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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