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

S. E. Pastukhova

Algebra i analiz, Tome 32 (2020) no. 1, pp. 51-77.

Résumé

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.

Détail

Keywords: variational problem with nonstandard coercivenes and growth conditions, a posteriori error estimates for approximate solutions, dual problem.

Bibliographie

[1] Zhikov V. V., “Voprosy skhodimosti, dvoistvennosti i usredneniya dlya funktsionalov variatsionnogo ischisleniya”, Izv. AN SSSR. Ser. mat., 47:5 (1983), 961–998 | MR | Zbl

[2] Zhikov V. V., “Usrednenie funktsionalov variatsionnogo ischisleniya i teorii uprugosti”, Izv. AN SSSR. Ser. mat., 50:4 (1986), 675–710 | MR

[3] Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Nauka, M., 1993 | MR

[4] Zhikov V. V., “On variational problems and nonlinear elliptic equations with nonstandard growth conditions”, J. Math. Sci., 173:5 (2011), 463–570 | DOI | MR | Zbl

[5] Kováčik O., Rákosník J., “On spaces $L^{p(x)}$ and $W^{k, p(x)}$”, Czechoslovak Math. J., 41:4 (1991), 592–618 | MR | Zbl

[6] Diening L., Harjulehto P., Hästö P., Ruzicka M., Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Math., 2017, Springer, Heidelberg, 2011 | DOI | MR | Zbl

[7] Zhikov V. V., “Lavrentiev phenomenon and homogenization for some variational problems”, Proc. Second Workshop “Composite Media and Homogenization Theory”, World Sci., Singapore, 1995 | MR

[8] Zhikov V. V., “On Lavrentiev's phenomenon”, Russ. J. Math. Phys., 3:2 (1995), 249–269 | MR | Zbl

[9] Zhikov V. V., “Ob effekte Lavrenteva”, Dokl. RAN, 345:1 (1995), 10–14 | MR | Zbl

[10] Zhikov V. V., “O plotnosti gladkikh funktsii v prostranstve Soboleva–Orlicha”, Zap. nauch. semin. POMI, 310, 2004, 67–81 | Zbl

[11] Zhikov V. V., Pastukhova S. E., “O povyshennoi summiruemosti gradienta reshenii ellipticheskikh uravnenii s peremennym pokazatelem nelineinosti”, Mat. sb., 199:12 (2008), 19–52 | DOI | Zbl

[12] Edmunds D. E., Rákosník J., “Density of smooth functions in $W^{k,p}(\Omega)$”, Proc. Roy. Soc. London. Ser. A, 437:1992 (1992), 229–236 | DOI | MR | Zbl

[13] Pastukhova S. E., “On consequences of the strong convergence in Lebesgue-Orlich spaces”, J. Math. Sci. (N.Y.), 235:3 (2018), 312–321 | DOI | MR | Zbl

[14] Ekland I., Temam R., Vypuklyi analiz i variatsionnye problemy, Mir, M., 1979

[15] Repin S., “A posteriori error estimation for nonlinear variational problems by duality theory”, Zap. nauch. semin. POMI, 243, 1997, 201–214 | MR | Zbl

[16] Bildhauer M., Repin S. I., “Estimates of the deviation from the minimizer for variational problems with power growth functionals”, Zap. nauch. semin. POMI, 336, 2006, 5–24 | MR | Zbl

[17] Repin S., “A posteriori error estimation for variational problems with uniformly convex functionals”, Math. Somp., 69:230 (2000), 481–500 | MR | Zbl

[18] Repin S., “Estimates of deviations from exact solution of the generalized Oseen problem”, Zap. nauch. semin. POMI, 410, 2013, 110–130 | MR | Zbl

[19] Repin S. I., “Otsenki otkloneniya ot tochnykh reshenii nekotorykh kraevykh zadach s usloviem neszhimaemosti”, Algebra i analiz, 16:5 (2004), 124–161

[20] Zhikov V. V., Yakubovich D. A., “Galerkin approximations in problems with p-Laplacian”, J. Math. Sci. (N.Y.), 219:1 (2016), 99–111 | DOI | MR | Zbl

[21] Pastukhova S. E., Yakubovich D. A., “O galerkinskikh priblizheniyakh v zadache Dirikhle s $p(x)$-laplasianom”, Mat. sb., 210:1 (2019), 1–15 | DOI | MR

[22] Pastukhova S. E., Yakubovich D. A., “Galerkin approximations in problems with anisotropic $p(~\cdot~)$-Laplacian”, Appl. Anal., 98:1-2 (2019), 345–361 | DOI | MR | Zbl

[23] Lindqvist P., Notes on the p-Laplace equation, Report. Univ. Jyväskylä Depart. Math. Stat. No 102, Univ. Jyväskylä, Jyväskylä, 2006 | MR | Zbl

[24] Pastukhova S. E., Khripunova A. S., “Gamma-closure of some classes of nonstandard convex integrands”, J. Math. Sci. (N.Y.), 177:1 (2011), 83–108 | DOI | MR | Zbl

[25] Pastukhova S. E., “Ob aposteriornykh otsenkakh tochnosti priblizhenii v variatsionnykh zadachakh so stepennymi funktsionalami”, Probl. mat. anal., 100 (2019), 133–144

[26] Edmunds D., Rákosník J., “Sobolev embeddings with variable exponent”, Studia Math., 143:3 (2000), 267–293 | DOI | MR | Zbl