Abstract
In this paper the regularity of the Lagrangiansf(x, ξ)=|ξ|α(x)(1<α 1≤α(x)≤α2< +∞) is studied. Our main result: Ifα(x) is Holder continuous, then the Lagrangianf(x, ξ)=f(x, ξ)=|ξ|α(x) is regular. This result gives a negative answer to a conjecture of V. Zhikov.
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Zhikov V V. Averaging of functional in the calculus of variations and elasticity.Math USSR Izvestija, (in Russian), 1986,50(4): 675–711.
Zhikov V V. On passing to the limit in nonlinear variational problem.Math Sbornik (in Russian), 1992,183(8): 47–84.
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Supported by the National Natural Science Foundation of China.
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Xianling, F. The regularity of Lagrangiansf(x, ξ)=254-01254-01254-01with Hölder exponents α(x). Acta Mathematica Sinica 12, 254–261 (1996). https://doi.org/10.1007/BF02106979
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DOI: https://doi.org/10.1007/BF02106979