Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSymmetry for positive critical points of Caffarelli-Kohn-Nirenberg inequalities
91
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider positive critical points of Caffarelli-Kohn-Nirenberg inequalities and prove a Liouville type result which allows us to give a complete classification of the solutions in a certain range of parameters, providing a symmetry result for positive solutions. The governing operator is a weighted $p$-Laplace operator, which we consider for a general $p in (1,d)$. For $p=2$, the symmetry breaking region for extremals of Caffarelli-Kohn-Nirenberg inequalities was completely characterized in [J. Dolbeault, M. Esteban, M. Loss, Invent. Math. 44 (2016)]. Our results extend this result to a general $p$ and are optimal in some cases.
rate research
Read More
In this paper, we consider the Caffarelli-Kohn-Nirenberg (CKN) inequality: begin{eqnarray*} bigg(int_{{mathbb R}^N}|x|^{-b(p+1)}|u|^{p+1}dxbigg)^{frac{2}{p+1}}leq C_{a,b,N}int_{{mathbb R}^N}|x|^{-2a}| abla u|^2dx end{eqnarray*} where $Ngeq3$, $a<frac{N-2}{2}$, $aleq bleq a+1$ and $p=frac{N+2(1+a-b)}{N-2(1+a-b)}$. It is well-known that up to dilations $tau^{frac{N-2}{2}-a}u(tau x)$ and scalar multiplications $Cu(x)$, the CKN inequality has a unique extremal function $W(x)$ which is positive and radially symmetric in the parameter region $b_{FS}(a)leq b<a+1$ with $a<0$ and $aleq b<a+1$ with $ageq0$ and $a+b>0$, where $b_{FS}(a)$ is the Felli-Schneider curve. We prove that in the above parameter region the following stabilities hold: begin{enumerate} item[$(1)$] quad stability of CKN inequality in the functional inequality setting $$dist_{D^{1,2}_{a}}^2(u, mathcal{Z})lesssim|u|^2_{D^{1,2}_a({mathbb R}^N)}-C_{a,b,N}^{-1}|u|^2_{L^{p+1}(|x|^{-b(p+1)},{mathbb R}^N)}$$ where $mathcal{Z}= { c W_taumid cinbbrbackslash{0}, tau>0}$; item[$(2)$]quad stability of CKN inequality in the critical point setting (in the class of nonnegative functions) begin{eqnarray*} dist_{D_a^{1,2}}(u, mathcal{Z}_0^ u)lesssimleft{aligned &Gamma(u),quad p>2text{ or } u=1, &Gamma(u)|logGamma(u)|^{frac12},quad p=2text{ and } ugeq2, &Gamma(u)^{frac{p}{2}},quad 1<p<2text{ and } ugeq2, endalignedright. end{eqnarray*} where $Gamma (u)=|div(|x|^{-a} abla u)+|x|^{-b(p+1)}|u|^{p-1}u|_{(D^{1,2}_a)^{}}$ and $$mathcal{Z}_0^ u={(W_{tau_1},W_{tau_2},cdots,W_{tau_ u})mid tau_i>0}.$$
We establish improv
We study the problem of prescribing $sigma_k$-curvature for a conformal metric on the standard sphere $mathbb{S}^n$ with $2 leq k < n/2$ and $n geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function $K$ near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of $K$ of flatness order $beta in [2,n)$. We prove among other things the following statements. (1) When $beta>n-2k$, the solution set is compact, has a nonzero total degree counting and is therefore non-empty. (2) When $ beta = n-2k$, there is an explicit positive constant $C(K)$ associated with $K$. If $C(K)>1$, the solution set is compact with a nonzero total degree counting and is therefore non-empty. If $C(K)<1$, the solution set is compact but the total degree counting is $0$, and the solution set is sometimes empty and sometimes non-empty. (3) When $frac{2}{n-2k}le beta < n-2k$, the solution set is compact, but the total degree counting is zero, and the solution set is sometimes empty and sometimes non-empty. (4) When $beta < frac{n-2k}{2}$, there exists $K$ for which there exists a blow-up sequence of solutions with unbounded energy. In this same range of $beta$, there exists also some $K$ for which the solution set is empty.
We establish theorems on the existence and compactness of solutions to the $sigma_2$-Nirenberg problem on the standard sphere $mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Mobius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Mobius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the $sigma_2$-Nirenberg problem, and a B^ocher type theorem for the associated fully nonlinear Mobius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the $sigma_2$-Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove $L^infty$ a priori estimates for solutions of the $sigma_2$-Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the $sigma_2$-Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators.
The objective of this paper is two-fold. First, we establish new sharp quantitative estimates for Faber-Krahn inequalities on simply connected space forms. In these spaces, geodesic balls uniquely minimize the first eigenvalue of the Dirichlet Laplacian among all sets of a fixed volume. We prove that for any open set $Omega$, [ lambda_1(Omega) - lambda_1(B) gtrsim |Omega Delta B|^2 + int |u_{Omega} - u_B|^2, ] where $B$ denotes the nearest geodesic ball to $Omega$ with $|B|=|Omega|$ and $u_Omega$ denotes the first eigenfunction with suitable normalization. On Euclidean space, this extends a result of Brasco-De Phillipis-Velichkov; the eigenfunction control largely builds upon on new regularity results for minimizers of critically perturbed Alt-Cafarelli type functionals in our companion paper. On the round sphere and hyperbolic space, the present results are the first sharp quantitative results with respect to any distance; here the local portion of the analysis is based on new implicit spectral analysis techniques. Second, we apply these sharp quantitative Faber-Krahn inequalities in order to establish a quantitative form of the Alt-Caffarelli-Friedman (ACF) monotonicity formula. A powerful tool in the study of free boundary problems, the ACF monotonicity formula is nonincreasing with respect to its scaling parameter for any pair of admissible subharmonic functions, and is constant if and only if the pair comprises two linear functions truncated to complementary half planes. We show that the energy drop in the ACF monotonicity formula from one scale to the next controls how close a pair of admissible functions is from a pair of complementary half-plane solutions. In particular, when the square root of the energy drop summed over all scales is small, our result implies the existence of tangents (unique blowups) of these functions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
