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New examples on Lavrentiev gap using fractals.(English)Zbl 1453.35082

The paper deals with the Lavrentiev gap, i.e., the phenomenon which occurs when the minimum of an integral functional \(\mathcal{G}\) taken over smooth functions differs from the one taken over the associated energy space. The Lavrentiev gap is clearly closely related to the (non-)density of smooth functions, i.e. to the fact that \(H^{1,p(\cdot)}(\Omega)\neq W^{1,p(\cdot)}(\Omega)\), where \(p(\cdot)\) is variable exponent. In [Izv. Akad. Nauk SSSR Ser. Mat., 50, No. 4, 675–710 (1986)],VV. Zhikov presented a two-dimensional checkboard example with a Lavrentiev gap. In such an example and in others, the dimension played a critical role, since the exponent presents a saddle point where it crossed the dimension \(d\). In the present paper, the authors provide new examples of variable exponents, such that the Lavrentiev gap occurs but which do not need to cross the dimensional threshold. They also show that \(H^{1,p(\cdot)}(\Omega)\neq W^{1,p(\cdot)}(\Omega)\) and the ambiguity of the notion of \(p(\cdot)\)-harmonicity.

MSC:

35J60 Nonlinear elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J20 Variational methods for second-order elliptic equations

Cite

References:

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[11]Hästö, P.A.: Counter-examples of regularity in variable exponent Sobolev spaces. In: The \(p\)-Harmonic Equation and Recent Advances in Analysis, volume 370 of Contemporary Mathematics, pp. 133-143. Amer. Math. Soc., Providence, RI (2005) ·Zbl 1084.46025
[12]Kokilashvili, V., Meskhi, A., Rafeiro, H., Samko, S.: Integral Operators in Non-standard Function Spaces. Vol. 1, volume 248 of Operator Theory: Advances and Applications. Birkhäuser/Springer, [Cham], Variable exponent Lebesgue and amalgam spaces (2016) ·Zbl 1385.47001
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[14]Lavrentiev, M., Sur quelques problemes du calcul des variations, Ann. Math. Pura Appl., 4, 107-124 (1926)
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[18]Zhikov, VV; Surnachev, MD, On density of smooth functions in weighted Sobolev spaces with variable exponents, St Petersb. Math. J., 27, 415-436 (2016) ·Zbl 1354.46039 ·doi:10.1090/spmj/1396
[19]Zhikov, VV, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50, 4, 675-710 (1986)
[20]Zhikov, VV, On weighted Sobolev spaces, Mat. Sb., 189, 8, 27-58 (1998) ·Zbl 0919.46026 ·doi:10.4213/sm344
[21]Zhikov, V. V.: On the density of smooth functions in Sobolev-Orlicz spaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI). 310(Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34]):67-81 (2004) ·Zbl 1086.46026
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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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