اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn existence and nonexistence of isoperimetric inequality with differents monomial weights
81
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the monomial weight $x^{A}=vert x_{1}vert^{a_{1}}ldotsvert x_{N}vert^{a_{N}}$, where $a_{i}$ is a nonnegative real number for each $iin{1,ldots,N}$, and we establish the existence and nonexistence of isoperimetric inequalities with different monomial weights. We study positive minimizers of $int_{partialOmega}x^{A}mathcal{H}^{N-1}(x)$ among all smooth bounded sets $Omega$ in $mathbb{R}^{N}$ with fixed Lebesgue measure with monomial weight $int_{Omega}x^{B}dx$.
قيم البحث
اقرأ أيضاً
We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequali
ties through the ABP method, we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromovs proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of Figalli-Maggi-Pratelli and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.
We prove a counterpart of the log-convex density conjecture in the hyperbolic plane.
The main purpose of this paper is to establish the existence, nonexistence and symmetry of nontrivial solutions to the higher order Brezis-Nirenberg problems associated with the GJMS operators $P_k$ on bounded domains in the hyperbolic space $mathbb{
H}^n$ and as well as on the entire hyperbolic space $mathbb{H}^n$. Among other techniques, one of our main novelties is to use crucially the Helgason-Fourier analysis on hyperbolic spaces and the higher order Hardy-Sobolev-Mazya inequalities and careful study of delicate properties of Greens functions of $P_k-lambda$ on hyperbolic spaces which are of independent interests in dealing with such problems. Such Greens functions allow us to obtain the integral representations of solutions and thus to avoid using the maximum principle to establish the symmetry of solutions.
We consider a nonlinear Dirichlet problem driven by the $(p,q)$-Laplacian and with a reaction which is parametric and exhibits the combined effects of a singular term and of a superdiffusive one. We prove an existence and nonexistence result for posi
tive solutions depending on the value of the parameter $lambda in overset{circ}{mathbb{R}}_+=(0,+infty)$.
We consider the nonlocal analogue of the Fisher-KPP equation. We do not assume that the Borel-measure is absolutely continuous with respect to the Lebesgue measure. We gives a sufficient condition for existence of traveling waves, and a necessary condition for existence of periodic traveling waves.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
