Local bounds, Harnack's inequality and Hölder continuity for divergence type elliptic equations with non-standard growth
We obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard p(x)-type growth. A model equation is the inhomogeneous p(x)-Laplacian. Namely, Δ<inf>p(x)</inf>u := div(|∇<inf>u</inf>|p(x)-2∇<inf>u</inf>) = f(x) in Ω fo...
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | JOUR |
| Materias: | |
| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00416932_v56_n1_p73_Wolanski |
| Aporte de: |
| Sumario: | We obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard p(x)-type growth. A model equation is the inhomogeneous p(x)-Laplacian. Namely, Δ<inf>p(x)</inf>u := div(|∇<inf>u</inf>|p(x)-2∇<inf>u</inf>) = f(x) in Ω for which we prove Harnack's inequality when f ∈ Lq<inf>0</inf> (Ω) if max {1, N/p<inf>1</inf>} < q<inf>0</inf> ≤ ∞. The constant in Harnack's inequality depends on u only through |||u|p(x)||p<inf>2</inf>-p<inf>1</inf><inf>L1(Ω)</inf>. Dependence of the constant on u is known to be necessary in the case of variable p(x). As in previous papers, log-Hölder continuity on the exponent p(x) is assumed. We also prove that weak solutions are locally bounded and Hölder continuous when f ∈ Lq<inf>0</inf>(x)(Ω) with q<inf>0</inf> ∈ c(Ω) and max{1,N/p(x)} < q<inf>0</inf>(x) in Ω. These results are then generalized to elliptic equations div A(x,u,∇<inf>u</inf>) = B(x,u,∇<inf>u</inf>) with p(x)-type growth. |
|---|