arXiv subjects

F. Brock

Publications and source records attributed to F. Brock.

3 recordsLinked to original sources

arXiv2016

Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to weights $|x|^α$

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.

math.FA↗

arXiv2015

Optimal Szegö-Weinberger type inequalities

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.

math.AP↗

arXiv2011

On isoperimetric inequalities with respect to infinite measures

We study isoperimetric problems with respect to infinite measures on $R ^n$. In the case of the measure $μ$ defined by $dμ= e^{c|x|^2} dx$, $c\geq 0$, we prove that, among all sets with given $μ-$measure, the ball centered at the origin has the smallest (weighted) $μ-$perimeter. Our results are then applied to obtain Polya-Szego-type inequalities, Sobolev embeddings theorems and a comparison result for elliptic boundary value problems.

math.AP↗