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Isabeau Birindelli

Publications and source records attributed to Isabeau Birindelli.

At least 19 recordsLinked to original sources
arXiv2026

Balance between degenerate elliptic operators and coercive Hamiltonians

For $p>1$, we consider the boundary value problem for fully nonlinear degenerate elliptic equations $-\lambda_i(D^2u)+|Du|^p+\gamma u=f(x)$ in bounded domains with Dirichlet or boundary blow-up conditions; here $\lambda_i(D^2u)$ denotes the $i$-th eigenvalue of the Hessian. We study existence and nonexistence of solutions together with the asymptotic behaviour of the solutions when $\gamma$ goes to zero. A priori Lipschitz estimates play an important role. The interplay between the operator's degeneracy and the superlinear growth of the Hamiltonian gives rise to phenomena that are very different depending on which of the two terms dominates, e.g. the ergodic dichotomy takes place only when $i=N$, while new phenomena arise for $i<N$ in which case, under mild conditions, solutions that blow up even in just one point do not exist, and conditions on the size of $f$ must be imposed for the existence of solutions to the Dirichlet problem with homogeneous boundary condition.

math.AP↗
arXiv2026

Fully nonlinear logistic equations with sanctuary

For the fully nonlinear stationary logistic equation ${\mathcal F}(x,D^2u)+\mu u=k(x)u^p$ with $p>1$ and $k(x)\geq 0$, in a bounded domain with Dirichlet boundary condition, we determine, in terms of $\mu$, the existence and uniqueness or the nonexistence of a positive solution. Furthermore, we study the asymptotic behavior of the solutions when $\mu$ approaches the boundary points of the existence range.

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arXiv2025

Radial solutions of truncated Laplacian equations in punctured balls

We consider equations involving the truncated laplacians and having lower order terms with singular potentials posed in punctured balls. We study both the principal eigenvalue problem and the problem of classification of solutions, in dependence of their asymptotic behaviour near the origin, for equations having also superlinear absorbing lower order terms. In the case of the maximising truncated Laplacian "Pk+", owing to the mild degeneracy of the operator, we obtain results which are analogous to the results for the Laplacian in dimension k. On the other hand, for minimising operator "Pk-" we show that the strong degeneracy in ellipticity of the operator produces radically different results.

math.AP↗
arXiv2024

Fully nonlinear elliptic PDEs in thin domains with oblique boundary condition

In this preprint we consider fully nonlinear equations in thin domains with oblique boundary condition, finding some new phenomena, in particular the limit equation contains "new terms" of the second, first and zeroth order which don't have an equivalent in the Neumann case treated in our previous work arXiv:2404.19577. The classical laplacian problem with Neumann boundary condition, goes back to the well known result of Hale and Raugel (1992).

math.AP↗
arXiv2024

Radial singular solutions of fully nonlinear equations in punctured balls

We study fully nonlinear uniformly elliptic equations having a singular reaction term with inverse quadratic potential and an absorbing superlinear term of p-power type. We consider equations posed in punctured balls centered at the origin, and we prove that all radial solutions are singular around the origin, by providing a complete classification in dependence of p of their asymptotic behavior near the singularity.

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arXiv2024

Liouville results for semilinear integral equations with conical diffusion

Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\quad\text{in $\mathbb R^N_+$}$$ are obtained, $-L$ being any symmetric and stable linear operator, positively homogeneous of degree $2s$, $s\in(0,1)$, whose spectral measure is absolutely continuous and positive only in a relative open set of the unit sphere of $\mathbb R^N$. The results are sharp: $u\equiv 0$ is the only nonnegative supersolution in the subcritical regime $1\leq p\leq\frac{N+s}{N-s}\,$, while nontrivial supersolutions exist, at least for some specific $-L$, as soon as $p>\frac{N+s}{N-s}$. \\ The arguments used rely on a rescaled test function's method, suitably adapted to such nonlocal setting with weak diffusion; they are quite general and also employed to obtain Liouville type results in the whole space.

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arXiv2023

Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls

This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions $( \barλ_γ, u_γ)$ of the equation $$F( D^2 u_γ)+ \bar λ_γ\frac{u_γ}{r^γ} = 0\ {\rm in} \ B(0,1)\setminus \{0\}, \ u_γ= 0 \ {\rm on} \ \partial B(0,1)$$ where $u_γ>0$ in $B(0,1)\setminus \{0\}$, and $γ>0$. We prove existence of radial solutions which are continuous on $\overline{ B(0,1)}$ in the case $γ<2$, existence of unbounded solutions in the case $γ= 2$ and a non existence result for $γ>2$. We also give the explicit value of $\bar λ_2$ in the case of Pucci's operators, which generalizes the Hardy--Sobolev constant for the Laplacian.

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arXiv2022

Propagation of minima for nonlocal operators

In this paper we state some sharp maximum principle, i.e. we characterize the geometry of the sets of minima for supersolutions of equations involving the $k$-\emph{th fractional truncated Laplacian} or the $k$-\emph{th fractional eigenvalue} which are fully nonlinear integral operators whose nonlocality is somehow $k$-dimensional.

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arXiv2021

Fractional truncated Laplacians: representation formula, fundamental solutions and applications

In this note we introduce some nonlinear extremal nonlocal operators that approximate the, so called, truncated Laplacians. For these operators we construct representation formulas that lead to the construction of what, with an abuse of notation, could be called "fundamental solutions". This, in turn, leads to Liouville type results. The interest is double: on one hand we wish to "understand" what is the right way to define the nonlocal version of the truncated Laplacians, on the other, we introduce nonlocal operators whose nonlocality is on one dimensional lines, and this dramatically changes the prospective, as is quite clear from the results obtained that often differs significantly with the local case or with the case where the nonlocality is diffused. Surprisingly this is true also for operators that approximate the Laplacian.

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arXiv2021

Mixed boundary value problems for fully nonlinear degenerate or singular equations

We prove existence, uniqueness and regularity results for mixed boundary value problems associated with fully nonlinear, possibly singular or degenerate elliptic equations. Our main result is a global Hölder estimate for solutions, obtained by means of the comparison principle and the construction of ad hoc barriers. The global Hölder estimate immediately yields a compactness result in the space of solutions, which could be applied in the study of principal eigenvalues and principal eigenfunctions of mixed boundary value problems.

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arXiv2019

Principal eigenvalues for k-Hessian operators by maximum principle methods

For fully nonlinear $k$-Hessian operators on bounded strictly $(k-1)$-convex domains $Ω$ in ${\mathbb R}^N$, a characterization of the principal eigenvalue associated to a $k$-convex and negative principal eigenfunction will be given as the supremum over values of a spectral parameter for which admissible viscosity supersolutions obey a minimum principle. The admissibility condition is phrased in terms of the natural closed convex cone $Σ_k$ in the space of symmetric N by N matrices, which is an elliptic set in the sense of Krylov [Trans. AMS, 1995] and which corresponds to using $k$-convex functions as admissibility constraints in the formulation of viscosity subsolutions and supersolutions. Moreover, the associated principal eigenfunction is constructed by an iterative viscosity solution technique, which exploits a compactness property which results from the establishment of a global Hölder estimate for the unique $k$-convex solutions of the approximating equations.

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arXiv2019

Positivity sets of supersolutions of degenerate elliptic equations and the strong maximum principle

We investigate positivity sets of nonnegative supersolutions of the fully nonlinear elliptic equations $F(x,u,Du,D^2u)=0$ in $Ω$, where $Ω$ is an open subset of ${\mathbb R}^N$, and the validity of the strong maximum principle for $F(x,u,Du,D^2u)=f$ in $Ω$, with $f\in\text{C}(Ω)$ being nonpositive. We obtain geometric characterizations of positivity sets $\left\{x\inΩ\,:\, u(x)>0\right\}$ of nonnegative supersolutions $u$ and establish the strong maximum principle under some geometric assumption on the set $\left\{x\inΩ\,:\, f(x)=0\right\}$.

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arXiv2019

A Liouville theorem for fully nonlinear problems with infinite boundary conditions and applications

We prove a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators. We then apply the result in order to obtain gradient boundary blow up rates for ergodic functions in bounded domains related to degenerate/singular operators, and, as a further consequence, we deduce the uniqueness of the ergodic functions.

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