带扰动的齐次发展方程的正则化效应
Regularizing effect of homogeneous evolution equations with perturbation
摘要：
自Aronson & Bénilan [C. R. Acad. Sci. Paris Sér.，1979]和Bénilan & Crandall [Johns Hopkins Univ. Press，1981]，众所周知，由非线性但齐次算子控制的一阶演化问题承认光滑效应，即每个相应的温和解在每个正时刻都是Lipschitz连续的。此外，如果基础巴拿赫空间具有Radon-Nikodrom性质，则这些温和解是a.e.可微，时间导数满足全局和逐点边界。本文证明了当齐次算子被Lipschitz连续映射扰动时，这些结果仍然成立。更准确地说，我们建立了整体$L^1$ Aronson-Bénilan型估计和逐点Aronson-Bénilan型估计。我们应用我们的理论来获得整体$L^q$ - $L^{\infty}$ -估计的时间导数的Dirichlet-to-Neumann算子与$p$ -Laplace-Beltrami算子和低阶项的紧黎曼流形与Lipschitz边界。
英文摘要：
Since the pioneering works by Aronson & Bénilan [C. R. Acad. Sci. Paris Sér., 1979] and Bénilan & Crandall [Johns Hopkins Univ. Press, 1981], it is well-known that first-order evolution problems governed by a nonlinear but homogeneous operator admit the smoothing effect that every corresponding mild solution is Lipschitz continuous at every positive time. Moreover, if the underlying Banach space has the Radon-Nikodým property, then these mild solution is a.e. differentiable, and the time-derivative satisfies global and point-wise bounds. In this paper, we show that these results remain true if the homogeneous operator is perturbed by a Lipschitz continuous mapping. More precisely, we establish global $L^1$ Aronson-Bénilan type estimates and point-wise Aronson-Bénilan type estimates. We apply our theory to derive global $L^q$-$L^{\infty}$-estimates on the time-derivative of the perturbed diffusion problem governed by the Dirichlet-to-Neumann operator associated with the $p$-Laplace-Beltrami operator and lower-order terms on a compact Riemannian manifold with a Lipschitz boundary.
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