A new weight class and Poincaré inequalities with the Radon measure. (English) Zbl 1279.26038
Summary: We first introduce and study a new family of weights, the \(A(\alpha, \beta, \gamma, E)\)-class which contains the well-known \(A_r(E)\)-weight as a proper subset. Then, as applications of the \(A(\alpha,\beta, \gamma;E)\)-class, we prove the local and global Poincaré inequalities with the Radon measure for the solutions of the non-homogeneous \(A\)-harmonic equation which belongs to a kind of the nonlinear partial differential equations.
MSC:
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 35J60 | Nonlinear elliptic equations |
| 31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |
| 58A10 | Differential forms in global analysis |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
References:
| [1] | doi:10.1016/j.jmaa.2007.05.078 · Zbl 1138.47037 · doi:10.1016/j.jmaa.2007.05.078 |
| [2] | doi:10.1006/jmaa.2000.6851 · Zbl 0959.58002 · doi:10.1006/jmaa.2000.6851 |
| [3] | doi:10.1016/S0022-247X(02)00331-1 · Zbl 1035.46024 · doi:10.1016/S0022-247X(02)00331-1 |
| [4] | doi:10.1155/S0161171202107046 · Zbl 1014.30014 · doi:10.1155/S0161171202107046 |
| [5] | doi:10.1016/j.camwa.2004.06.006 · Zbl 1155.31303 · doi:10.1016/j.camwa.2004.06.006 |
| [6] | doi:10.1016/S0022-247X(03)00036-2 · Zbl 1021.31004 · doi:10.1016/S0022-247X(03)00036-2 |
| [7] | doi:10.1006/jmaa.1998.6096 · Zbl 0918.26013 · doi:10.1006/jmaa.1998.6096 |
| [8] | doi:10.1016/S0022-247X(03)00216-6 · Zbl 1027.30053 · doi:10.1016/S0022-247X(03)00216-6 |
| [9] | doi:10.1007/BF00411477 · Zbl 0793.58002 · doi:10.1007/BF00411477 |
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