论文标题
混合锯齿蛋白型问题中的定量对称性，以实现约束的扭转刚度
Quantitative symmetry in a mixed Serrin-type problem for a constrained torsional rigidity
论文作者
论文摘要
我们考虑在半球$ b _+$中包含的域$ω$中的混合边界值问题，并在其边界的部分$ \ bar {t} $中，与$ \ partial b _+$的弯曲部分共同。问题与某种约束的扭转刚度有关。在这种情况下，相关的解决方案$ u $满足$ t $的steklov条件，并且在$σ= \partialΩ\ setMinus \ bar {t} \ subset b _+$上的均质dirichlet条件。我们提供了一个整体身份，该身份将$ω$的第二个衍生物（对称函数）与$σ$上的正常衍生$u_ν$相关联。这种身份的第一个重要结果是在$σ$上$u_ν$的相当弱的过度确定整体条件下的刚度结果：实际上，事实证明，$σ$必须是符合$ t $正交的球形上限。该结果返回了J. Guo和C. Xia在更强的条件下获得的结果，即$u_ν$的值在$σ$上是恒定的。第二个重要的结果是一组稳定性界限，如果$u_ν$偏离了norm $ l^1（σ）$的常数，则定量衡量$σ$与球形盖的一致性相去甚远。
We consider a mixed boundary value problem in a domain $Ω$ contained in a half-ball $B_+$ and having a portion $\bar{T}$ of its boundary in common with the curved part of $\partial B_+$. The problem has to do with some sort of constrained torsional rigidity. In this situation, the relevant solution $u$ satisfies a Steklov condition on $T$ and a homogeneous Dirichlet condition on $Σ= \partialΩ\setminus \bar{T} \subset B_+$. We provide an integral identity that relates (a symmetric function of) the second derivatives of the solution in $Ω$ to its normal derivative $u_ν$ on $Σ$. A first significant consequence of this identity is a rigidity result under a quite weak overdetermining integral condition for $u_ν$ on $Σ$: in fact, it turns out that $Σ$ must be a spherical cap that meets $T$ orthogonally. This result returns the one obtained by J. Guo and C. Xia under the stronger pointwise condition that the values of $u_ν$ be constant on $Σ$. A second important consequence is a set of stability bounds, which quantitatively measure how $Σ$ is far uniformly from being a spherical cap, if $u_ν$ deviates from a constant in the norm $L^1(Σ)$.
