Loss of compactness phenomena for Dirichlet energy

在整个数学史上，几何和分析之间的联系使人们对这两个领域有了深入的了解，并导致了许多新的发展，从而解决了重大问题，包括高原问题和庞加莱猜想。这些结果之后，研究这些领域之间的相互作用仍然活跃，有许多开放的problems.A几何分析的一个重要领域是研究变分法中的几何问题，因为许多几何对象可以被定义为适当定义的泛函的临界点。自然发生泛函的两个例子是流形之间映射的Dirichlet能量和曲面的面积，它们的临界点分别是调和映射和极小曲面。变分法中的一个问题是证明临界点的存在性。两个重要的方法来做到这一点是进化的梯度流和直接使用最小化序列。在这两种情况下，人们的目标是保持拓扑性质;然而，这需要一个成功的紧性理论，可以证明一个主要的障碍。在Dirichlet能量的背景下，梯度流是由Eells-Sampson在20世纪60年代引入的，他证明了负弯曲目标的全局定义光滑解的存在性。然而，在20世纪70年代，Eells-Wood的工作表明，一般情况下，人们必须期待奇点的形成。随后，这个问题的损失的紧凑性激发了研究什么类型的收敛可以预期举行。在1985年，斯特鲁威证明了调和映射流的解是光滑的，最多可以远离由能量集中引起的200多个奇点。对几乎调和映射行为的进一步研究已经产生了一个紧性理论，证明了远离气泡形成的许多点的强收敛性，并建立了“气泡树收敛”，其中能量的损失由气泡来解释。尽管有这些结果，但关于奇点附近流动的最佳描述和这种子序列的最佳紧性理论仍然有许多悬而未决的问题，例如爆破尺度与切空间衰减之间的定量关系。在过去的几年里，在回答几乎调和映射流的精细结构问题上取得了进展，例如Del-Pino等人关于奇点的位置和形成率的工作。到目前为止，致密性损失现象的精细分析仍然是一个具有许多挑战性的开放问题。该项目旨在促进更好地了解这些问题，特别是能源景观的精细结构及其在临界点附近的影响和对梯度流动态的影响。为了研究这一点，我们将首先考虑高度对称的设置，其中奇点是已知的发生，旨在获得最佳的结果，为这个特定的设置和使用这种情况下，测试方法开发的研究更一般的设置。为了做到这一点，我们将结合使用现代数学的技术，如Teichmuller理论和Lojasiewicz估计，以及使用Rupflin最近的作品作为灵感的明确例子。此外，我们希望开发新的技术，将适用于其他非线性偏微分方程，因此，有兴趣的怀尔德领域的mathematics.The核心部分，这个研究项目福尔斯属于EPSRC数学分析研究领域，因为我们的目标是使用和开发技术，从功能分析以及椭圆和抛物线偏微分方程理论。也有多个连接到EPSRC几何和拓扑研究领域，因为紧凑性现象的损失与拓扑现象和几何结构的退化密切相关，并且我们打算研究的问题受到几何的启发。

Throughout the history of mathematics, the link between geometry and analysis has given insight into both fields and led to many new developments, in turn solving major problems, including the Plateau problem and the Poincaré conjecture. Following these results, research into the interplay between these fields remains active, with many open problems.An important area of geometric analysis is the study of geometric problems in the calculus of variations, as many geometric objects can be defined as critical points of suitably defined functionals. Two examples of naturally occurring functionals are the Dirichlet energy of maps between manifolds and the area of surfaces, whose critical points are harmonic maps and minimal surfaces, respectively. One problem in the calculus of variations is proving the existence of critical points. Two important methods to do this are evolution by gradient flow and directly using minimising sequences. In both cases, one aims to preserve topological properties; however, this requires a successful compactness theory that can prove a major hurdle. In the context of Dirichlet energy, the gradient flow was introduced by Eells-Sampson in the 1960s, who proved the existence of globally defined smooth solutions for negatively curved targets. However, the work of Eells-Wood in the 1970s showed that, in general, one must expect the formation of singularities.Subsequently, this question of loss of compactness has inspired research into what type of convergence can be expected to hold. In 1985, Struwe showed that solutions to the harmonic map flow were smooth away from at most finitely many singularities caused by a concentration of energy. Further research into the behaviour of almost harmonic maps has yielded a compactness theory proving strong convergence away from finitely many points where bubbles form and establishing "bubble tree convergence," where the loss of energy is accounted for by the bubbles.Despite these results, there are still many open problems concerning the optimal description of the flow near singularities and the optimal compactness theory for such a subsequence, such as the quantitative relationship between the blow-up scale and the decay of the tangent space. In the past few years, there has been progress on answering questions on the fine structure of almost harmonic map flows, such as the work of Del-Pino et al. on the location and rate of formation of singularities. To date, the fine analysis of loss of compactness phenomena remains an open problem with many challenges. This project aims to contribute to a better understanding of these questions, in particular the fine structure of the energy landscape and its impact near critical points and on the dynamics of gradient flows. In order to investigate this, we will firstly consider highly symmetric settings, where singularities are known to occur, aiming to obtain optimal results for this particular setting and use this case to test methods developed to study more general settings. To do so, we will look to use a combination of techniques from modern mathematics such as Teichmuller theory and Lojasiewicz estimates, together with explicit examples using the recent works by Rupflin as inspiration. Furthermore, we expect to develop novel techniques that would be applicable to other non-linear PDEs and, hence, be of interest to the wilder field of mathematics.The core part of this research project falls within the EPSRC Mathematical Analysis research area, as we aim to use and develop techniques from functional analysis as well as elliptic and parabolic PDE theory. There are also multiple connections to the EPSRC Geometry and Topology research area, as loss of compactness phenomena are closely connected to topological phenomena and degeneracy of geometric structure, and as the problem we intend to study is inspired by geometry.