Bubbling and Regularity Issues in Geometric Non-linear Analysis
Bubbling and Regularity Issues in Geometric Non-linear Analysis
复制标题
几何非线性分析中的冒泡和规律性问题
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
T. Rivière
中科院分区：
文献类型：
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作者：
T. Rivière
在物理和几何中出现的许多椭圆型和抛物型变分问题（金兹堡-朗道方程、调和映射、杨-米尔斯场、ω -瞬子、Yamabe方程、一般的几何流……）都具有一个临界维度，其中不变性群（相似、共形群）起作用。在所有这些不同的情况下，这个共同的特征产生了相同的非线性效果。人们观察到在空间中几乎线性的状态和主要非线性状态之间的严格分裂，它有两个主要特征：它需要量子化的能量，并且沿着特殊几何兴趣的可整流对象（测地线，最小曲面，j全纯曲线，特殊拉格朗日流形，平均曲率流……）产生。
Numerous elliptic and parabolic variational problems arising in physics and geometry (Ginzburg-Landau equations, harmonic maps, Yang-Mills fields, Omega-instantons, Yamabe equations, geometric flows in general...) possess a critical dimension in which an invariance group (similitudes, conformal groups) acts. This common feature generates, in all these different situations, the same non-linear effect. One observes a strict splitting in space between an almost linear regime and a dominantly non-linear regime which has two major characteristics : it requires a quantized amount of energy and arises along rectifiable objects of special geometric interest (geodesics, minimal surfaces, J-holomorphic curves, special lagrangian manifolds, mean-curvature flows...).