Symmetry in an overdetermined fourth order elliptic boundary value problem
Symmetry in an overdetermined fourth order elliptic boundary value problem
复制标题
超定四阶椭圆边值问题中的对称性
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发表时间：
1986
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通讯作者：
A. Bennett
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文献类型：
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作者：
A. Bennett
设$ $为$mathbb{R}^N $中的有界定义域，其边值问题有经典解：$ $ (u) = - 1$ $在$ $；={{偏u} /{偏n}}$ on $偏Omega；u = c$（常数）在$偏上。我们证明$ $必须是一个开放的球并且u必须是围绕$ $的中心径向对称的。这个结果类似于Serrin (Arch。老鼠。动力机械。分析的。， 43(1971)，第304-318页)和Weinberger (Arch。老鼠。动力机械。分析的。$Delta u = - 1$在$Omega,u = 0$和${{偏u} /{偏n}} = c$在$partial Omega $上的问题。我们的结果是由四阶椭圆方程的一个极大值原理和格林定理的几个应用得到的。然后，我们利用积分恒等式得到了开球的两个特征——第一个依赖于我们的结果，第二个依赖于Serrin和Weinberger的结果。
Let $Omega $ be a bounded domain in $mathbb{R}^N $ for which the following boundary value problem has a classical solution: $Delta (Delta u) = - 1$ in $Omega ; = {{partial u} / {partial n}}$ on $partial Omega ;Delta u = c$ (constant) on $partial Omega $. We show that $Omega $ must be an open ball and that u must be radially symmetric about the center of $Omega $. This result is analogous to that of Serrin (Arch. Rat. Mech. Anal., 43 (1971), pp. 304–318) and Weinberger (Arch. Rat. Mech. Anal., 43 (1971), pp. 319–320) for the problem $Delta u = - 1$ in $Omega ,u = 0$ and ${{partial u} / {partial n}} = c$ on $partial Omega $. Our result is obtained from a maximum principle for fourth order elliptic equations and several applications of Green’s theorem. We then obtain two characterizations of open balls by means of integral identities–the first depends on our result and the second on that of Serrin and Weinberger.