A posteriori estimates of the deviation from exact solutions to variational problems under nonstandard coerciveness and growth conditions,St. Petersburg Mathematical Journal

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A posteriori estimates of the deviation from exact solutions to variational problems under nonstandard coerciveness and growth conditions
St. Petersburg Mathematical Journal ( IF 0.8 ) Pub Date : 2021-01-11 , DOI: 10.1090/spmj/1637
S. E. Pastukhova

Abstract:A posteriori estimates are proved for the accuracy of approximations of solutions to variational problems with nonstandard power functionals. More precisely, these are integral functionals with power type integrands having a variable exponent $p(\,\cdot \,)$ . It is assumed that $p(\,\cdot \,)$ is bounded away from one and infinity. Estimates in the energy norm are obtained for the difference of the approximate and exact solutions. The majorant

in these estimates depends only on the approximation

and the data of the problem, but is independent of the exact solution

. It is shown that

vanishes as

tends to

and

only if

. The superquadratic and subquadratic cases (which means that $p(\,\cdot \,)\ge 2$ , or $p(\,\cdot \,)\le 2$ , respectively) are treated separately.

中文翻译：

后验估计在非标准矫顽力和增长条件下从精确解到变分问题的偏差

摘要：后验估计证明了具有非标准幂函数的变分问题的近似解的准确性。更准确地说，它们是具有幂指数的幂型被积的积分泛函。假设它是一个无穷远的边界。对于近似和精确解的差异，获得了能量范数的估计。该majorant上述估计只依赖于近似和问题的数据，但是独立的精确解。证明消失趋于且仅在发生时消失。超二次和次二次情况（即或 $p（\，\ cdot \，）$ $p（\，\ cdot \，）$

$p（\，\ cdot \，）\ ge 2$ $p（\，\ cdot \，）\ le 2$ 分别）。

更新日期：2021-01-14

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