Methods are presented for the construction of nondecomposable positive definite integral Hermitian forms over the ring of integers Rm of an imaginary quadratic field ?(√?m). Using our methods, one can construct expli...Methods are presented for the construction of nondecomposable positive definite integral Hermitian forms over the ring of integers Rm of an imaginary quadratic field ?(√?m). Using our methods, one can construct explicitly an n-ary nondecomposable positive definite Hermitian Rm-lattice ( L, h) with given discriminant 2 for every n?2 (resp. n?13 or odd n?3) and square-free m = 12 k + t with k?1 and t∈ (1,7) (resp. k?1 and t = 2 or k?0 and t∈ 5,10,11). We study also the case for discriminant different from 2.展开更多
We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space：where rn is any positive integer satisfying 0 〈 2m 〈 n. We first prove ...We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space：where rn is any positive integer satisfying 0 〈 2m 〈 n. We first prove that the positive solutions of （0.1） are super polyharmonic, i.e.,where x＊ = （x1,... ,Xn-1, --Xn） is the reflection of the point x about the plane Rn-1. Then, we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of （0.3）, in which α can be any real number between 0 and n. By some Pohozaev type identities in integral forms, we prove a Liouville type theorem--the non-existence of positive solutions for （0.1）.展开更多
文摘Methods are presented for the construction of nondecomposable positive definite integral Hermitian forms over the ring of integers Rm of an imaginary quadratic field ?(√?m). Using our methods, one can construct explicitly an n-ary nondecomposable positive definite Hermitian Rm-lattice ( L, h) with given discriminant 2 for every n?2 (resp. n?13 or odd n?3) and square-free m = 12 k + t with k?1 and t∈ (1,7) (resp. k?1 and t = 2 or k?0 and t∈ 5,10,11). We study also the case for discriminant different from 2.
文摘We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space：where rn is any positive integer satisfying 0 〈 2m 〈 n. We first prove that the positive solutions of （0.1） are super polyharmonic, i.e.,where x＊ = （x1,... ,Xn-1, --Xn） is the reflection of the point x about the plane Rn-1. Then, we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of （0.3）, in which α can be any real number between 0 and n. By some Pohozaev type identities in integral forms, we prove a Liouville type theorem--the non-existence of positive solutions for （0.1）.
