The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to Hörmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We first prove L p (p ≥ 2) estimates for gradients of weak solutions by using a priori estimates and a known reverse Hölder inequality, and consider regularity to the corresponding nondiagonal homogeneous degenerate elliptic systems. Then we get higher Morrey and Campanato estimates for gradients of weak solutions to original systems and Hölder estimates for weak solutions.Regularity for solutions to elliptic systems in Euclidean spaces has been studied by many authors and a lot of important conclusions were got. Campanato in [2] obtained gradient estimates for weak solutions to linear elliptic system with discontinuous coefficients.For related articles, we quote [1,14] and the references therein. Huang in [18] derived Morrey estimates for uniformly elliptic systems by applying Campanato's technique. Zheng and Feng in [28] established Hölder estimates for weak solutions to quasilinear elliptic systems by reverse Hölder inequality and Dirichlet growth theorem, where the coefficients belong to L ∞ ∩ V MO, and low terms satisfy controlled

In this paper, we establish gradient estimates in Morrey spaces and Hölder continuity for weak solutions of the following degenerate elliptic systemwhere X 1 , · · · , X q are real smooth vector fields satisfying Hörmander's condition, coefficients a αβ i j ∈ V M O X ∩ L ∞ ( ), α, β = 1, 2, · · · , q, i, j = 1, 2, · · · , N , X * α is the transposed vector field of X α .

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In this paper, we study the degenerate parabolic variational inequality problem in a bounded domain. First, the weak solutions of the variational inequality are defined. Second, the existence and uniqueness of the solutions in the weak sense are proved by using the penalty method and the reduction method.

In this paper, we discuss the variation of the numbers of the isomorphic classes of stable lattices when the weight and the level vary in a Hida deformation by using the Kubota-Leopoldt p-adic L-function. Then in Corollary 1.7, we give a sufficient condition for the numbers of the isomorphic classes of stable lattices in Hida deformation to be infinite.is the canonical character and−1 the Teichmüller character. Let O ⊂ Q p be a commutative ring which is finite flat over Z p and let ψ be a Dirichlet character modulo M . We denote by S k (Γ 0 (M ) , ψ, O) the space of cusp forms of weight k, level M , Neben character ψ and Fourier coefficients in O. We also denote by the same symbol ψ the corresponding character of Gal (Q(µ M )/Q) ∼ = (Z/M ) × . For a group ∆ M which is isomorphic to Gal (Q(µ M )/Q), a Z p -module M which has a Z p -linear action of ∆ M and a character ε of ∆ M , we denote by M ε = M ⊗ Zp[∆M ]