We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators. In particular, we show that there exists a compactly supported nonnegative Sobolev function $u_0$ that attains the infimum (which will be a positive real number)
of the set [ left{ intint_{{u > 0}times{u>0}} frac{|u(x) - u(y)|^2}{|x - y|^{n + 2 sigma}}d x d y : u in mathring H^sigma(mathbb{R}^n), int_{mathbb{R}^n} u^2 = 1, |{u > 0 }| leq 1right}. ] Unlike the corresponding problem for the usual fractional Laplacian, where the domain of the integration is $mathbb{R}^n times mathbb{R}^n$, symmetrization techniques may not apply here. Our approach is instead based on the direct method and new a priori diameter estimates. We also present several remaining open questions concerning the regularity and shape of the minimizers, and the form of the Euler-Lagrange equations.
We prove that any positive solution of the Yamabe equation on an asymptotically flat $n$-dimensional manifold of flatness order at least $frac{n-2}{2}$ and $nle 24$ must converge at infinity either to a fundamental solution of the Laplace operator on
the Euclidean space or to a radial Fowler solution defined on the entire Euclidean space. The flatness order $frac{n-2}{2}$ is the minimal flatness order required to define ADM mass in general relativity; the dimension $24$ is the dividing dimension of the validity of compactness of solutions to the Yamabe problem. We also prove such alternatives for bounded solutions when $n>24$. We prove these results by establishing appropriate asymptotic behavior near an isolated singularity of solutions to the Yamabe equation when the metric has a flatness order of at least $frac{n-2}{2}$ at the singularity and $n<24$, also when $n>24$ and the solution grows no faster than the fundamental solution of the flat metric Laplacian at the singularity. These results extend earlier results of L. Caffarelli, B. Gidas and J. Spruck, also of N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, when the metric is conformally flat, and work of C.C. Chen and C. S. Lin when the scalar curvature is a non-constant function with appropriate flatness at the singular point, also work of F. Marques when the metric is not necessarily conformally flat but smooth, and the dimension of the manifold is three, four, or five, as well as recent similar results by the second and third authors in dimension six.
We study extinction profiles of solutions to fast diffusion equations with some initial data in the Marcinkiewicz space. The extinction profiles will be the singular solutions of their stationary equations.
We study a Sobolev critical fast diffusion equation in bounded domains with the Brezis-Nirenberg effect. We obtain extinction profiles of its positive solutions, and show that the convergence rates of the relative error in regular norms are at least
polynomial. Exponential decay rates are proved for generic domains. Our proof makes use of its regularity estimates, a curvature type evolution equation, as well as blow up analysis. Results for Sobolev subcritical fast diffusion equations are also obtained.
We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes.
Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity.
We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $sigma in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math. 187, no. 2,
395-506, 2012) for the scalar curvature flow on the standard unit sphere.
We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the cel
ebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup*inf type Harnack inequality of Schoen for integral equations.
We prove the existence of entire solutions of the Monge-Amp`ere equations with prescribed asymptotic behavior at infinity of the plane, which was left by Caffarelli-Li in 2003. The special difficulty of the problem in dimension two is due to the glob
al logarithmic term in the asymptotic expansion of solutions at infinity. Furthermore, we give a PDE proof of the characterization of the space of solutions of the Monge-Amp`ere equation $det abla^2 u=1$ with $kge 2$ singular points, which was established by Galvez-Martinez-Mira in 2005. We also obtain the existence in higher dimensional cases with general right hand sides.
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the scalar-flat conformal
class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) $nge 12$ and $partial M$ has a nonumbilic point; or (ii) $nge 10$, $partial M$ is umbilic and the Weyl tensor does not vanish at some boundary point.
We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains.
