论文标题
矢量规范在球形层中的不平等
Inequalities for the norms of vector functions in a spherical layer
论文作者
论文摘要
我们考虑矢量在域同构中的函数到由两次连续可区分的表面界定的球形层。对域施加了其他限制，该域允许使用简单的方法进行证明。在外部和内边界上，矢量的正常和切向成分分别为零。对于此类功能，通过其笛卡尔组件的平方梯度之和的积分从上方估算了平方矢量域的积分。最后一个积分是通过平方发散和转子的总和的积分来估计的。这些不等式允许定义两个规范，等于矢量的笛卡尔组件的总和作为空间的元素$ w_2^{（1）}（ω）$。还估计了平方矢量边界上的积分。在所有证明的不平等现象中，常数仅由域的形状决定，并且不取决于特定的向量函数。对于调查混合椭圆边界值问题的操作员来说，不平等是必要的。
We consider the vector functions in a domain homeomorphic to a spherical layer bounded by twice continuously differentiable surfaces. Additional restrictions are imposed on the domain, which allow to conduct proofs using simple methods. On the outer and inner boundaries, the normal and the tangential components of the vector are zero, respectively. For such functions, the integral over the domain of the squared vector is estimated from above via the integral of the sum of squared gradients of its Cartesian components. The last integral is estimated through the integral of the sum of the squared divergence and rotor. These inequalities allow to define two norms equivalent to the sum of the norms of the Cartesian components of vector functions as the elements of the space $W_2^{(1)}(Ω)$. The integrals over the boundaries of the squared vector are also estimated. The constants in all proved inequalities are determined only by the shape of the domain and do not depend on a specific vector function. The inequalities are necessary for investigating the operator of a mixed elliptic boundary value problem.
