arXiv subjects
Publications and source records attributed to Tianling Jin.
We prove optimal pointwise Schauder estimates in the spatial variables for solutions of linear parabolic integro-differential equations. Optimal Hölder estimates in space-time for those spatial derivatives are also obtained.
We study conformal metrics on $R^3$, i.e., metrics of the form $g_u=e^{2u}|dx|^2$, which have constant $Q$-curvature and finite volume. This is equivalent to studying the non-local equation $$ (-Δ)^\frac32 u = 2 e^{3u}$$ in $R^3$ $$V:=\int_{\mathbb{R}^3}e^{3u}dx<\infty,$$ where $V$ is the volume of $g_u$. Adapting a technique of A. Chang and W-X. Chen to the non-local framework, we show the existence of a large class of such metrics, particularly for $V\le 2π^2=|S^3|$. Inspired by previous works of C-S. Lin and L. Martinazzi, who treated the analogue cases in even dimensions, we classify such metrics based on their behavior at infinity.
In this paper, we examine the boundary $L^2$ term of the sharp Sobolev trace inequality $\|u\|_{L^{q}(\pa M)}^2\leq S \|\nabla_g u\|_{L^2(M)}^2 +A(M,g)\|u\|^2_{L^2(\pa M)}$ on Riemannian manifolds $(M,g)$ with boundaries $\pa M$, where $q=\frac{2(n-1)}{n-2}$, $S$ is the best constant and $A(M,g)$ is some positive constant depending only on $M$ and $g$. We obtain a sharp trace inequality involving the mean curvature in a remainder term, which would fail in general once the mean curvature is replaced by any smaller function.
This paper is a continuation of our earlier work "[T. Jin, Y.Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, to appear in J. Eur. Math. Soc.]", where compactness results were given on a fractional Nirenberg problem. We prove two existence results stated there. We also obtain a fractional Aubin inequality.
In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations $(-Δ)^σu=u^{\frac{n+2σ}{n-2σ}}$ with an isolated singularity, where $σ\in (0,1)$. We prove that all the solutions are asymptotically radially symmetric. When $σ=1$, these have been proved in \cite{CGS} by Caffarelli, Gidas and Spruck.
In this paper, we study existence, regularity, classification, and asymptotical behaviors of solutions of some Monge-Ampère equations with isolated and line singularities. We classify all solutions of $\det \nabla^2 u=1$ in $\R^n$ with one puncture point. This can be applied to characterize ellipsoids, in the same spirit of Serrin's overdetermined problem for the Laplace operator. In the case of having $k$ non-removable singular points for $k>1$, modulo affine equivalence the set of all generalized solutions can be identified as an explicit orbifold of finite dimension. We also establish existence of global solutions with general singular sets, regularity properties, and optimal estimates of the second order derivatives of generalized solutions near the singularity consisting of a point or a straight line. The geometric motivation comes from singular semi-flat Calabi-Yau metrics.
We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres $(\Sn, [g_{\Sn}])$, it converges to the standard sphere up to a Möbius diffeomorphism. This result allows us to obtain extinction profiles of solutions of some fractional porous medium equations. In the end, we use this fractional fast diffusion equation, together with its extinction profile and some estimates of its extinction time, to improve a Sobolev inequality via a quantitative estimate of the remainder term.
In this paper, we give a new proof of a celebrated theorem of Jörgens which states that every classical convex solution of \[ \det\nabla^2 u (x)=1\quad {in} \mathbb{R}^2 \] has to be a second order polynomial. Our arguments do not use complex analysis, and can be applied to establish such Liouville type theorems for solutions of a class of degenerate Monge-Ampère equations. We prove that every convex generalized (or Alexandrov) solution of \[ \det \nabla^2 u(x_1,x_2)=|x_1|^α \quad {in} \mathbb{R}^2, \] where $α>-1$, has to be \[ u(x_1,x_2)= \frac{a}{(α+2)(α+1)}|x_1|^{2+α}+\frac{a b^2}{2}x_1^2 +bx_1x_2+ \frac{1}{2a} x_2^2+\ell(x_1,x_2) \] for some constants $a>0$, $b$ and a linear function $\ell(x_1,x_2)$. This work is motivated by the Weyl problem with nonnegative Gauss curvature.
We establish some sharp weighted trace inequalities $W^{1,2}(ρ^{1-2σ}, M)\hookrightarrow L^{\frac{2n}{n-2σ}}(\pa M)$ on $n+1$ dimensional compact smooth manifolds with smooth boundaries, where $ρ$ is a defining function of $M$ and $σ\in (0,1)$. This is stimulated by some recent work on fractional (conformal) Laplacians and related problems in conformal geometry, and also motivated by a conjecture of Aubin.
We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem.
We construct a function $u \in W^{1,1}_{\mathrm{loc}} (B(0,1))$ which is a solution to $\Div (A \nabla u)=0$ in the sense of distributions, where $A$ is continuous and $u \not \in W^{1,p}_{\mathrm{loc}} (B(0,1))$ for $p > 1$. We also give a function $u \in W^{1,1}_{\mathrm{loc}} (B(0,1))$ such that $u \in W^{1,p}_{\mathrm{loc}}(B(0,1))$ for every $p < \infty$, $u$ satisfies $\Div (A \nabla u)=0$ with $A$ continuous but $u \not \in W^{1, \infty}_{\mathrm{loc}}(B(0,1))$.