Sobolev and mean‐value inequalities on generalized submanifolds of Rn
Sobolev and mean‐value inequalities on generalized submanifolds of Rn
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DOI：
10.1002/cpa.3160260305
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发表时间：
1973-05
影响因子：
3
通讯作者：
J. H. Michael;L. M. Simon
中科院分区：
文献类型：
--
作者：
J. H. Michael;L. M. Simon
普通索博列夫不等式已为人所知多年，其在偏微分方程理论中的价值也是众所周知的。 1967年，Miranda在[4]中获得了最小图的Sobolev不等式。 Bombieri、De Giorgi 和 Miranda 使用这种新不等式的改进版本来导出最小表面方程解的梯度界限（参见 [2]）。 Bombieri 在 [I] 和 Michael 中给出了更简单的不等式证明（参见 [7] 的附录）。在本文的定理2.1中，将建立一般的Sobolev不等式。这种一般不等式是在所谓的广义流形上获得的。作为结果的特殊情况，我们推导出普通的 Sobolev 不等式、平均曲率方程弱解图上的 Sobolev 不等式以及 R"（任意余维）的任意 C2 子流形上的 Sobolev 不等式。在 R" 的 C2 子流形的情况下，定理 2.1 中导出的类型的不等式首先在 [7] 中获得，使用完全不同的 （以及更长的）争论。事实上，[7] 中建立的不等式比此处获得的不等式具有更粗糙的性质。
The ordinary Sobolev inequality has been known for many years and its value in the theory of partial differential equations is well known. In 1967, Miranda in [4] obtained a Sobolev inequality for minimal graphs. A refined version of this new inequality was used by Bombieri, De Giorgi and Miranda to derive gradient bounds for solutions to the minimal surface equation (see [2]). A simpler proof of the inequality was given by Bombieri in [I] and by Michael (see the appendix of [7]). In Theorem 2.1 of the present paper a general Sobolev inequality will be established. This general inequality is obtained on what might be termed a generalized manifold. As special cases of the result we derive the ordinary Sobolev inequality, a Sobolev inequality on graphs of weak solutions to the mean curvature equation, and a Sobolev inequality on arbitrary C2 submanifolds of R"(of arbitrary co-dimension). In the case of C2 submanifolds of R" an inequality of the type derived in Theorem 2.1 was first obtained in [7], using a completely different (and much longer) argument. In fact, the inequality established in [7] was of a coarser nature than the one obtained here.