Geodesic
Estimates of the deviation from the minimizer for variational problems with power growth functionals
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 37, Tome 336 (2006), pp. 5-24

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The paper is concerned with the derivation of directly computable estimates of the difference between approximate solutions and the minimizer of the variational problem $$ J_\alpha[w]:=\int_\Omega\Big[\frac1\alpha|\nabla w|^\alpha-fw\Big]\,\mathrm dx\to\min. $$ If the functional has a superquadratic growth, then the estimate is given in terms of the natural energy norm. For problems with subquadratic growth it is more convenient to derive such estimates in terms of the dual variational problem. The estimates are obtained for the Dirichlet, Neumann and mixed boundary conditions. 
@article{ZNSL_2006_336_a0,
     author = {M. Bildhauer and S. I. Repin},
     title = {Estimates of the deviation from the minimizer for variational problems with power growth functionals},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--24},
     publisher = {mathdoc},
     volume = {336},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_336_a0/}
}
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M. Bildhauer; S. I. Repin. Estimates of the deviation from the minimizer for variational problems with power growth functionals. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 37, Tome 336 (2006), pp. 5-24. http://geodesic.mathdoc.fr/item/ZNSL_2006_336_a0/