A simple proof of the regularity theorem for the variational inequality of the obstacle problem
A simple proof of the regularity theorem for the variational inequality of the obstacle problem
复制标题
障碍问题变分不等式正则定理的简单证明
DOI：
10.1016/0362-546x(86)90119-7
复制
发表时间：
1986
影响因子：
1.4
通讯作者：
Björn Gustafsson
中科院分区：
文献类型：
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作者：
Björn Gustafsson
S~2上Sobolev空间(如文[8])的E@*‘(Q)等，以及Ni(Q)和HB(Q)之间的标准对偶的(-，a>；)。在不作其他说明的情况下，函数空间中元素之间的开集上的等式和(非严格)不等式被解释为分布意义上的。变分不等式(L)-(2)被称为障碍问题的最简单形式的变分不等式([6，第二章，第61节)，并且它的唯一解的存在是众所周知的事实(16，第二章，定理6.21)。(L)-(2)解的正则性也是众所周知的，并且存在许多不同的证明。第一个证明(通过“惩罚”)出现在(7，定理3)中。11)。另见[6，第四章，第21节。另外两个证明在13，Corollaire II.31和[2，定理1.11]中。估计(3)在这些论文中没有明确提到，但它或多或少地隐含在其中(它来自于[7，推论3.11，在任何情况下都是众所周知的)。
E@*‘(Q) etc. for Sobolev spaces on S2 (as in [8]) and (-, a> for the standard duality pairing between Ni (Q) and Hb (Q). When nothing else is stated, equalities and (nonstrict) inequalities on open sets between elements in function spaces are to be interpreted in the sense of distributions.The variational inequality (l)-(2) is known as the variational inequality of the obstacle problem in its simplest form ([6, Chapter II, Section 61) and the existence of a unique solution of it is a widely known fact (16, Chapter II, theorem 6.21). The regularity of the solution of (l)-(2) is also well-known and a number of different proofs of it exist. The first proof (by “penalization”) appeared in (7, theorem3. 11). See also [6, Chapter IV, Section 21. Two other proofs are in 13, corollaire II. 31 and [2, theoreme 1.11. The estimate (3) is not mentioned explicitly in these papers but it is more or less implicit in them (it follows eg from [7, corollary 3.11 and is in any case well-known).