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Giulio Galise

Publications and source records attributed to Giulio Galise.

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.

math.AP↗
arXiv2025

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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arXiv2025

Liouville type results for the fractional truncated Laplacians in a half-space

Existence issues of viscosity supersolutions in the half-space $\mathbb R^N_+$, for a class of fully nonlinear integral equations involving the fractional truncated Laplacians and a power-like nonlinearity in the unknown function, are addressed in this paper, the aim being to obtain estimates on the threshold exponents separating the existence from the nonexistence regimes.

math.AP↗
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.

math.AP↗
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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arXiv2019

New concentration phenomena for a class of radial fully nonlinear equations

We study radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values, showing that new concentration phenomena occur. Finally we define a suitable weighted energy for these solutions and compute its limit value.

math.AP↗
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

Existence through convexity for the truncated Laplacians

We study the Dirichlet problem on a bounded convex domain of $\mathbb R^N$, with zero boundary data, for truncated Laplacians ${\mathcal P}_k^\pm$, with $k<N$. We establish a necessary and sufficient condition (Theorem 1) in terms of the "flatness" of domains for existence of a solution for general inhomogeneous term. This result, in particular, shows that the strict convexity of the domain is sufficient for the solvability of the Dirichlet problem. The result and related ideas are applied to the solvability of the Dirichlet problem for the operator ${\mathcal P}_k^+$ with lower order term when the domain is strictly convex and the existence of principal eigenfunctions for the operator ${\mathcal P}_1^+$. An existence theorem is presented with regard to the principal eigenvalue for the Dirichlet problem with zero-th order term for the operator ${\mathcal P}_1^+$. A nonexistence result is established for the operator ${\mathcal P}_k^+$ with first order term when the domain has a boundary portion which is nearly flat. Furthermore, when the domain is a ball, we study the Dirichlet problem, with a constant inhomogeneous term and a possibly sign-changing first order term, and the associated eigenvalue problem.

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arXiv2018

On positive solutions of fully nonlinear degenerate Lane-Emden type equations

We prove existence and uniqueness results of positive viscosity solutions of fully nonlinear degenerate elliptic equations with power-like zero order perturbations in bounded domains. The principal part of such equations is either $\mathcal{P}^-_{k}(D^2u)$ or $\mathcal{P}^+_{k}(D^2u)$, some sort of \lq\lq truncated Laplacians\rq\rq, given respectively by the smallest and the largest partial sum of $k$ eigenvalues of the Hessian matrix. New phenomena with respect to the semilinear case occur. Moreover, for $\mathcal{P}^-_{k}$, we explicitely find the critical exponent $p$ of the power nonlinearity that separates the existence and nonexistence range of nontrivial solutions with zero Dirichlet boundary condition.

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arXiv2018

Towards a reversed Faber-Krahn inequality for the truncated Laplacian

We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for a class of very degenerate elliptic operators, with the aim to show that, at least for square type domains having fixed volume, the symmetry of the domain maximize the principal eigenvalue, contrary to what happens for the Laplacian.

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