A posteriori estimates of the deviation from exact solutions to variational problems under nonstandard coerciveness and growth conditions

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 $ M$ in these estimates depends only on the approximation $ v$ and the data of the problem, but is independent of the exact solution $ u$. It is shown that $ M=M(v)$ vanishes as $ v$ tends to $ u$ and $ M(v)=0$ only if $ v=u$. The superquadratic and subquadratic cases (which means that $ p( \cdot )\ge 2$, or $ p( \cdot )\le 2$, respectively) are treated separately.