We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energy quantization principle for sequences of solutions of boundary value problems that have bounded energy and whose energy densities satisfy nonlinear bounds on the Laplacian and normal derivative: One obtains local uniform bounds on the complement of finitely many points, where some minimum quantum of energy concentrates.
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Katrin Wehrheim, Energy quantization and mean value inequalities for nonlinear boundary value problems. J. Eur. Math. Soc. 7 (2005), no. 3, pp. 305–318DOI 10.4171/JEMS/30