Searcharxiv⌕ Search

arXiv subjects

Tianling Jin

Publications and source records attributed to Tianling Jin.

At least 37 records · Page 2Linked to original sources
arXiv2022

Optimal regularity and fine asymptotics for the porous medium equation in bounded domains

We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time $T^*$. More precisely, we show that solutions are $C^{2,α}(\overlineΩ)$ in space, with $α=\frac1m$, and $C^\infty$ in time (uniformly in $x\in \overlineΩ$), for $t>T^*$. Furthermore, this allows us to refine the asymptotics of solutions for large times, improving the best known results so far in two ways: we establish a faster rate of convergence $O(t^{-1-γ})$, and we prove that the convergence holds in the $C^{1,α}(\overlineΩ)$ topology.

math.AP↗
arXiv2021

On a Rayleigh-Faber-Krahn inequality for the regional fractional Laplacian

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 \[ \left\{ \int\int_{\{u > 0\}\times\{u>0\}} \frac{|u(x) - u(y)|^2}{|x - y|^{n + 2 σ}}d x d y : u \in \mathring H^σ(\mathbb{R}^n), \int_{\mathbb{R}^n} u^2 = 1, |\{u > 0 \}| \leq 1\right\}. \] 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.

math.AP↗
arXiv2021

Local estimates for conformal $Q$-curvature equations

We derive local estimates of positive solutions to the conformal $Q$-curvature equation $$ (-Δ)^m u = K(x) u^{\frac{n+2m}{n-2m}} ~~~~~~ in ~ Ω\backslash Λ$$ near their singular set $Λ$, where $Ω\subset \mathbb{R}^n$ is an open set, $K(x)$ is a positive continuous function on $Ω$, $Λ$ 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 $Λ$, we prove that $u(x) \leq C [{dist}(x, Λ)]^{-(n-2m)/2}$ when the upper Minkowski dimension of $Λ$ is less than $(n-2m)/2$.

math.AP↗
arXiv2021

Optimal boundary regularity for fast diffusion equations in bounded domains

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.

math.AP↗
arXiv2019

An existence theorem on the isoperimetric ratio over scalar-flat conformal classes

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 larger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) $9\le n\le 11$ and $\partial M$ has a nonumbilic point; or (ii) $7\le n\le 9$, $\partial M$ 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.

math.DG↗
arXiv2019

Dini and Schauder estimates for nonlocal fully nonlinear parabolic equations with drifts

We obtain Dini and Schauder type estimates for concave fully nonlinear nonlocal parabolic equations of order $σ\in (0,2)$ 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.

math.AP↗
arXiv2019

Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities

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 celebrated 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.

math.AP↗
arXiv2017

On the isoperimetric quotient over scalar-flat conformal classes

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) $n\ge 12$ and $\partial M$ has a nonumbilic point; or (ii) $n\ge 10$, $\partial M$ is umbilic and the Weyl tensor does not vanish at some boundary point.

math.AP↗
arXiv2017

Minimizers for the fractional Sobolev inequality on domains

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.

math.AP↗
arXiv2016

Hölder gradient estimates for a class of singular or degenerate parabolic equations

We prove interior Hölder estimate for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation $$ u_t=|\nabla u|^κ\mbox{div} (|\nabla u|^{p-2}\nabla u), $$ where $p\in (1,\infty)$ and $κ\in (1-p,\infty).$ This includes the from $L^\infty$ to $C^{1,α}$ regularity for parabolic $p$-Laplacian equations in both divergence form with $κ=0$, and non-divergence form with $κ=2-p$. This work is a continuation of a paper by the last two authors \cite{JS}.

math.AP↗
arXiv2016

Hölder gradient estimates for parabolic homogeneous p-Laplacian equations

We prove interior Hölder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous $p$-Laplacian equation \[ u_t=|\nabla u|^{2-p} \mbox{ div} (|\nabla u|^{p-2}\nabla u), \] where $1<p<\infty$. This equation arises from tug-of-war-like stochastic games with noise. It can also be considered as the parabolic $p$-Laplacian equation in non divergence form.

math.AP↗
arXiv2014

The Nirenberg problem and its generalizations: A unified approach

Making use of integral representations, we develop a unified approach to establish blow up profiles, compactness and existence of positive solutions of the conformally invariant equations $P_σ(v)= Kv^{\frac{n+2σ}{n-2σ}}$ on the standard unit sphere $\mathbb{S}^n$ for all $σ\in (0,n/2)$, where $P_σ$ is the intertwining operator of order $2σ$. Finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures. When $σ=1$, it is the prescribing scalar curvature problem or the Nirenberg problem, and when $σ=2$, it is the prescribing $Q$-curvature problem.

math.AP↗
arXiv2014

Schauder estimates for nonlocal fully nonlinear equations

In this paper, we establish pointwise Schauder estimates for solutions of nonlocal fully nonlinear elliptic equations by perturbative arguments. A key ingredient is a recursive Evans-Krylov theorem for nonlocal fully nonlinear translation invariant equations.

math.AP↗