# American Institute of Mathematical Sciences

July  2016, 21(5): 1635-1649. doi: 10.3934/dcdsb.2016015

## Interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations

 1 Department of Mathematics,Shanghai University, Shanghai 200444, China 2 LMAM, School of Mathematical Sciences, Peking University, Bejing 100871

Received  September 2013 Revised  March 2014 Published  April 2016

In this paper we present a new proof for the interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations, whose prototype is the $p$-Laplace equation.
Citation: Fengping Yao, Shulin Zhou. Interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1635-1649. doi: 10.3934/dcdsb.2016015
##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd edition),, Academic Press, (2003). Google Scholar [2] S. Byun, F. Yao and S. Zhou, Gradient Estimates in Orlicz space for nonlinear elliptic Equations,, J. Funct. Anal., 255 (2008), 1851. doi: 10.1016/j.jfa.2008.09.007. Google Scholar [3] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations,, Ann. of Math., 130 (1989), 189. doi: 10.2307/1971480. Google Scholar [4] Y. Chen and L. Wu, Second Order Elliptic Partial Differential Equations and Elliptic Systems,, American Mathematical Society, (1998). Google Scholar [5] A. Cianchi and V. Maz'ya, Global Lipschitz regularity for a class of quasilinear elliptic equations,, Comm. Partial Differential Equations, 36 (2011), 100. doi: 10.1080/03605301003657843. Google Scholar [6] E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827. doi: 10.1016/0362-546X(83)90061-5. Google Scholar [7] F. Duzaar and G. Mingione, Local Lipschitz regularity for degenerate elliptic systems,, Ann. Inst. H. Poincaré, 27 (2010), 1361. doi: 10.1016/j.anihpc.2010.07.002. Google Scholar [8] F. Duzaar and G. Mingione, Gradient estimates via linear and nonlinear potentials,, J. Funct. Anal., 259 (2010), 2961. doi: 10.1016/j.jfa.2010.08.006. Google Scholar [9] L. C. Evans, A new proof of local $C^{1,\alpha}$ regularity for solutions of certain degenerate elliptic p.d.e.,, J. Differential Equations, 45 (1982), 356. doi: 10.1016/0022-0396(82)90033-X. Google Scholar [10] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,, Princeton University Press, (1983). Google Scholar [11] D. Gilbarg and N. Trudinger, Elliptic Partial Diferential Equations of Second Order (3rd edition),, Springer-Verlag, (1998). Google Scholar [12] J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations,, Indiana Univ. Math. J., 32 (1983), 849. doi: 10.1512/iumj.1983.32.32058. Google Scholar [13] G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations,, Comm. Partial Differential Equations, 16 (1991), 311. doi: 10.1080/03605309108820761. Google Scholar [14] J. Malý, D. Swanson and W. Ziemer, Fine behavior of functions whose gradients are in an Orlicz space,, Studia Math., 190 (2009), 33. doi: 10.4064/sm190-1-2. Google Scholar [15] M. Shaw and L. Wang, Hölder and Lp estimates for Db on CR manifolds of arbitrary codimension,, Math. Ann., 331 (2005), 297. doi: 10.1007/s00208-004-0583-5. Google Scholar [16] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126. doi: 10.1016/0022-0396(84)90105-0. Google Scholar [17] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems,, Acta Math., 138 (1977), 219. doi: 10.1007/BF02392316. Google Scholar [18] L. Wang, Compactness methods for certain degenerate elliptic equations,, J. Differential Equations, 107 (1994), 341. doi: 10.1006/jdeq.1994.1016. Google Scholar [19] L. Wang, Hölder estimates for subelliptic operators,, J. Funct. Anal., 199 (2003), 228. doi: 10.1016/S0022-1236(03)00093-4. Google Scholar [20] L. Wang, F. Yao, S. Zhou and H. Jia, Optimal regularity for the poisson equation,, Proc. Amer. Math. Soc., 137 (2009), 2037. doi: 10.1090/S0002-9939-09-09805-0. Google Scholar

show all references

##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd edition),, Academic Press, (2003). Google Scholar [2] S. Byun, F. Yao and S. Zhou, Gradient Estimates in Orlicz space for nonlinear elliptic Equations,, J. Funct. Anal., 255 (2008), 1851. doi: 10.1016/j.jfa.2008.09.007. Google Scholar [3] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations,, Ann. of Math., 130 (1989), 189. doi: 10.2307/1971480. Google Scholar [4] Y. Chen and L. Wu, Second Order Elliptic Partial Differential Equations and Elliptic Systems,, American Mathematical Society, (1998). Google Scholar [5] A. Cianchi and V. Maz'ya, Global Lipschitz regularity for a class of quasilinear elliptic equations,, Comm. Partial Differential Equations, 36 (2011), 100. doi: 10.1080/03605301003657843. Google Scholar [6] E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827. doi: 10.1016/0362-546X(83)90061-5. Google Scholar [7] F. Duzaar and G. Mingione, Local Lipschitz regularity for degenerate elliptic systems,, Ann. Inst. H. Poincaré, 27 (2010), 1361. doi: 10.1016/j.anihpc.2010.07.002. Google Scholar [8] F. Duzaar and G. Mingione, Gradient estimates via linear and nonlinear potentials,, J. Funct. Anal., 259 (2010), 2961. doi: 10.1016/j.jfa.2010.08.006. Google Scholar [9] L. C. Evans, A new proof of local $C^{1,\alpha}$ regularity for solutions of certain degenerate elliptic p.d.e.,, J. Differential Equations, 45 (1982), 356. doi: 10.1016/0022-0396(82)90033-X. Google Scholar [10] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,, Princeton University Press, (1983). Google Scholar [11] D. Gilbarg and N. Trudinger, Elliptic Partial Diferential Equations of Second Order (3rd edition),, Springer-Verlag, (1998). Google Scholar [12] J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations,, Indiana Univ. Math. J., 32 (1983), 849. doi: 10.1512/iumj.1983.32.32058. Google Scholar [13] G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations,, Comm. Partial Differential Equations, 16 (1991), 311. doi: 10.1080/03605309108820761. Google Scholar [14] J. Malý, D. Swanson and W. Ziemer, Fine behavior of functions whose gradients are in an Orlicz space,, Studia Math., 190 (2009), 33. doi: 10.4064/sm190-1-2. Google Scholar [15] M. Shaw and L. Wang, Hölder and Lp estimates for Db on CR manifolds of arbitrary codimension,, Math. Ann., 331 (2005), 297. doi: 10.1007/s00208-004-0583-5. Google Scholar [16] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126. doi: 10.1016/0022-0396(84)90105-0. Google Scholar [17] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems,, Acta Math., 138 (1977), 219. doi: 10.1007/BF02392316. Google Scholar [18] L. Wang, Compactness methods for certain degenerate elliptic equations,, J. Differential Equations, 107 (1994), 341. doi: 10.1006/jdeq.1994.1016. Google Scholar [19] L. Wang, Hölder estimates for subelliptic operators,, J. Funct. Anal., 199 (2003), 228. doi: 10.1016/S0022-1236(03)00093-4. Google Scholar [20] L. Wang, F. Yao, S. Zhou and H. Jia, Optimal regularity for the poisson equation,, Proc. Amer. Math. Soc., 137 (2009), 2037. doi: 10.1090/S0002-9939-09-09805-0. Google Scholar
 [1] Wendong Wang, Liqun Zhang. The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1261-1275. doi: 10.3934/dcds.2011.29.1261 [2] Rong Dong, Dongsheng Li, Lihe Wang. Regularity of elliptic systems in divergence form with directional homogenization. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 75-90. doi: 10.3934/dcds.2018004 [3] Yun Yang. Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5133-5152. doi: 10.3934/dcds.2015.35.5133 [4] Christian Bonatti, Sylvain Crovisier, Katsutoshi Shinohara. The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited. Journal of Modern Dynamics, 2013, 7 (4) : 605-618. doi: 10.3934/jmd.2013.7.605 [5] Matteo Cozzi. On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5769-5786. doi: 10.3934/dcds.2015.35.5769 [6] M. Matzeu, Raffaella Servadei. A variational approach to a class of quasilinear elliptic equations not in divergence form. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 819-830. doi: 10.3934/dcdss.2012.5.819 [7] Ilaria Fragalà, Filippo Gazzola, Gary Lieberman. Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains. Conference Publications, 2005, 2005 (Special) : 280-286. doi: 10.3934/proc.2005.2005.280 [8] Giuseppe Riey. Regularity and weak comparison principles for double phase quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4863-4873. doi: 10.3934/dcds.2019198 [9] Libin Wang. Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Communications on Pure & Applied Analysis, 2003, 2 (1) : 77-89. doi: 10.3934/cpaa.2003.2.77 [10] Gary Lieberman. Nonlocal problems for quasilinear parabolic equations in divergence form. Conference Publications, 2003, 2003 (Special) : 563-570. doi: 10.3934/proc.2003.2003.563 [11] Raphaël Danchin, Piotr B. Mucha. Divergence. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1163-1172. doi: 10.3934/dcdss.2013.6.1163 [12] Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 165-176. doi: 10.3934/dcdss.2020009 [13] Dian Palagachev, Lubomira G. Softova. Quasilinear divergence form parabolic equations in Reifenberg flat domains. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1397-1410. doi: 10.3934/dcds.2011.31.1397 [14] Yong Zhou, Jishan Fan. Regularity criteria for a magnetohydrodynamic-$\alpha$ model. Communications on Pure & Applied Analysis, 2011, 10 (1) : 309-326. doi: 10.3934/cpaa.2011.10.309 [15] Keonhee Lee, Kazumine Moriyasu, Kazuhiro Sakai. $C^1$-stable shadowing diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 683-697. doi: 10.3934/dcds.2008.22.683 [16] Lan Wen. A uniform $C^1$ connecting lemma. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 257-265. doi: 10.3934/dcds.2002.8.257 [17] Hermann Köenig, Vitali Milman. Derivative and entropy: the only derivations from $C^1(RR)$ to $C(RR)$. Electronic Research Announcements, 2011, 18: 54-60. doi: 10.3934/era.2011.18.54 [18] V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413 [19] Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1 [20] Flavio Abdenur, Lorenzo J. Díaz. Pseudo-orbit shadowing in the $C^1$ topology. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 223-245. doi: 10.3934/dcds.2007.17.223

2018 Impact Factor: 1.008