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Giuseppina di Blasio

Publications and source records attributed to Giuseppina di Blasio.

13 recordsLinked to original sources
arXiv2025

Weighted Robin eigenvalue problems and nonlinear elliptic equations with general growth in the gradient

We prove an existence result for Robin boundary value problems modeled on \[ \begin{cases} Δu + |\nabla u|^2 + λf(x) = 0 & \text{in } Ω \\ \frac{\partial u}{\partial ν} + βu = 0 & \text{on } \partialΩ\end{cases} \] where $Ω$ is a bounded, sufficiently smooth open set in $\mathbb R^N$, $f(x)$ belongs to the Marcinkiewicz space $M^{\frac N2}$ and {$β>0$}, under a smallness assumption on the datum $λ$. In order to study such problem, we will show several properties of the weighted, singular Robin eigenvalue problem \[ λ_{1,f,γ}(Ω)= \inf_{ψ\in H^{1},\;\int_Ωfψ^{2}=1}\left\{\int_Ω|\nabla ψ|^{2}dx+γ\int_{\partialΩ}ψ^{2}\right\}. \]

math.AP↗
arXiv2024

Existence and regularity results for a class of non-uniformly elliptic Robin problems

In this paper, we study the existence and the summability of solutions to a Robin boundary value problem whose prototype is the following: $$ \begin{cases} -\text{div}(b(|u|)\nabla u)=f &\text{in }Ω,\\[.2cm] \displaystyle\frac{\partial u}{\partial ν}+βu=0 &\text{on }\partialΩ\end{cases} $$ where $Ω$ is a bounded Lipschitz domain in $\mathbb R^N$, $N>2$, $β>0$, $b(s)$ is a positive function which may vanish at infinity and $f$ belongs to a suitable Lebesgue space. The presence of such a function $b$ in the principal part of the operator prevents it from being uniformly elliptic when $u$ is large.

math.AP↗
arXiv2020

Eigenvalues of the Finsler $p$-Laplacian on varying domains

We study the dependence of the first eigenvalue of the Finsler $p$-Laplacian and the corresponding eigenfunctions upon perturbation of the domain and we generalize a few results known for the standard $p$-Laplacian. In particular, we prove a Frechét differentiability result for the eigenvalues, we compute the corresponding Hadamard formulas and we prove a continuity result for the eigenfunctions. Finally, we briefly discuss a well-known overdetermined problem and we show how to deduce the Rellich-Pohozaev identity for the Finsler $p$-Laplacian from the Hadamard formula.

math.AP↗
arXiv2017

Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle

In this paper we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue $λ_{F}(p,Ω)$ of the anisotropic $p$-Laplacian, $1<p<+\infty$. Our aim is to enhance how, by means of the $\mathcal P$-function method, it is possible to get several sharp estimates for $λ_{F}(p,Ω)$ in terms of several geometric quantities associated to the domain. The $\mathcal P$-function method is based on a maximum principle for a suitable function involving the eigenfunction and its gradient.

math.AP↗
arXiv2016

Comparison results for nonlinear anisotropic parabolic problems

Comparison results for solutions to the Dirichlet problems for a class of nonlinear, anisotropic parabolic equations are established. These results are obtained through a semi-discretization method in time after providing estimates for solutions to anisotropic elliptic problems with zero-order terms.

math.AP↗
arXiv2015

Anisotropic Hardy inequalities

We study some Hardy-type inequalities involving a general norm in $R^n$ and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed.

math.AP↗
arXiv2015

Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian

In this paper we prove existence results and asymptotic behavior for strong solutions $u\in W^{2,2}_{\textrm{loc}}(Ω)$ of the nonlinear elliptic problem \begin{equation} \tag{P} \label{abstr} \left\{ \begin{array}{ll} -Δ_{H}u+H(\nabla u)^{q}+λu=f&\text{in }Ω,\\ u\rightarrow +\infty &\text{on }\partialΩ, \end{array} \right. \end{equation} where $H$ is a suitable norm of $\mathbb R^{n}$, $Ω$ is a bounded domain, $Δ_{H}$ is the Finsler Laplacian, $1 0$ and $f$ is a suitable function in $L^{\infty}_{\textrm{loc}}$. Furthermore, we are interested in the behavior of the solutions when $λ\rightarrow 0^{+}$, studying the so-called ergodic problem associated to \eqref{abstr}. A key role in order to study the ergodic problem will be played by local gradient estimates for \eqref{abstr}.

math.AP↗
arXiv2011

Isoperimetric estimates for the first Neumann eigenvalue of Hermite differential equations

We provide isoperimetric Szegö-Weinberger type inequalities for the first nontrivial Neumann eigenvalue $μ_{1}(Ω)$ in Gauss space, where $Ω$ is a possibly unbounded domain of $\mathbb{R}^{N}$. Our main result consists in showing that among all sets of $\mathbb{R}^{N}$ symmetric about the origin, having prescribed Gaussian measure, $μ_{1}(Ω)$ is maximum if and only if $Ω$ is the euclidean ball centered at the origin.

math.AP↗