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María del Mar González

Publications and source records attributed to María del Mar González.

17 recordsLinked to original sources
arXiv2026

A geometric approach to nonlocal 2-Hessian equations

We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, $\inf_{A\in {A}_2}Δ^s (u\circ A)(A^{-1}x)$. We characterize the class of coefficient matrices ${A}_2$, which determines the behavior of the operator, and provide a detailed geometric description of the possible degeneracies. Our main theorem shows that the nonlocal 2-Hessian equation remains uniformly elliptic for strictly positive right-hand sides, which leads to regularity estimates. The results hold under weaker hypotheses than those previously considered in the literature, and for the full range $s\in(0,1)$. In particular, no convexity on the solutions is required. Our hypotheses can be interpreted as nonlocal counterparts of the local notions of semiconcavity and 2-convexity. Moreover, all the results are stable as $s\to1$, recovering the local case. The geometric methods developed here are new, even in the local setting, and may be relevant to a broader class of nonlocal fully nonlinear equations and curvature-type problems.

math.AP↗
arXiv2026

Traveling-wave behavior for Fisher-KPP equations in the hyperbolic space

We study the Cauchy problem in the hyperbolic space for the heat equation with a Fisher-KPP type forcing term. Depending on the relative strength of diffusion, measured by the infimum of the spectrum of the Laplace-Beltrami operator, as compared to the growth due to the forcing term, solutions may propagate or vanish as time passes. We prove new results concerning this dichotomy that include the critical case where diffusion and reaction are of the same order. If the initial datum possesses some symmetry (invariance under a cohomogeneity one subgroup of the group of isometries of the hyperbolic space), the problem reduces to a unidimensional one. In the case of propagation, the solution to this unidimensional problem converges in shape to an Euclidean traveling wave of minimal speed in an appropriate moving frame. The choice of this frame depends on the subgroup of isometries (elliptic, hyperbolic or parabolic) under which the initial datum is invariant. In contrast with the Euclidean case, the asymptotic spreading speed (in due coordinates) depends on the dimension, while the coefficient of the logarithmic correction in the location of the front does not, no matter the underlying isometry.

math.AP↗
arXiv2026

A gluing construction of singular solutions for a fully non-linear equation in conformal geometry

In this paper we study the $σ_2$--Yamabe equation, $n>4$, for solutions with a prescribed singular set $Λ$ given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than $(n-\sqrt{n}-2)/2$. The $σ_2$--curvature in conformal geometry is defined as the second elementary symmetric polynomial of the eigenvalues of the Schouten tensor, which yields a fully non-linear PDE for the conformal factor. We show that the classical gluing method, used by Mazzeo-Pacard (JDG 1996) for the scalar curvature problem, can be used in the fully non-linear setting. This is a consequence of the conformal properties of the $σ_2$ equation, which imply that the linearized operator has good mapping properties in weighted spaces.

math.DG↗
arXiv2026

The Pohozaev identity for the Spectral Fractional Laplacian

In this paper, we prove a Pohozaev identity for the Spectral Fractional Laplacian (SFL). This identity allows us to establish non-existence results for the semilinear Dirichlet problem $(-Δ|_Ω)^su = f(u)$ in star-shaped domains. The first such identity for non-local operators was established by Ros-Oton and Serra in 2014 for the Restricted Fractional Laplacian (RFL). However, the SFL differs fundamentally from the RFL, and the integration by parts strategy of Ros-Oton and Serra cannot be applied. Instead, we develop a novel spectral approach that exploits the underlying quadratic structure. Our main result expresses the identity as a Schur product of the classical Pohozaev quadratic form and a transition matrix that depends on the eigenvalues of the Laplacian and the fractional exponent.

math.AP↗
arXiv2024

Spectral properties of Lévy 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 Lévy 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 Lévy 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

Concentration limit for non-local dissipative convection-diffusion kernels on the hyperbolic space

We study a non-local evolution equation on the hyperbolic space $\mathbb{H}^N$. We first consider a model for particle transport governed by a non-local interaction kernel defined on the tangent bundle and invariant under the geodesic flow. We study the relaxation limit of this model to a local transport problem, as the kernel gets concentrated near the origin of each tangent space. Under some regularity and integrability conditions on the kernel, we prove that the solution of the rescaled non-local problem converges to that of the local transport equation. Then, we construct a large class of interaction kernels that satisfy those conditions. We also consider a non-local, non-linear convection-diffusion equation on $\mathbb{H}^N$ governed by two kernels, one for each of the diffusion and convection parts, and we prove that the solution converges to the solution of a local problem as the kernels get concentrated. We prove and then use in this sense a compactness tool on manifolds inspired by the work of Bourgain-Brezis-Mironescu.

math.AP↗
arXiv2022

A Steklov version of the Torsional Rigidity

Motivated by the connection between the first eigenvalue of the Dirichlet-Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of $W^{1,2}(Ω)$ into $L^1(\partialΩ)$. We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simply connected sets and for convex sets in any dimension.

math.AP↗
arXiv2022

Bulk-Boundary eigenvalues for Bilaplacian problems

We initiate the study of a bulk-boundary eigenvalue problem for the Bilaplacian with a particular third order boundary condition that arises from the study of dynamical boundary conditions for the Cahn-Hilliard equation. First we consider continuity properties under parameter variation (in which the parameter also affects the domain of definition of the operator). Then we look at the ball and the annulus geometries (together with the punctured ball), obtaining the eigenvalues as solutions of a precise equation involving special functions. An interesting outcome of our analysis in the annulus case is the presence of a bifurcation from the zero eigenvalue depending on the size of the annulus.

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↗
arXiv2020

Optimal configuration and symmetry breaking phenomena in the composite membrane problem with fractional Laplacian

We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain $Ω\subset \mathbb{R}^n$, $α>0$ and $0<A<|Ω|$, find a subset $D\subset Ω$ of area $A$ such that the first Dirichlet eigenvalue of the operator $(-Δ)^s+αχ_D$ is as small as possible. The solution $D$ is called as an optimal configuration for the data $(Ω,α,A)$. Looking at the well-known extension definition for the fractional Laplacian, in the case $s=1/2$ this is essentially the composite membrane problem for which the mass is concentrated at the boundary as one is trying to maximize the Steklov eigenvalue. We prove existence of solutions and study properties of optimal configuration $D$. This is a free boundary problem which could be formulated as a two-sided unstable obstacle problem. Moreover, we show that for some rotationally symmetric domains (thin annuli), the optimal configuration is not rotational symmetric, which implies the non-uniqueness of the optimal configuration $D$. On the other hand, we prove that for a convex domain $Ω$ having reflection symmetries, the optimal configuration possesses the same symmetries, which implies uniqueness of the optimal configuration $D$ in the ball case.

math.AP↗
arXiv2020

Admissible surfaces in progressive addition lenses

Progressive addition lenses contain a surface of spatially-varying curvature, which provides variable optical power for different viewing areas over the lens. We derive complete compatibility equations that provide the exact magnitude of cylinder along lines of curvatures on any arbitrary PAL smooth surface. These equations reveal that, contrary to current knowledge, cylinder, and its derivative, does not only depend on principal curvature and its derivatives along the principal line but also on the geodesic curvature and its derivatives along the line orthogonal to the principal line. We quantify the relevance of the geodesic curvature through numerical computations. We also derive an extended and exact Minkwitz theorem only restricted to be applied along lines of curvatures, but excluding umbilical points.

math.DG↗
arXiv2016

$Γ$-convergence of variational functionals with boundary terms in Stein manifolds

Let $Ω$ be an open subset of a Stein manifold $Σ$ and let $M$ be its boundary. It is well known that $M$ inherits a natural contact structure. In this paper we consider a family of variational functionals $F_\varepsilon$ defined by the sum of two terms: a Dirichlet-type energy associated with a sub-Riemannian structure in $Ω$ and a potential term on the boundary $M$. We prove that the functionals $F_\varepsilon$ $Γ$-converge to the intrinsic perimeter in $M$ associated with its contact structure. Similar results have been obtained in the Euclidean space by Alberti, Bouchitté, Seppecher. We stress that already in the Euclidean setting the situation is not covered by the classical Modica-Mortola Theorem because of the presence of the boundary term. We recall also that Modica-Mortola type results (without a boundary term) have been proved in the Euclidean space for sub-Riemannian energies by Monti and Serra Cassano.

math.AP↗
arXiv2015

Isolated singularities for a semilinear equation for the fractional Laplacian arising in conformal geometry

We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are: first, to set the problem into a natural geometric framework, and second, to perform some kind of phase portrait study for this non-local ODE.

math.AP↗
arXiv2013

Instability and bifurcation in a trend depending price formation model

A well-known model due to J.-M. Lasry and P.L. Lions that presents the evolution of prices in a market as the evolution of a free boundary in a diffusion equation is suggested to be modified in order to show instabilities for some values of the parameters. This loss of stability is associated to the appearance of new types of solutions, namely periodic solutions, due to a Hopf bifurcation and representing price oscillations, and traveling waves, that represent either inflationary or deflationary behavior.

math.AP↗
arXiv2012

Layer solutions for the fractional Laplacian on hyperbolic space: existence, uniqueness and qualitative properties

We investigate the equation $$(-Δ_{\mathbb H^n})^γ w=f(w)\quad in \mathbb H^{n},$$ where $(-Δ_{\mathbb H^n})^γ$ corresponds to the fractional Laplacian on hyperbolic space for $γ\in (0,1)$ and $f$ is a smooth nonlinearity that typically comes from a double well potential. We prove the existence of heteroclinic connections in the following sense; a so-called layer solution is a smooth solution of the previous equation converging to $\pm 1$ at any point of the two hemispheres $S_\pm \subset \partial_\infty \mathbb H^n$ and which is strictly increasing with respect to the signed distance to a totally geodesic hyperplane $Π.$ We prove that under additional conditions on the nonlinearity uniqueness holds up to isometry. Then we provide several symmetry results and qualitative properties of the layer solutions. Finally, we consider the multilayer case, at least when $γ$ is close to one.

math.AP↗