Doubling Properties of Caloric Functions

关于$1/2$-对称热容量的半可加性

On the semi-additivity of the $1/2$-symmetric caloric capacity

本文研究$1/2$-对称热容量，建立其在$\mathbb{R}^{n+1}$中的半可加性，并显式计算矩形容量以展示各向异性。

Blow-ups of caloric measure in time varying domains and applications to two-phase problems

We develop a method to study the structure of the common part of the boundaries of disjoint and possibly non-complementary time-varying domains in $\mathbb{R}^{n+1}$, $n \geq 2$, at the points of mutual absolute continuity of their respective caloric measures. Our set of techniques, which is based on parabolic tangent measures, allows us to tackle the following problems: 1) Let $Ω_1$ and $Ω_2$ be disjoint domains in $\mathbb{R}^{n+1}$, $n \geq 2$, which are quasi-regular for the heat equation and regular for the adjoint heat equation, and their complements satisfy a mild non-degeneracy hypothe…

The hyperbolic Yang--Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions

This is the second part in a four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the hyperbolic Yang--Mills equation in the $(4+1)$-dimensional space-time. This paper provides the key gauge-dependent analysis of the hyperbolic Yang--Mills equation. We consider topologically trivial solutions in the caloric gauge, which was defined in the first paper arXiv:1709.08599 using the Yang--Mills heat flow. In this gauge, we establish a strong form of local well-posedness, where the time of existence is bounded from below by the energy c…

On Some Quantitative Unique Continuation Properties of Fractional Schrödinger Equations: Doubling, Vanishing Order and Nodal Domain Estimates

In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schrödinger equations) on a compact, smooth Riemannian manifold, $(M,g)$, without boundary. Moreover, with only slight modifications these results generalize to equations with $C^1$ potentials. Here Carleman estimates are a key tool. These yield a quantitative three balls inequality which implies quantitative bulk and boundary doubling estimates and hence leads to the control of the maximal order of vanishing. Using the boundary…

On doubling inequalities for elliptic systems

We prove doubling inequalities for solutions of elliptic systems with an iterated Laplacian as diagonal principal part and for solutions of the Lame' system of isotropic linearized elasticity. These inequalities depend on global properties of the solutions.

关于一阶Hamilton-Jacobi方程中变量倍增技术

On the doubling of variables technique in first order Hamilton-Jacobi equations

本文研究了一阶Hamilton-Jacobi方程中变量倍增技术的应用，通过调节惩罚项将正则性假设转化为几何性质，并应用于Wasserstein空间中的方程。

抛物型均匀可修正性由有界热函数的Carleson测度估计刻画

Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions

本文证明抛物型均匀可修正性等价于热方程有界解的Carleson测度估计及热测度的日冕分解，在温和假设下建立了新的刻画。

Hölder-Zygmund Estimates for Degenerate Parabolic Systems

We consider energy solutions of the inhomogeneous parabolic $p$-Laplacien system $\partial_t u-\text{div}(|D u|^{p-2}D u)=-\text{div} g$. We show in the case $p\geq 2$ that if the right hand side $g$ is locally in $L^\infty(\text{BMO})$, then $u$ is locally in $L^\infty(\mathcal{C}^1)$, where $\mathcal{C}^1$ is the 1-Hölder--Zygmund space. This is the borderline case of the Calderón-Zygmund theorey. We provide local quantitative estimates. We also show that finer properties of $g$ are conserved by $D u$, e.g.\ Hölder continuity. Moreover, we prove a new decay for gradients of $p$-caloric solut…

Boundary estimates for a degenerate parabolic equation with partial Dirichlet boundary conditions

We study the boundary regularity properties and derive a priori pointwise supremum estimates of weak solutions and their derivatives in terms of suitable weighted $L^2$-norms for a class of degenerate parabolic equations that satisfy homogeneous Dirichlet boundary conditions on certain portions of the boundary. Such equations arise in population genetics in the study of models for the evolution of gene frequencies. Among the applications of our results is the description of the structure of the transition probabilities and of the hitting distributions of the underlying gene frequencies process…

抛物型非线性非局部方程的梯度估计

Gradient estimates for parabolic nonlinear nonlocal equations

本文建立了一类抛物型非线性非局部测度数据问题解的逐点梯度估计，并证明其解的一阶正则性及齐次方程解的Hölder连续空间梯度。

The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces

We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality, $1 < p< \infty$. We apply these key tools to establish a fine version of the Kellogg property, characterize finely continuous functions by means of quasicontinuous functions, and show that capacitary measures associated with Cheeger supersolutions are supported by the fine boundary of the set.

Extension and trace results for doubling metric measure spaces and their hyperbolic fillings

In this paper we study connections between Besov spaces of functions on a compact metric space $Z$, equipped with a doubling measure, and the Newton--Sobolev space of functions on a uniform domain $X_\varepsilon$. This uniform domain is obtained as a uniformization of a (Gromov) hyperbolic filling of $Z$. To do so, we construct a family of hyperbolic fillings in the style of the work of Bonk and Kleiner and the work of Bourdon and Pajot. Then for each parameter $β>0$ we construct a lift $μ_β$ of the doubling measure $ν$ on $Z$ to $X_\varepsilon$, and show that $μ_β$ is doubling and supports a …

紧致加倍度量测度空间中反射分数阶 $p$-Laplace 型非齐次方程 Dirichlet 边值问题的适定性

Well-posedness of Dirichlet boundary value problems for reflected fractional $p$-Laplace-type inhomogeneous equations in compact doubling metric measure spaces

研究紧致加倍度量测度空间中反射分数阶 p-Laplace 型非齐次方程的 Dirichlet 问题，证明了存在性、唯一性、比较与稳定性，并建立了内部正则性和 Kellogg 型性质。

Bounded geometry and $p$-harmonic functions under uniformization and hyperbolization

The uniformization and hyperbolization transformations formulated by Bonk, Heinonen and Koskela in \emph{"Uniformizing Gromov Hyperbolic Spaces"}, Astérisque {\bf 270} (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local $p$-Poincaré inequality, then the transformed mea…