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 derive local estimates of positive solutions to the conformal $Q$-curvature equation $$ (-Delta)^m u = K(x) u^{frac{n+2m}{n-2m}} ~~~~~~ in ~ Omega backslash Lambda $$ near their singular set $Lambda$, where $Omega subset mathbb{R}^n$ is an open se
t, $K(x)$ is a positive continuous function on $Omega$, $Lambda$ is a closed subset of $mathbb{R}^n$, $2 leq m < n/2$ and $m$ is an integer. Under certain flatness conditions at critical points of $K$ on $Lambda$, we prove that $u(x) leq C [{dist}(x, Lambda)]^{-(n-2m)/2}$ when the upper Minkowski dimension of $Lambda$ is less than $(n-2m)/2$.
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.
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $partial M$, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly la
rger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) $9le nle 11$ and $partial M$ has a nonumbilic point; or (ii) $7le nle 9$, $partial M$ is umbilic and the Weyl tensor does not vanish identically on the boundary. This is a continuation of the work cite{Jin-Xiong} by the second named author and Xiong.
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 obtain Dini and Schauder type estimates for concave fully nonlinear nonlocal parabolic equations of order $sigmain (0,2)$ with rough and non-symmetric kernels, and drift terms. We also study such linear equations with only measurable coefficients
in the time variable, and obtain Dini type estimates in the spacial variable. This is a continuation of the work [10, 11] by the first and last authors.
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.
