The aim of the paper is to study properties of solutions to the nonlinear fractional subLaplace equations on the Heisenberg group. Based on the method of moving planes to the Heisenberg group, we prove the Liouville p...The aim of the paper is to study properties of solutions to the nonlinear fractional subLaplace equations on the Heisenberg group. Based on the method of moving planes to the Heisenberg group, we prove the Liouville property of solutions on a half space and the symmetry and monotonicity of the solutions on the whole group respectively.展开更多
In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity...In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity for weak solutions.After deriving the parabolic Sobolev inequality,(1,1)type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma,the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Hölder space.As a consequence,we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.展开更多
In this paper we prove some Liouville type results for the p-sub-Laplacian on the group of Heisenberg type. A strong maximum principle and a Hopf type principle concerning p-sub-Laplacian are established.
基金National Natural Science Foundation of China(Grant No.11771354)and the National Natural Science Basic Research plan in Shaanxi Province of China(Grant No.2016JM1023).
文摘The aim of the paper is to study properties of solutions to the nonlinear fractional subLaplace equations on the Heisenberg group. Based on the method of moving planes to the Heisenberg group, we prove the Liouville property of solutions on a half space and the symmetry and monotonicity of the solutions on the whole group respectively.
基金This work is supported by the National Natural Science Foundation of China(No.1177-1354).
文摘In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity for weak solutions.After deriving the parabolic Sobolev inequality,(1,1)type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma,the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Hölder space.As a consequence,we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.
文摘In this paper we prove some Liouville type results for the p-sub-Laplacian on the group of Heisenberg type. A strong maximum principle and a Hopf type principle concerning p-sub-Laplacian are established.
