arXiv subjects

Gabrielle Nornberg

Publications and source records attributed to Gabrielle Nornberg.

15 recordsLinked to original sources

arXiv2026

The Landis Conjecture For Nonlocal Elliptic Operators: Polynomial Decay

We obtain a unique continuation result at infinity for fully nonlinear elliptic integro-differential operators of order 2s which satisfy the maximum and minimum principles in bounded subdomains, under the decay assumption $o(|x|^{-(N+2s)})$ at infinity. Our result is new even in the case of the fractional Laplacian, as it unveils the nonlocal nature of the decay in Landis conjecture, evolving from exponential to polynomial.

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arXiv2026

Strong maximum principle for fully nonlinear nonlocal problems

In this paper, we study solvability and qualitative properties of nonnegative solutions for a sublinear nonlocal problem with fully nonlinear structure in the form $$ \mathcal{M}^{\pm}[u]+a(x)u^{q}(x)=0 \; \text{ in }Ω,\qquad u\geq 0 \; \text{ in }Ω. $$ Here $Ω\subset \mathbb{R}^n$ is a bounded $C^{1,1}$ convex domain, $\mathcal{M}^{ \pm}$ stands for nonlocal Pucci extremal operators defined in a class $\mathcal{L}_*$ of homogeneous kernels, $q\in(0,1)$, and $a$ is a possibly sign-changing weight. We introduce a new nonlocal hypothesis on the negative part of the solution outside the domain, which together with the negative part of the potential, influences the formation of dead cores and cannot be removed. Our approach relies on uniform bounds from below of the maximum of nontrivial solutions through Liouville theorems, and on a Hopf lemma for viscosity solutions driven by fully nonlinear operators, which we also prove.

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arXiv2025

Existence and a priori bounds for fully nonlinear PDEs with a harmonic map-like structure

In this paper, we study a new class of fully nonlinear uniformly elliptic equations with a so-called harmonic map-like structure, whose model case is given by \begin{equation*} \mathcal{M}^{\pm}_{λ,Λ}(D^2u) \pm b(x) |Du| \pm β(u)\langle M(x) Du,Du \rangle \pm c(x) u = f(x)\; \textrm{ in } Ω, \end{equation*} where $Ω\subset \mathbb{R}^n$ is a bounded $C^{1,1}$ domain, $\mathcal{M}^{\pm}$ are the Pucci extremal operators, $β(s) = s^k$ for some $k \in \mathbb{N} $ odd, $b \in L^{q}_{+}(Ω)$, $c,f \in L^p(Ω)$, and $n \leq p \leq q$, $q>n$. We obtain existence results under a smallness regime on the coefficients, along with some classical results such as the Aleksandrov--Bakelman--Pucci estimate and the comparison principle, as well as a priori bounds for the respective Dirichlet problem in the noncoercive case. We also establish multiplicity results and qualitative behavior, which seem to be new in the case of the Laplacian operator.

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arXiv2025

The moving patch model with fractional diffusion

In this paper we study the following one-dimensional reaction-diffusion problem $$ u_t+(-Δ)^s u=f(x-c t, u) \;\:\textrm{ in } \mathbb{R}\times (0,+\infty), $$ where $s>\frac{1}{2}$, $c \in \mathbb{R}$ is a prescribed velocity, and $f$ is of KPP type, which describes the evolution of a population in an advective environment subjected to nonlocal diffusion. We suppose the environment is such that it is only advantageous in a bounded ``patch", outside of which the species dies at an asymptotically constant rate. We first derive an optimal solvability criteria for the corresponding traveling waves problem $$Δ^s u+c u^{\prime}+f(x, u)=0 \;\:\textrm{ in } \mathbb{R},$$ through the first eigenvalue of the associated linearized elliptic operator with drift. Then we use this criteria to establish the long time behavior of the solution to the parabolic problem, for any continuous bounded nonnegative initial data, leading the species either through their extinction or survival. Moreover, assuming that for $c=0$ the population survives, we show that there exist two positive critical speeds $c^{*}$ and $c^{**}$ such that for all $|c| c^{**}$.

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arXiv2024

Fundamental solutions and critical Lane-Emden exponents for nonlinear integral operators in cones

In this article we study the fundamental solutions or "$α$-harmonic functions" for some nonlinear positive homogeneous nonlocal elliptic problems in conical domains, such as \begin{eqnarray*}\label{ecbir1a1} {\mathcal F }(u)=0\ \ \hbox{in} \ \ \mathcal{C}_ω,\quad u=0\ \ \hbox{in} \ \ \mathbb{R}^n\setminus \mathcal{C}_ω,\ \ \end{eqnarray*} where $ω$ is a proper $C^2$ domain in $S^{N-1}$ for $ N\geq 2$, $\mathcal{C}_ω:=\{x\,:\,x\neq 0, {|x|^{-1}}x\in ω\}$ is the cone-like domain related to $ω$, and ${\mathcal F }$ is an extremal fully nonlinear integral operator. We prove the existence of two fundamental solutions that are homogeneous and do not change signs in the cone; one is bounded at the origin and the other at infinity. As an application, we use the fundamental solutions obtained to prove Liouville type theorems in cones for supersolutions of the Lane-Emden-Fowler equation in the form \begin{eqnarray*}\label{eq 0.2} {\mathcal F }(u)+u^p = 0\ \ \hbox{in} \ \ \mathcal{C}_ω, \quad u=0\ \ \hbox{in} \ \ \mathbb{R}^n\setminus \mathcal{C}_ω.\ \ \end{eqnarray*} We also prove a generalized Hopf type lemma in domains with corners. Most of our results are new even when ${\mathcal F }$ is the fractional Laplacian operator.

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arXiv2021

Principal spectral curves for Lane-Emden fully nonlinear type systems and applications

In this paper we exploit the phenomenon of two principal half eigenvalues in the context of fully nonlinear Lane-Emden type systems with possibly unbounded coefficients and weights. We show that this gives rise to the existence of two principal spectral curves on the plane. We also construct a possible third spectral curve related to a second eigenvalue and an anti-maximum principle, which are novelties even for Lane-Emden systems involving linear operators. As applications, we derive a maximum principle in small domains for these systems, as well as existence and uniqueness of positive solutions in the sublinear regime. Most of our results are new even in the scalar case, in particular for a class of Isaac's operators with unbounded coefficients, whose $W^{2,\varrho}$ regularity estimates we also prove.

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arXiv2021

Radial solutions for Hénon type fully nonlinear equations in annuli and exterior domains

In this note we study existence of positive radial solutions in annuli and exterior domains for a class of nonlinear equations driven by Pucci extremal operators subject to a Hénon type weight. Our approach is based on the shooting method applied to the corresponding ODE problem, energy arguments, and the associated flow of an autonomous quadratic dynamical system.

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arXiv2021

Existence, nonexistence and uniqueness for Lane-Emden type fully nonlinear systems

We study existence, nonexistence, and uniqueness of positive radial solutions for a class of nonlinear systems driven by Pucci extremal operators under a Lane-Emden coupling configuration. Our results are based on the analysis of the associated quadratic dynamical system and energy methods. For both regular and exterior domain radial solutions we obtain new regions of existence and nonexistence. Besides, we show an exclusion principle for regular solutions, either in $\mathbb{R}^N$ or in a ball, by exploiting the uniqueness of trajectories produced by the flow. In particular, for the standard Lane-Emden system involving the Laplacian operator, we prove that the critical hyperbola of regular radial positive solutions is also the threshold for existence and nonexistence of radial exterior domain solutions with Neumann boundary condition. As a byproduct, singular solutions with fast decay at infinity are also found.

math.AP↗

arXiv2021

On unique continuation principles for some elliptic systems

In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions for the Lane-Emden system posed in a ball, in the critical and supercritical regimes. Some of our results also apply to general fully nonlinear operators, such as Pucci's extremal operators, being new even for scalar equations.

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arXiv2020

Regularity estimates for fully nonlinear elliptic PDEs with general Hamiltonian terms and unbounded ingredients

We develop an optimal regularity theory for $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form whose gradient growth is described through a Hamiltonian function with measurable and possibly unbounded coefficients. Our approach treats both superlinear and sublinear gradient regimes in a unified way. We show $C^{0,α}$, $C^{0,\textrm{Log-Lip}}$, $C^{1,α}$, $ C^{1,\textrm{Log-Lip}}$ and $C^{2,α}$ regularity estimates, by displaying the growth allowed to the Hamiltonian in order to deal with an unbounded nonlinear gradient coefficient, whose integrability in turn gets worse as we approach the quadratic regime. Moreover, we find proper compatibility conditions for which our regularity results depend intrinsically on the integrability of the underlying source term. As a byproduct of our methods, we prove a priori BMO estimates; sharp regularity to associated recession and flat profiles under relaxed convexity assumptions; improved regularity for a class of singular PDEs; and a Perron type result under unbounded ingredients.

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arXiv2020

A dynamical system approach to a class of radial weighted fully nonlinear equations

In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. In particular we recover known results on regular solutions for the fully nonlinear non weighted problem by alternative proofs. We also slightly improve the classification of the solutions for the extremal operator $\mathcal{M}^-$.

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arXiv2020

Symmetry properties of positive solutions for fully nonlinear elliptic systems

We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such as $$ F_i \,(x,Du_i,D^2u_i) +f_i \,(x,u_1, \ldots , u_n,Du_i)=0, \;\; 1 \leq i \leq n, $$ in a bounded domain $Ω$ in $\mathbb{R}^N$ with Dirichlet boundary condition $u_1=\ldots,u_n=0$ on $\partialΩ$. Here, $f_i $'s are nonincreasing with the radius $r=|x|$, and satisfy a cooperativity assumption. In addition, each $f_i $ is the sum of a locally Lipschitz with a nondecreasing function in the variable $u_i$, and may have superlinear gradient growth. We show that symmetry occurs for systems with nondifferentiable $f_i$'s by developing a unified treatment of the classical moving planes method in the spirit of Gidas-Ni-Nirenberg. We also present different applications of our results, including uniqueness of positive solutions for Lane-Emden systems in the subcritical case in a ball, and symmetry for a class of systems with natural growth in the gradient.

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arXiv2019

A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth

We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as $$ -F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle =λc_{i1}(x) u_1 + \cdots + λc_{in}(x) u_n +h_i(x), $$ for $i=1,\cdots,n$, in a bounded $C^{1,1}$ domain $Ω\subset \mathbb{R}^N$ with Dirichlet boundary conditions; here $n\geq 1$, $λ\in\mathbb{R}$, $c_{ij},\, h_i \in L^\infty(Ω)$, $c_{ij}\geq 0$, $M_i$ satisfies $0<μ_1 I\leq M_i\leq μ_2 I$, and $F_i$ is an uniformly elliptic Isaacs operator. We obtain uniform a priori bounds for systems, under a weak coupling hypothesis that seems to be optimal. As an application, we also establish existence and multiplicity results for these systems, including a branch of solutions which is new even in the scalar case.

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arXiv2019

$C^{1,α}$ regularity for fully nonlinear elliptic equations with superlinear growth in the gradient

We extend the Caffarelli-Świech-Winter $C^{1,α}$ regularity estimates to $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form with superlinear growth in the gradient and unbounded coefficients. As an application, in addition to the usual $W^{2,p}$ results, we prove the existence of positive eigenvalues for proper operators with nonnegative unbounded weight, in particular for Pucci's operators with unbounded coefficients.

math.AP↗

arXiv2018

A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as $$ -F(x,u,Du,D^2u) =λc(x)u+\langle M(x)D u, D u \rangle +h(x) $$ in a bounded domain with a Dirichlet boundary condition, here $λ\in\mathbb{R}$, $c,\, h \in L^p(Ω)$, $p>n\geq 1$, $c\gneqq 0$ and the matrix $M$ satisfies $0<μ_1 I\leq M\leq μ_2 I$. Recently this problem was studied in the "coercive" case $λc\le0$, where uniqueness of solutions can be expected, and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle. We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.

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