arXiv subjects
Publications and source records attributed to F. Chiacchio.
By means of a suitable weighted rearrangement, we obtain various apriori bounds for the solutions to a Robin problem. Among other things, we derive a family of Faber-Krahn type inequalities.
We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on $L^1$ comparison) between solutions to an elliptic partial differential equation on a smooth bounded set $Ω$ with a rather general boundary condition, and solutions to a suitable related problem defined on a ball having the same volume as $Ω$. This includes for instance mixed problems where Dirichlet boundary conditions are prescribed on part of the boundary, while Robin boundary conditions are prescribed on its complement.
Let $γ$ be a smooth, non-closed, simple curve whose image is symmetric with respect to the $y$-axis, and let $D$ be a planar domain consisting of the points on one side of $γ$, within a suitable distance $δ$ of $γ$. Denote by $μ_1^{odd}(D)$ the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the $y$-axis. If $γ$ satisfies some simple geometric conditions, then $μ_1^{odd}(D)$ can be sharply estimated from below in terms of the length of $γ$, its curvature, and $δ$. Moreover, we give explicit conditions on $δ$ that ensure $μ_1^{odd}(D)=μ_1(D)$. Finally, we can extend our bound on $μ_1^{odd}(D)$ to a certain class of three-dimensional domains. In both the two- and three-dimensional settings, our domains are generically non-convex.
We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $Ω$ in $\mathbb{R} ^N$ with fixed Lebesgue measure, $\int_{\partial Ω} |x|^k \, \mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Szëgo principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems.
Denote with $μ_{1}(Ω;e^{h\left(|x|\right)})$ the first nontrivial eigenvalue of the Neumann problem \begin{equation*} \left\{\begin{array}{lll} -\text{div}\left(e^{h\left(|x|\right)}\nabla u\right) =μe^{h\left(|x|\right)}u & \text{in} & Ω& & \frac{\partial u}{\partial ν}=0 & \text{on} & \partial Ω, \end{array} \right. \end{equation*} where $Ω$ is a bounded and Lipschitz domain in $\mathbb{R}^{N}$. Under suitable assumption on $h$ we prove that the ball centered at the origin is the unique set maximizing $μ_{1}(Ω;e^{h\left(|x|\right)})$ among all Lipschitz bounded domains $Ω$ of $\mathbb{R}^{N}$ of prescribed $e^{h\left(|x|\right)}dx$-measure and symmetric about the origin. Moreover, an example in the model case $h\left(|x|\right) =|x|^{2},$ shows that, in general, the assumption on the symmetry of the domain cannot be dropped. In the one-dimensional case, i.e. when $Ω$ reduces to an interval $(a,b),$ we consider a wide class of weights (including both Gaussian and anti-Gaussian). We then describe the behavior of the eigenvalue as the interval $(a,b)$ slides along the $x$-axis keeping fixed its weighted length.
In this paper, generalizing to the non smooth case already existing results, we prove that, for any convex planar set $Ω$, the first non-trivial Neumann eigenvalue $μ_1(Ω)$ of the Hermite operator is greater than or equal to 1. Furthermore, and this is our main result, under some additional assumptions on $Ω$, we show that $μ_1(Ω)=1$ if and only if $Ω$ is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.
In this paper we prove a sharp lower bound for the first nontrivial Neumann eigenvalue $μ_1(Ω)$ for the $p$-Laplace operator in a Lipschitz, bounded domain $Ω$ in $\R^n$. Our estimate does not require any convexity assumption on $Ω$ and it involves the best isoperimetric constant relative to $Ω$.
This paper deals with the Neumann eigenvalue problem for the Hermite operator defined in a convex, possibly unbounded, planar domain $Ω$, having one axis of symmetry passing through the origin. We prove a sharp lower bound for the first eigenvalue $μ_1^{odd}(Ω)$ with an associated eigenfunction odd with respect to the axis of symmetry. Such an estimate involves the first eigenvalue of the corresponding one-dimensional problem. As an immediate consequence, in the class of domains for which $μ_1(Ω)=μ_1^{odd}(Ω)$, we get an explicit lower bound for the difference between $μ(Ω)$ and the first Neumann eigenvalue of any strip.