非齐次奇异抛物$p$-Laplace方程解的局部行为
Local behavior of solutions to inhomogeneous singular parabolic $p$-Laplace equations
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中文总结 AI 辅助
本文通过时间重标度与$L^\infty$比较估计，将非齐次奇异抛物$p$-Laplace方程解的正则性证明归结为空间水平集测度的椭圆型衰减估计，从而克服了内蕴尺度方法中的主要困难。
中文摘要 AI 辅助
已知通过内蕴尺度方法证明$p$-Laplace型拟线性奇异抛物方程解的Hölder正则性的一个主要困难在于建立水平集时空测度的衰减估计。本文考虑一个带非负时间无关强迫项和Dirichlet边界数据的非齐次奇异抛物$p$-Laplace方程。通过结合时间重标度论证与$L^\infty$比较估计，我们推导出在任意固定正时间之后解关于时间的Lipschitz正则性，从而将问题归结为关于空间水平集测度的椭圆型衰减估计，该估计在时间上本质一致。这一归结使我们能够解决上述障碍。
英文摘要
It is known that a major difficulty in proving Hölder regularity for solutions to quasilinear singular parabolic equations of $p$-Laplace type via the method of intrinsic scaling is to establish the decay estimate of the space-time measure of level sets. In this paper, we consider an inhomogeneous singular parabolic $p$-Laplace equation with nonnegative time-independent forcing and Dirichlet data. By combining a time-rescaling argument with $L^\infty$ comparison estimates, we derive Lipschitz regularity in time after any fixed positive elapsed time, which reduces the problem to an elliptic-type decay estimate for the spatial measure of level sets that is essentially uniform in time. This reduction enables us to address the above obstacle.
发表机构
- Hebei Normal University（河北师范大学）
- Beijing Normal University（北京师范大学）
- University of Science and Technology Beijing（北京科技大学）
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