POTENTIAL ESTIMATES IN PARABOLIC OBSTACLE PROBLEMS
POTENTIAL ESTIMATES IN PARABOLIC OBSTACLE PROBLEMS
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DOI：
10.5186/aasfm.2012.3730
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发表时间：
2012-08
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影响因子：
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通讯作者：
Christoph Scheven
中科院分区：
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作者：
Christoph Scheven
对于具有二次增长的抛物型障碍问题，我们根据给定数据的势给出了解及其梯度的逐点估计。作为应用，如果数据满足相应的洛伦兹空间正则性，我们推导出洛伦兹空间估计。此外，我们还讨论了正则性理论中的一种边界情况，梯度的有界性和连续性问题以及解本身的问题。在目前的工作中，我们建立了抛物线障碍问题解的点估计。障碍问题在各种应用中发挥着突出的作用，例如在力学或控制理论中，参见（4,22），但在其他数学领域，如势理论中，障碍问题的解被证明是超解的近似值（16,18,23）。在这里，我们处理与这类方程相关的障碍问题
For parabolic obstacle problems with quadratic growth, we give pointwise estimates both for the solutions and their gradients in terms of potentials of the given data. As applications, we derive Lorentz space estimates if the data satisfies the corresponding Lorentz space regularity. Moreover, we discuss a borderline case in the regularity theory, the question of boundedness and continuity of the gradients as well as of the solutions itself. In the present work, we establish pointwise estimates by potentials for solutions to parabolic obstacle problems. Obstacle problems play a prominent role in various applications, for example in mechanics or control theory, cf. (4, 22), but also in other fields of mathematics such as potential theory, where solutions to obstacle problems prove useful as approximations of super-solutions (16, 18, 23). Here, we treat obstacle problems that are related to equations of the type