Identities And Inequalities Of Anisotropic Laplace Operator And Its Applications

With the development of science and technology,more and more attention is paid on the anisotropic elliptic equations,which describe the dynamics of fluids in the anisotropic media when the conductivities of the media are different in each direction.This paper is devoted to the study of some identities and inequalities of the anisotropic Laplace operator.In the first part,we establish anisotropic Pohozaev identities on the bounded smooth domain and the whole space for the solutions to the anisotropic elliptic problems respectively.Nonexistence of nontrivial solutions are proved by using these identities.In the second part,we obtain an anisotropic Picone identity for the anisotrop-ic Laplace operator;as applications,a Sturmian comparison principle to the anisotropic elliptic equation and an anisotropic Hardy inequality are deriveded.We also obtain a nonlinear Picone identity to the pseudo p-Laplace operator;some applications are given including a Liouville theorem to the singular pseudo p-Laplace system,a Sturmian comparison principle to the pseudo p-Laplace equa-tion,a new Hardy inequality with weight and remainder term,and nonexistence of positive super-solutions to the pseudo p-Laplace equation.In the third part,an anisotropic Caccioppoli inequality of the weak solutions to the homogeneous anisotropic elliptic problem on the cube is concluded,which can be regarded as a reverse anisotropic Poincar?e inequality.The logarithmic anisotropic Sobolev inequality is proved by using the anisotropic Sobolev inequal-ity,and the logarithmic weighted anisotropic Sobolev inequality is also obtained.In the fourth part,we consider the weighted anisotropic integral functional by using the weighted anisotropic Sobolev inequality and the iteration lemma,and prove that the higher integrability for the minimizers when the boundary datum has the higher integrability.We also investigate the boundednesses of exponential form and L~?(?)form of the minimizers.Furthermore,the similar results for the minimizers of the obstacle problem to the weighted anisotropic integral functional are given as well as.In the final part,we concern an anisotropic elliptic problem by using the vari-ational method and prove existence of nontrivial weak solutions and nonexistence of nontrivial weak solutions by combining an anisotropic Poincar?e inequality.