)^{\frac 1 2}$ are described by the D\'avila--del Pino--Wei system. A second important goal of the paper is to establish that hyperplanes are the only stable $s$-minimal hypersurfaces in $\mathbb R^4$, for $s\in(0,1)$ sufficiently close to $1$. This is done by exploiting suitable modifications of the results described above. In this application, it is crucially used that our curvature and separations estimates hold without any assumption on area bounds (in contrast to the analogous estimates for Allen-Cahn).

math.DG↗

arXiv2023

Singular solutions for space-time fractional equations in a bounded domain

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann--Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense.

math.AP↗

arXiv2023

Boundary regularity of an isotropically censored nonlocal operator

In a bounded domain, we consider a variable range nonlocal operator, which is maximally isotropic in the sense that its radius of interaction equals the distance to the boundary. We establish $C^{1,\alpha}$ boundary regularity and existence results for the Dirichlet problem.

math.AP↗

arXiv2021

From fractional Lane-Emden-Serrin equation -- existence, multiplicity and local behaviors via classical ODE -- to fractional Yamabe metrics with singularity of "maximal" dimension

Point singularities of solutions to the classical Lane-Emden-Serrin equation have a polyhomogeneous asymptotic expansion whose logarithmic corrections are determined by a first order ODE. Surprisingly, we are able to discover such an ODE for the fractional Lane-Emden-Serrin equation, and therefore give a short classification for the precise local behavior of its solutions up to the second order involving a double logarithm. This seems to be the first time that a nonlocal equation is associated to a genuinely local ODE in one dimension. New non-existence, existence and multiplicity results for the corresponding Dirichlet problem are also discussed. Moreover, we construct complete $s$-fractional Yamabe metrics in $\mathbb{R}^n$ which are singular along a smooth submanifold of dimension $(n-2s)/2$, via direct integral asymptotic analysis with global geometric weights. This covers the missing borderline case as suggested by the deep work of Schoen and Yau. While such dimension is maximal in the class of distributional solutions, we conjecture the existence of complete metrics, understood in a suitably generalized sense, with prescribed singularities of strictly higher dimensions.

math.AP↗

arXiv2020

Lamellar phase solutions for diblock copolymers with nonlocal diffusions

For a diblock copolymer with total chain length $\gamma>0$ and mass ratio $m\in(-1,1)$, we consider the problem of minimizing the doubly nonlocal free energy $$ \mathcal{E}_{\varepsilon}(u) =\mathcal{H}(u) +\frac{1}{\varepsilon^{2s}} \int_{\Omega}W(u)\,dx +\frac{1}{2}\int_{\Omega} \left|(-\gamma^{2}\Delta)^{-\frac{1}{2}}(u-m)\right|^2\,dx $$ in a domain $\Omega$, where $\mathcal{H}(u)$ is a fractional $H^s$-norm with $s\in(0,\frac12)$, and $W$ is a double-well potential. This arises in the study of micro-phase separation phenomena for diblock copolymers with nonlocal diffusions. On the unit interval, we identify the $\Gamma$-limit as $\varepsilon\to0^+$, and also find explicit isolated local minimizers associated the lamellar morphology phase in the case $m=0$, provided that the chain is sufficiently short or the nonlocal interaction is sufficiently strong (i.e. as $s\to0^+$). We stress that such extra condition is new for the nonlocal case and is not present in the classical model. The proof, while elementary, requires a careful analysis of the nonlocal integrals.

math.AP↗

arXiv2020

Singular solutions for fractional parabolic boundary value problems

The standard problem for the classical heat equation posed in a bounded domain $\Omega$ of $\mathbb R^n$ is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial and boundary value problem can still be solved on the condition that the non-zero boundary data must be singular, i.e., the solution $u(t,x)$ blows up as $x$ approaches $\partial \Omega$ in a definite way. In this paper we construct a theory of existence and uniqueness of solutions of the parabolic problem with singular data taken in a very precise sense, and also admitting initial data and a forcing term. When the boundary data are zero we recover the standard fractional heat semigroup. A general class of integro-differential operators may replace the classical fractional Laplacian operators, thus enlarging the scope of the work. As further results on the spectral theory of the fractional heat semigroup, we show that a one-sided Weyl-type law holds in the general class, which was previously known for the restricted and spectral fractional Laplacians, but is new for the censored (or regional) fractional Laplacian. This yields bounds on the fractional heat kernel.

math.AP↗

arXiv2020

Blow-up phenomena in nonlocal eigenvalue problems: when theories of $L^1$ and $L^2$ meet

We develop a linear theory of very weak solutions for nonlocal eigenvalue problems $\mathcal L u = \lambda u + f$ involving integro-differential operators posed in bounded domains with homogeneous Dirichlet exterior condition, with and without singular boundary data. We consider mild hypotheses on the Green's function and the standard eigenbasis of the operator. The main examples in mind are the fractional Laplacian operators. Without singular boundary datum and when $\lambda$ is not an eigenvalue of the operator, we construct an $L^2$-projected theory of solutions, which we extend to the optimal space of data for the operator $\mathcal L$. We present a Fredholm alternative as $\lambda$ tends to the eigenspace and characterise the possible blow-up limit. The main new ingredient is the transfer of orthogonality to the test function. We then extend the results to singular boundary data and study the so-called large solutions, which blow up at the boundary. For that problem we show that, for any regular value $\lambda$, there exist "large eigenfunctions" that are singular on the boundary and regular inside. We are also able to present a Fredholm alternative in this setting, as $\lambda$ approaches the values of the spectrum. We also obtain a maximum principle for weighted $L^1$ solutions when the operator is $L^2$-positive. It yields a global blow-up phenomenon as the first eigenvalue is approached from below. Finally, we recover the classical Dirichlet problem as the fractional exponent approaches one under mild assumptions on the Green's functions. Thus "large eigenfunctions" represent a purely nonlocal phenomenon.

math.AP↗

arXiv2020

Uniqueness of entire ground states for the fractional plasma problem

We establish uniqueness of vanishing radially decreasing entire solutions, which we call ground states, to some semilinear fractional elliptic equations. In particular, we treat the fractional plasma equation and the supercritical power nonlinearity. As an application, we deduce uniqueness of radial steady states for nonlocal aggregation-diffusion equations of Keller-Segel type, even in the regime that is dominated by aggregation.

math.AP↗

arXiv2019

An analytic construction of singular solutions related to a critical Yamabe problem

We answer affirmatively a question of Aviles posed in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted $L^\infty$ spaces that handles multiple occurrences of criticality, without the need of derivative estimates. The above solution constitutes an \emph{Ansatz} for the Yamabe problem with a prescribed singular set of maximal dimension $(n-2)/2$, for which, using the same machinery, we provide an alternative construction to the one given by Pacard. His linear theory uses $L^p$-theory on manifolds, while our approach studies the equations in the ambient space and is therefore suitable for generalization to nonlocal problems. In a forthcoming paper, we will prove analogous results in the fractional setting.

math.AP↗

arXiv2019

ODE-methods in non-local equations

Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing scheme of our previous article "On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program"; we survey and improve those, and present new applications as well. First, from the explicit symbol of the conformal fractional Laplacian, a variation of constants formula is obtained for fractional Hardy operators. We thus develop, in addition to a suitable extension in the spirit of Caffarelli--Silvestre, an equivalent formulation as an infinite system of second order constant coefficient ODEs. Classical ODE quantities like the Hamiltonian and Wro\'nskian may then be utilized. As applications, we obtain a Frobenius theorem and establish new Poho\vzaev identities. We also give a detailed proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane-Emden equation.

math.AP↗

arXiv2018

Convergence of the fractional Yamabe flow for a class of initial data

This work is a follow-up on the work of the second author with P. Daskalopoulos and J.L. V\'{a}zquez. In this latter work, we introduced the Yamabe flow associated to the so-called fractional curvature and prove some existence result of mild (semi-group) solutions. In the present work, we continue this study by proving that for some class of data one can prove actually convergence of the flow in a more general context. We build on the approach simplified in the book of M. Struwe.

math.AP↗

arXiv2018

Bound state solutions for the supercritical fractional Schr\"odinger equation

We prove the existence of positive solutions for the supercritical nonlinear fractional Schr\"odinger equation $(-\Delta)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1), \ n>2s$. We show that if $V(x)=o(|x|^{-2s})$ as $|x|\to +\infty$, then for $p>\frac{n+2s-1}{n-2s-1}$, this problem admits a continuum of solutions. More generally, for $p>\frac{n+2s}{n-2s}$, conditions for solvability are also provided. This result is the extension of the work by Davila, Del Pino, Musso and Wei to the fractional case. Our main contributions are: the existence of a smooth, radially symmetric, entire solution of $(-\Delta)^s w=w^p$ in $\mathbb R^n$, and the analysis of its properties. The difficulty here is the lack of phase-plane analysis for a nonlocal ODE; instead we use conformal geometry methods together with Schaaf's argument as in the paper by Ao, Chan, DelaTorre, Fontelos, Gonz\'alez and Wei on the singular fractional Yamabe problem.

math.AP↗

arXiv2018

On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program

We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, and we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of an ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. No traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we examine a fractional Schr\"{o}dinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.

math.AP↗