局部Burkholder泛函、拟凸性与几何函数论

对称矩阵上Burkholder函数的拟凸性

Quasiconvexity of the Burkholder function on symmetric matrices

该研究运用Morse理论证明Burkholder函数及Székelyhidi引入的分离函数在对称矩阵上具有拟凸性，由此得到Beurling–Ahlfors变换作用于实值函数的若干尖锐不等式。

Automatic quasiconvexity of homogeneous isotropic rank-one convex integrands

We consider the class of non-negative rank-one convex isotropic integrands on $\mathbb{R}^{n\times n}$ which are also positively $p$-homogeneous. If $p \leq n = 2$ we prove, conditional on the quasiconvexity of the Burkholder integrand, that the integrands in this class are quasiconvex at conformal matrices. If $p \geq n = 2$, we show that the positive part of the Burkholder integrand is polyconvex. In general, for $p \geq n$, we prove that the integrands in the above class are polyconvex at conformal matrices. Several examples imply that our results are all nearly optimal.

Towards a Theory of Multi-Parameter Geometrical Variational Problems: Fibre Bundles, Differential Forms and Riemannian Quasiconvexity

We are concerned with the relaxation and existence theories of a general class of geometrical minimisation problems, with action integrals defined via differential forms over fibre bundles. We find natural algebraic and analytic conditions which give rise to a relaxation theory. Moreover, we propose the notion of ``Riemannian quasiconvexity'' for cost functions whose variables are differential forms on Riemannian manifolds, which extends the classical quasiconvexity condition in the Euclidean settings. The existence of minimisers under the Riemannian quasiconvexity condition has been establish…

Characterizations of symmetric polyconvexity

Symmetric quasiconvexity plays a key role for energy minimization in geometrically linear elasticity theory. Due to the complexity of this notion, a common approach is to retreat to necessary and sufficient conditions that are easier to handle. This article focuses on symmetric polyconvexity, which is a sufficient condition. We prove a new characterization of symmetric polyconvex functions in the two- and three-dimensional setting, and use it to investigate relevant subclasses like symmetric polyaffine functions and symmetric polyconvex quadratic forms. In particular, we provide an example of …

Local and semilocal Poincaré inequalities on metric spaces

We consider several local versions of the doubling condition and Poincaré inequalities on metric spaces. Our first result is that in proper connected spaces, the weakest local assumptions self-improve to semilocal ones, i.e. holding within every ball. We then study various geometrical and analytical consequences of such local assumptions, such as local quasiconvexity, self-improvement of Poincaré inequalities, existence of Lebesgue points, density of Lipschitz functions and quasicontinuity of Sobolev functions. It turns out that local versions of these properties hold under local assumptions, …

度量空间上有界p-调和函数与拟调和函数在局部假设下的刘维尔定理及可去集

Liouville theorems and removable sets for bounded $p$-harmonic and quasiharmonic functions on metric spaces under local assumptions

本文针对满足局部p-庞加莱不等式等条件的度量空间，刻画了有界p-调和函数与拟调和函数的可去紧集，揭示可去性与刘维尔型定理等价，确定了相关关键性质并给出刘维尔定理的初等证明，结果适用于流形等空间。