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Gyula Csato

Publications and source records attributed to Gyula Csato.

5 recordsLinked to original sources
arXiv2020

Extremals for the Singular Moser-Trudinger Inequality via n-Harmonic Transplantation

The Moser-Trudinger embedding has been generalized in [Adimurthi A.; Sandeep K., A singular Moser-Trudinger embedding and its applications, \textit{NoDEA Nonlinear Differential Equations Appl.}, 13 (2007), no. 5-6, 585--603] to the following weighted version: if $Ω\subset\mathbb{R}^n$ is bounded, $ω_{n-1}$ is the $\mathcal{H}^{n-1}$ measure of the unit sphere, then for $α>0$ and $β\in [0,n)$, $$ \sup_{u\in\mathcal{B}_1}\int_Ω\frac{e^{α|u|^{n/(n-1)}}}{|x|^β}\leq C \ \Leftrightarrow \ \fracα{α_n}+\fracβ{n}\leq1,\qquad $$ where $α_n=n\cnn$ and $\mathcal{B}_1 = \left\{ u \in W_0^{1, n}(Ω) \ | \ \int_Ω |\nabla u |^n \leq1 \right\}$. We prove that the supremum is attained on any domain $Ω$. The paper also fills in the gaps in the proof of [Lin K.C., Extremal functions for Moser's inequality, \textit{Trans. of. Am. Math. Soc.}, 384 (1996), 2663--2671], which deals with the case $β=0.$

math.AP↗
arXiv2017

On the best constant in {G}affney inequality

We discuss the value of the best constant in Gaffney inequality namely $$ \lVert \nabla ω\rVert_{L^{2}}^{2}\leq C\left( \lVert dω\rVert_{L^{2}}^{2}+\lVert δω\rVert_{L^{2}% }^{2}+\lVert ω\rVert_{L^{2}}^{2}\right) $$ when either $ν\wedgeω=0$ or $ν\,\lrcorner\,ω=0$ on $\partialΩ.$

math.FA↗
arXiv2017

On the isoperimetric problem with perimeter density r^p

In this paper the author studies the isoperimetric problem in $\re^n$ with perimeter density $|x|^p$ and volume density $1.$ We settle completely the case $n=2,$ completing a previous work by the author: we characterize the case of equality if $0\leq p\leq 1$ and deal with the case $-\infty<p<-1$ (with the additional assumption $0\inΩ$). In the case $n\geq 3$ we deal mainly with the case $-\infty<p<0,$ showing among others that the results in $2$ dimensions do not generalize for the range $-n+1<p<0.$

math.DG↗
arXiv2015

The singular Moser- Trudger Inequality on simply connected domains

In this paper the authors complete their study of the singular Moser-Trudinger embedding [G. Csato and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations, DOI 10.1007/s00526-015-0867-5], abbreviated [CR]. The proof in [CR] is however far too technical and complicated for simply connected domains. Here we give a much simpler and more self-contained proof using complex analysis, which also generalizes the corresponding proof given by Flucher for such domains. This should make [CR] more easily accessible.

math.AP↗