Survey on gradient estimates for nonlinear elliptic equations in various function spaces

S.-S. Byun ; D. K. Palagachev ; L. G. Softova

Algebra i analiz, Tome 31 (2019) no. 3, pp. 10-35.

Résumé

Very general nonvariational elliptic equations of $ p$-Laplacian type are treated. An optimal Calderón-Zygmund theory is developed for such a nonlinear elliptic equation in divergence form in the setting of various function spaces including Lebesgue spaces, Orlicz spaces, weighted Orlicz spaces, and variable exponent Lebesgue spaces. The addressed arguments also apply to Morrey spaces, Lorentz spaces and generalized Orlicz spaces.

Détail

Keywords: gradient estimate, nonlinear elliptic equation, $L^p$ space, weighted Lebesgue space, Orlicz space, BMO, Muckenhoupt weight, Reifenberg flat domain.

Bibliographie

[1] Acerbi E., Mingione G., “Gradient estimates for the $p(x)$-Laplacean system”, J. Reine Angew. Math., 584 (2005), 117–148

[2] Acerbi E., Mingione V., “Gradient estimates for a class of parabolic systems”, Duke Math. J., 136:2 (2007), 285–320

[3] Barbu V., Nonlinear semigroups and differential equations in Banach Spaces, Editura Acad. Repub. Soc. Rom., Bucharest; Noordhoff Internat. Publ., Leiden, 1976

[4] Baroni P., Bogelein V., “Calderón–Zygmund estimates for parabolic $p(x,t)$-Laplacian systems”, Rev. Mat. Iberoam., 30:4 (2014), 1355–1386

[5] Bogelein V., “Calderón–Zygmund theory for nonlinear parabolic systems”, Calc. Var. Partial Differential Equations, 51:3–4 (2014), 555–596

[6] Bui T., Duong X., “Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains”, Calc. Var. Partial Differential Equations, 56:2 (2017), 47

[7] Byun S.-S., “Gradient estimates in Orlicz spaces for nonlinear elliptic equations with BMO nonlinearity in nonsmooth domains”, Forum Math., 23:4 (2011), 693–711

[8] Byun S.-S., Cho Y., Palagachev D. K., “Global weighted estimates for nonlinear elliptic obstacle problems over Reifenberg domains”, Proc. Amer. Math. Soc., 13:6 (2015), 2527–2541

[9] Byun S.-S., Cho Y., Wang L., “Calderón–Zygmund theory for nonlinear elliptic problems with irregular obstacles”, J. Funct. Anal., 263:10 (2012), 3117–3143

[10] Byun S.-S., Kim Y., “Elliptic equations with measurable nonlinearities in nonsmooth domains”, Adv. Math., 288 (2016), 152–200

[11] Byun S.-S., Ok J., “On $W^{1,q(~\cdot~)}$-estimates for elliptic equations of $p(x)$-Laplacian type”, J. Math. Pures Appl. (9), 106:3 (2016), 512–545

[12] Byun S.-S., Ok J., “Nonlinear parabolic equations with variable exponent growth in nonsmooth domains”, SIAM J. Math. Anal., 48:5 (2016), 3148–3190

[13] Byun S.-S., Ok J., Palagachev D. K., Softova L., “Parabolic systems with measurable coefficients in weighted Orlicz spaces”, Commun. Contemp. Math., 18:2 (2016), 1550018

[14] Byun S.-S., Palagachev D. K., “Morrey regularity of solutions to quasilinear elliptic equations over Reifenberg flat domains”, Calc. Var. Partial Differential Equations, 49:1–2 (2014), 37–76

[15] Byun S.-S., Palagachev D. K., “Weighted $L^p$-estimates for elliptic equations with measurable coefficients in nonsmooth domains”, Potential Anal., 41:1 (2014), 51–79

[16] Byun S.-S., Palagachev D. K., Ryu S., “Weighted $W^{1,p}$ estimates for solutions of non-linear parabolic equations over non-smooth domains”, Bull. Lond. Math. Soc., 45:4 (2013), 765–778

[17] Byun S.-S., Palagachev D. K., Shin P., “Sobolev–Morrey regularity of solutions to general quasilinear elliptic equations”, Nonlinear Anal., 147 (2016), 176–190

[18] Byun S.-S., Palagachev D. K., Softova L., “Global gradient estimates in weighted Lebesgue spaces for parabolic operators”, Ann. Acad. Sci. Fenn. Math., 41:1 (2016), 67–83

[19] Byun S.-S., Park J., “Global weighted Orlicz estimates for parabolic measure data problems: application to estimates in variable exponent spaces”, J. Math. Anal. Appl., 67:2 (2018), 1194–1207

[20] Byun S.-S., Ryu S., “Weighted Orlicz estimates for general nonlinear parabolic equations over nonsmooth domains”, J. Funct. Anal., 272:10 (2017), 4103–4121

[21] Byun S.-S., Ryu S., “Global weighted estimates for the gradient of solutions to nonlinear elliptic equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30:2 (2013), 291–313

[22] Byun S.-S., Softova L. G., “Gradient estimates in generalized Morrey spaces for parabolic operators”, Math. Nachr., 288:14–15 (2015), 1602–1614

[23] Byun S.-S., Softova L. G., “Parabolic obstacle problem with measurable data in generalized Morrey spaces”, Z. Anal. Anwend., 35:2 (2016), 153–171

[24] Byun S.-S., Wang L., “Elliptic equations with BMO coefficients in Reifenberg domains”, Comm. Pure Appl. Math., 57:10 (2004), 1283–1310

[25] Byun S.-S., Wang L., “Parabolic equations in Reifenberg domains”, Arch. Ration. Mech. Anal., 176:2 (2005), 271–301

[26] Byun S.-S., Wang L., “Nonlinear gradient estimates for elliptic equations of general type”, Calc. Var. Partial Differential Equations, 45:3–4 (2012), 403–419

[27] Byun S.-S., Yao F., Zhou S., “Gradient estimates in Orlicz space for nonlinear elliptic equations”, J. Funct. Anal., 45:8 (2008), 1851–1873

[28] Caffarelli L. A., Peral I., “On $W^{1,p}$ estimates for elliptic equations in divergence form”, Comm. Pure Appl. Math., 51:1 (1998), 1–21

[29] Calderon A. P., Zygmund A., “On the existence of certain singular integrals”, Acta Math., 88 (1952), 85–139

[30] Cianchi A., “Optimal Orlicz–Sobolev embeddings”, Rev. Mat. Iberoam., 20:2 (2004), 427–474

[31] Cruz-Uribe D. V., Fiorenza A., Variable Lebesgue spaces. Foundations and harmonic analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Heidelberg, 2013

[32] Cruz-Uribe D. V., Hästö P., “Extrapolation and interpolation in generalized Orlicz spaces”, Trans. Amer. Math. Soc., 370:6 (2018), 4323–4349

[33] Cruz-Uribe D. V., Wang L.-A. D., “Extrapolation and weighted norm inequalities in the variable Lebesgue spaces”, Trans. Amer. Math. Soc., 369:2 (2017), 1205–1235

[34] DiBenedetto E., Degenerate parabolic equations, Universitext, Springer, New York, 1993

[35] Diening L., Harjulehto P., Hästö P., Ružička M., Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Math., 2017, Springer, Heidelberg, 2011

[36] Diening L., Kaplicky P., Schwarzacher S., “BMO estimates for the $p$-Laplacian”, Nonlinear Anal., 75:2 (2012), 637–650

[37] Diening L., Schwarzacher S., “Global gradient estimates for the $p(x)$-Laplacian”, Nonlinear Anal., 106 (2014), 70–85

[38] Di Fazio G., “$L^p$ estimates for divergence form elliptic equations with discontinuous coefficients”, Boll. Un. Mat. Ital. A (7), 10:2 (1996), 409–420

[39] Fiorenza A., Krbec M., “Indices of Orlicz spaces and some applications”, Comm. Math. Univ. Carolin., 38:3 (1997), 433–451

[40] Guliyev V. S., Softova L. G., “Generalized Morrey estimates for the gradient of divergence form parabolic operators with discontinuous coefficients”, J. Differential Equations, 259:6 (2015), 2368–2387

[41] Giaquinta M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. of Math. Stud., 105, Princeton Univ. Press, Princeton, NJ, 1983

[42] Giusti E., Direct methods in the calculus of variations, World Sci. Publ. Co., Inc., River Edge, NJ, 2003

[43] Grafakos L., Classical Fourier analysis, Graduate Texts in Math., 249, Third ed., Springer, New York, 2014

[44] Iwaniec T., “Projections onto gradient fields and $L^p$-estimates for degenerated elliptic operators”, Studia Math., 75:3 (1983), 293–312

[45] Iwaniec T., Sbordone C., “Riesz transforms and elliptic PDEs with VMO coefficients”, J. Anal. Math., 74 (1998), 183–212

[46] John F., Nirenberg L., “On functions of bounded mean oscillation”, Comm. Pure Appl. Math., 14 (1961), 415–426

[47] Kerman R. A., Torchinsky A.,, “Integral inequalities with weights for the Hardy maximal function”, Studia Math., 71:3 (1981/82), 277–284

[48] Kinnunen J., Zhou Sh., “A local estimate for nonlinear equations with discontinuous coefficients”, Comm. Partial Differential Equations, 24:11–12 (1999), 2043–2068

[49] Kokilashvili V., Krbec M., Weighted inequalities in Lorentz and Orlicz spaces, World Sci. Publ. Co., Inc., River Edge, NJ, 1991

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

[51] Krasnoselskii M. A., Rutitskii Ya. B., Vypuklye funktsii i prostranstva Orlicha, GITTL, M., 1958

[52] Krylov N. V., Safonov M. V., “Nekotoroe svoistvo reshenii parabolicheskikh uravnenii s izmerimymi koeffitsientami”, Izv. AN SSSR. Ser. mat., 44:1 (1980), 161–175

[53] Kuusi T., Mingione G., “Universal potential estimates”, J. Funct. Anal., 262:10 (2012), 4205–4269

[54] Kuusi T., Mingione G., “Guide to nonlinear potential estimates”, Bull. Math. Sci., 4:1 (2014), 1–82

[55] Kuusi T., Mingione G., “Linear potentials in nonlinear potential theory”, Arch. Ration. Mech. Anal., 207:1 (2013), 215–246

[56] Kuusi T., Mingione G., “The Wolff gradient bound for degenerate parabolic equations”, J. Eur. Math. Soc. (JEMS), 16:4 (2014), 835–892

[57] Kuusi T., Mingione G., “Vectorial nonlinear potential theory”, J. Eur. Math. Soc. (JEMS), 20:4 (2018), 929–1004

[58] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973

[59] Landis E. M., Uravneniya vtorogo poryadka ellipticheskogo i parabolicheskogo tipov, Nauka, M., 1971

[60] Mengesha T., Phuc N. C., “Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains”, J. Differential Equations, 250:5 (2011), 2485–2507

[61] Meyers N. G., “An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations”, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189–206

[62] Milakis E., Toro T., “Divergence form operators in Reifenberg flat domains”, Math. Z., 264:1 (2010), 15–41

[63] Morrey C. B., Multiple shntegrals in the salculus of mariations, Grundlehren Math. Wiss., 130, Springer-Verlag, Inc., New York, 1966

[64] Muckenhoupt B., “Weighted norm inequalities for the Hardy maximal function”, Trans. Amer. Math. Soc., 165 (1972), 207–226

[65] Orlicz W., “Über konjugierte Exponentenfolgen”, Studia Math., 3 (1931), 200–211

[66] Palagachev D. K., Softova L., “The Calderón–Zygmund property for quasilinear divergence form equations over Reifenberg flat domains”, Nonlinear Anal., 74:5 (2011), 1721–1730

[67] Rao M. M., Ren Z. D., Theory of Orlicz spaces, Monogr. Textbooks Pure Appl. Math., 146, Marcel Dekker, Inc., New York, 1991

[68] Reifenberg E. R., “Solution of the Plateau problem for $m$-dimensional surfaces of varying topological type”, Acta Math., 104 (1960), 1–92

[69] Sarason D., “Functions of vanishing mean oscillation”, Trans. Amer. Math. Soc., 207 (1975), 391–405

[70] Sharapudinov I. I., “O topologii prostranstva $L^{p(t)}([0;1])$”, Mat. zametki, 26:4 (1978), 613–632

[71] Stein E. M., Harmonic analysis: real-variable methods, orthogonality and oscillatory integrals, Princeton Math. Ser., 43, Princeton Univ. Press, Princeton, NJ, 1993

[72] Torchinsky A., Real-variable methods in harmonic analysis, Pure Appl. Math., 123, Acad. Press, Inc., Orlando, FL, 1986

[73] Toro T., “Doubling and flatness: geometry of measures”, Notices Amer. Math. Soc., 44:9 (1997), 1087–1094

[74] Tsenov I. V., “Obobschenie zadachi o nailuchshem priblizhenii funktsii v prostranstve $L^s$”, Uchen. zap. Dagestan. gos. un-ta, 7 (1961), 25–37

[75] Wang L., “A geometric approach to the Calderón–Zygmund estimates”, Acta Math. Sin. (Engl. Ser.), 19:2 (2003), 381–396

[76] Wang L., Yao F., Zhou S., Jia H., “Optimal regularity theory for the Poisson equation”, Proc. Amer. Math. Soc., 137:6 (2009), 2037–2047

[77] Yao F., “Local gradient estimates for the $p(x)$-Laplacian elliptic equations”, Math. Inequal. Appl., 17:1 (2014), 259–268

[78] Zhang C., “Global weighted estimates for the nonlinear parabolic equations with non-standard growth”, Calc. Var. Partial Differential Equations, 55:5 (2016), 109

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

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

[81] Zhikov V. V., “On Lavrentiev's phenomenon”, Russian J. Math. Phys., 3:2 (1995), 249–269

[82] Zhikov V. V., “Otsenki tipa Meiersa dlya resheniya nelineinoi sistemy Stoksa”, Differ. uravn., 33:1 (1997), 107–114