57 research outputs found

### On the second solution to a critical growth Robin problem

We investigate the existence of the second mountain-pass solution to a Robin problem, where the equation is at critical growth and depends on a positive parameter $\lambda$. More precisely, we determine existence and nonexistence regions for this type of solutions, depending both on $\lambda$ and on the parameter in the boundary conditions

### Improved higher order Poincar\'e inequalities on the hyperbolic space via Hardy-type remainder terms

The paper deals about Hardy-type inequalities associated with the following
higher order Poincar\'e inequality:
$\left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty}
\setminus \{0\}} \frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^2
\ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2
\ dv_{\mathbb{H}^{N}} }\,,$ where $0 \leq l < k$ are integers and $\mathbb{H}^{N}$ denotes the
hyperbolic space. More precisely, we improve the Poincar\'e inequality
associated with the above ratio by showing the existence of $k$ Hardy-type
remainder terms. Furthermore, when $k = 2$ and $l = 1$ the existence of further
remainder terms are provided and the sharpness of some constants is also
discussed. As an application, we derive improved Rellich type inequalities on
upper half space of the Euclidean space with non-standard remainder terms.Comment: 17 page

### Sharp Poincar\'e-Hardy and Poincar\'e-Rellich inequalities on the hyperbolic space

We study Hardy-type inequalities associated to the quadratic form of the
shifted Laplacian $-\Delta_{\mathbb H^N}-(N-1)^2/4$ on the hyperbolic space
${\mathbb H}^N$, $(N-1)^2/4$ being, as it is well-known, the bottom of the
$L^2$-spectrum of $-\Delta_{\mathbb H^N}$. We find the optimal constant in the
resulting Poincar\'e-Hardy inequality, which includes a further remainder term
which makes it sharp also locally. A related inequality under suitable
curvature assumption on more general manifolds is also shown. Similarly, we
prove Rellich-type inequalities associated with the shifted Laplacian, in which
at least one of the constant involved is again sharp.Comment: Final version. To appear in JF

### Energy transfer between modes in a nonlinear beam equation

We consider the nonlinear nonlocal beam evolution equation introduced by
Woinowsky- Krieger. We study the existence and behavior of periodic solutions:
these are called nonlinear modes. Some solutions only have two active modes and
we investigate whether there is an energy transfer between them. The answer
depends on the geometry of the energy function which, in turn, depends on the
amount of compression compared to the spatial frequencies of the involved
modes. Our results are complemented with numerical experiments, overall, they
give a complete picture of the instabilities that may occur in the beam. We
expect these results to hold also in more complicated dynamical systemComment: Journal-Mathematiques-Pures-Appliquees, 201

### A Minimaxmax Problem for Improving the Torsional Stability of Rectangular Plates

We use a gap function in order to compare the torsional performances of
different reinforced plates under the action of external forces. Then, we
address a shape optimization problem, whose target is to minimize the torsional
displacements of the plate: this leads us to set up a minimaxmax problem, which
includes a new kind of worst-case optimization. Two kinds of reinforcements are
considered: one aims at strengthening the plate, the other aims at weakening
the action of the external forces. For both of them, we study the existence of
optima within suitable classes of external forces and reinforcements. Our
results are complemented with numerical experiments and with a number of open
problems and conjectures

### An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds

We prove \emph{optimal} improvements of the Hardy inequality on the
hyperbolic space. Here, optimal means that the resulting operator is critical
in the sense of [J.Funct.Anal. 266 (2014), pp. 4422-89], namely the associated
inequality cannot be further improved. Such inequalities arise from more
general, \emph{optimal} ones valid for the operator $P_{\lambda}:=
-\Delta_{\mathbb{H}^N} - \lambda$ where $0 \leq \lambda \leq
\lambda_{1}(\mathbb{H}^N)$ and $\lambda_{1}(\mathbb{H}^N)$ is the bottom of the
$L^2$ spectrum of $-\Delta_{\mathbb{H}^N}$, a problem that had been studied in
[J.Funct.Anal. 272 (2017), pp. 1661-1703 ] only for the operator
$P_{\lambda_{1}(\mathbb{H}^N)}$. A different, critical and new inequality on
$\mathbb{H}^N$, locally of Hardy type, is also shown. Such results have in fact
greater generality since there are shown on general Cartan-Hadamard manifolds
under curvature assumptions, possibly depending on the point.
Existence/nonexistence of extremals for the related Hardy-Poincar\'e
inequalities are also proved using concentration-compactness technique and a
Liouville comparison theorem. As applications of our inequalities we obtain an
improved Rellich inequality and we derive a quantitative version of
Heisenberg-Pauli-Weyl uncertainty principle for the operator $P_\lambda.$Comment: Final Versio

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