具有次临界增长的椭圆障碍问题解的正则性
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摘要
利用一个改进的p-调和逼近引理,首先证明了具有次临界增长的p-Laplace型拟线性椭圆障碍问题解的梯度的Morrey正则性。进一步地,利用Hlder连续函数的积分刻划引理得到了解的Hlder连续性。利用该方法避免了证明梯度的反向不等式,从而简化了证明。
Based on a modification of p-harmonic approximation argument,the gradients of solutions to the quasilinear elliptic p-Laplace type obstacle problems with subcritical growth enjoy the Morrey regularity are proved. Then the Hlder continuity of solutions is obtained by using the integral characterization of Hlder continuous functions. Making use of this method,one can simplify the proof avoiding the proof of a suitable reverse Hlder inequality for the gradient.
Based on a modification of p-harmonic approximation argument,the gradients of solutions to the quasilinear elliptic p-Laplace type obstacle problems with subcritical growth enjoy the Morrey regularity are proved. Then the Hlder continuity of solutions is obtained by using the integral characterization of Hlder continuous functions. Making use of this method,one can simplify the proof avoiding the proof of a suitable reverse Hlder inequality for the gradient.
引文
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[11]CHEN Yazhe,WU Lancheng.Second order elliptic equations and elliptic systems[M].Rhode Island:American Mathematical Society,1998.
[12]KINNUNEN J,ZHOU Shulin.A local estimate for nonlinear equations with discontinuous coefficients[J].Communications in Partial Differential Equations,1999,24(11/12):2043-2068.
[2]CHOE H J,LEWIS J.On the obstacle problem for quasilinear elliptic equations of p Laplacian type[J].SIAM Journal on M athematical Analysis,1991,22(3):623-638.
[3]CHOE H J.A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle problems[J].Archive for Rational M echanics and Analysis,1991,114(4):383-394.
[4]DUZAAR F,GASTEL A.Nonlinear elliptic systems with Dini continuous coefficients[J].Archiv der Mathematik,2002,78(1):58-73.
[5]QIU Yalin.Optimal partial regularity of second order nonlinear elliptic systemswith Dini continuous coefficients for the superquadratic case[J].Nonlinear Analysis,2012,75(8):3574-3590.
[6]BOGELEIN V,DUZAAR F,HABERMANN J,et al.Partial H9lder continuity for discontinuous elliptic problems with VMOcoefficients[J].Proceedings of the London M athematical Society,2011,103(3):371-404.
[7]DUZAAR F,GROTOWSKI J F.Optimal interior partial regularity for nonlinear elliptic systems:the method of A-harmonic approximation[J].M anuscripta M athematica,2000,103(3):267-298.
[8]DUZAAR F,MINGIONE G.The p-harmonic approximation and the regularity of p-harmonic maps[J].Calculus of Variations and Partial Differential Equations,2004,20(3):235-256.
[9]DUZAAR F,MINGIONE G.Regularity for degenerate elliptic problems via p-harmonic approximation[J].Annales de l'Institut Henri Poincaré(C)Non,Analyse Linear,2004,21(5):735-766.
[10]YU Haiyan,ZHENG Shenzhou.Optimal partial regularity for quasilinear elliptic systems with VMO coefficients based on A-harmonic approximations[J].Electronic Journal of Differential Equations,2015,2015(16):1-12.
[11]CHEN Yazhe,WU Lancheng.Second order elliptic equations and elliptic systems[M].Rhode Island:American Mathematical Society,1998.
[12]KINNUNEN J,ZHOU Shulin.A local estimate for nonlinear equations with discontinuous coefficients[J].Communications in Partial Differential Equations,1999,24(11/12):2043-2068.