Continuity at boundary points of solutions of quasilinear elliptic equations with a~non-standard growth condition

Yu. A. Alkhutov ; O. V. Krasheninnikova

Izvestiya. Mathematics , Tome 68 (2004) no. 6, pp. 1063-1117.

Résumé

We study the behaviour at boundary points of a solution of the Dirichlet problem with continuous boundary function for the Euler equation generated by the Lagrangian $|\nabla u|^{p(x)}/p(x)$ with variable$p=p(x)$ that has logarithmic modulus of continuity and satisfies the condition $1$. We obtain a regularity criterion for a boundary point of Wiener type, an estimate for the modulus of continuity of the solution near a regular boundary point, and geometric conditions for regularity.

Bibliographie

[1] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1964 | MR

[2] Zhikov V. V., “Voprosy skhodimosti, dvoistvennosti i usredneniya dlya funktsionalov variatsionnogo ischisleniya”, Izv. AN SSSR. Ser. matem., 47:5 (1983), 961–995 | MR

[3] Zhikov V. V., “O postanovke kraevykh zadach dlya integralov vida $|\xi|^{\alpha(x)}$”, UMN, 41:4 (1986), 187–188 | MR

[4] Zhikov V. V., “Usrednenie funktsionalov variatsionnogo ischisleniya i teorii uprugosti”, Izv. AN SSSR. Ser. matem., 50:4 (1986), 675–711 | MR

[5] Giaquinta M., “Growth Conditions and Regularity, a Counterexample”, Manuscripta Math., 59 (1987), 245–248 | DOI | MR | Zbl

[6] Marcellini P., Un Exemple de Solution Discontinue d'un Probléme Variationell dans le cas scalaire, Preprint No11, Instituto Matematico U. Dini. Universitá di Firenze, 1987

[7] R. uz̆hic̆ka M., Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., 1748, 2000

[8] Zhikov V. V., “On some variational problems”, Russian J. of Math. Physics, 5:1 (1996), 105–116 | MR

[9] Lavrentieff M., “Sur quelques problémes du calcul des variations”, Ann. Mat. Pura Appl., 4 (1926), 7–28 | MR | Zbl

[10] Zhikov V. V., “On Lavrentiev's Phenomenon”, Russian J. of Math. Physics, 3:2 (1994), 249–269 | MR

[11] Zhikov V. V., “Meyers type estimates for the solution of a nonlinear Stokes system”, Diff. Eq., 33:1 (1997), 108–115 | MR | Zbl

[12] Leray J., Lions J.-L., “Quelques résultats de Vi$\check{\text{s}}$ik sur les problémes elliptiques non linéaires par les méthodes de Minty-Brouder”, Bull. Soc. Math. France, 93 (1965), 97–107 | MR | Zbl

[13] Ladyzhenskaya O. A., Uraltseva N. N., “Kvazilineinye ellipticheskie uravneniya i variatsionnye zadachi so mnogimi nezavisimymi peremennymi”, UMN, 16:1 (1961), 19–90 | MR | Zbl

[14] Marcellini P., “Regularity for Elliptic Equations with General Growth Conditions”, J. Diff. Equations, 105 (1993), 296–333 | DOI | MR | Zbl

[15] Fan X., A Class of De Giorgi Type and Holder Continuity of Minimizers of Variational with $m(x)$-Growth Condition, Lanzhou Univ., China, 1995 | Zbl

[16] Alkhutov Yu. A., “Neravenstvo Kharnaka i gëlderovost reshenii nelineinykh ellipticheskikh uravnenii s nestandartnym usloviem rosta”, Diff. uravneniya, 33:12 (1997), 1651–1660 | MR | Zbl

[17] Chiado Piat V., Coscia A., “Hölder continuity of minimiizers of functionals with variable growth exponent”, Manuscripta Math., 93 (1997), 283–299 | DOI | MR | Zbl

[18] Acerbi E., Fusco N. A., “Transmission problem in the calculus of variation”, Calc. Var., 2 (1994), 1–16 | DOI | MR | Zbl

[19] Krasheninnikova O. V., “O nepreryvnosti v tochke reshenii ellipticheskikh uravnenii s nestandartnym usloviem rosta”, Tr. Matem. in-ta im. V. A. Steklova RAN, 236 (2002), 204–211 | MR | Zbl

[20] Kondratev V. A., Landis E. M., “Kachestvennaya teoriya lineinykh differentsialnykh uravnenii v chastnykh proizvodnykh vtorogo poryadka”, Itogi nauki i tekhniki. Sovr. problemy matematiki. Fundamentalnye napravleniya, 32, VINITI, M., 1988, 99–215 | MR

[21] Heinonen J., Kilpeläinen T., Martio O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, 1993 | MR | Zbl

[22] Wiener N., “Certain notions in potential theory”, J. Math. Phys., 3 (1924), 24–51

[23] Littman W., Stampacchia G., Weinberger H. F., “Regular points for elliptic equations with discontinuous coefficients”, Ann. Scuola Norm. Sup. Pisa, 17:3 (1963), 43–77 | MR | Zbl

[24] Mazya V. G., “O nepreryvnosti v granichnoi tochke reshenii kvazilineinykh ellipticheskikh uravnenii”, Vestn. LGU. Ser. matem., 25:13 (1970), 42–55 | MR | Zbl

[25] Gariepy R., Ziemer W. P., “A regularity condition at the boundary for solutions of quasilinear elliptic equations”, Arch. Rational Mech. Anal., 67 (1977), 25–39 | DOI | MR | Zbl

[26] Skrypnik I. V., “Kriterii regulyarnosti granichnoi tochki dlya kvazilineinykh ellipticheskikh uravnenii”, DAN SSSR, 274:5 (1984), 1040–1044 | MR

[27] Kilpeläinen T., Malý J., “The Wiener test and potential estimates for quasilinear elliptic equations”, Acta Math., 172 (1994), 137–161 | DOI | MR | Zbl

[28] Lindkvist P., Martio O., “Two theorems of N. Wiener for solutions of quasilinear elliptic equations”, Acta Math., 155 (1985), 153–171 | DOI | MR

[29] Choquet G., “Theory of capacities”, Annales de l'Institut Fourier, 155:5 (1953–1954), 131–295 | MR

[30] Kondratev V. A., “O razreshimosti pervoi kraevoi zadachi dlya silno ellipticheskikh uravnenii”, Tr. MMO, 16, URSS, M., 1967, 209–292 | MR | Zbl

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

[32] Trudinger N. S., “On Harnack type inequalities and their application to quasilinear elliptic equations”, Comm. Pure Appl. Math., 20 (1967), 721–747 | DOI | MR | Zbl

[33] Stampacchia G., “Le probléme de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinuous”, Ann. Inst. Fourier, 15 (1965), 189–258 | MR | Zbl

[34] Lindkvist P., “On the equation $\operatorname{div}(|\nabla u|^{p-2}\nabla u)+\lambda|u|^{p-2}u=0$”, Proc. Amer. Math. Soc., 109 (1990), 157–164 | DOI | MR

[35] Gehring F. W., “The $L^p$-integrability of the partial derivatives of a quasiconformal mapping”, Acta Math., 130 (1973), 265–277 | DOI | MR | Zbl

[36] Giaquinta M., Modica G., “Regularity results for some classes of higher order nonlinear elliptic systems”, J. für die reine und angewandte Math., 311/312 (1979), 145–169 | MR | Zbl

[37] Schwartz L., Théorie des Distributions, V. 1, Hermann, Paris, 1950 | MR | Zbl

[38] John G., Nirenberg L., “On functions of bounded mean oscillation”, Comm. Pure Appl. Math., 14 (1961), 415–426 | DOI | MR | Zbl

[39] Fabes E. B., Stroock D. W., “The $L^p$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations”, Duke Math. J., 51:4 (1984), 99–1016 | DOI | MR

[40] Mazya V. G., Prostranstva Soboleva, Izd-vo LGU, L., 1985 | Zbl