In this paper, we propose and analyze a mixed $H^1$-conforming finite element method for solving Maxwell's equations in terms of electric field and Lagrange multiplier, where the multiplier is introduced accounting for the divergence constraint. We mainly focus on the case that the physical domain is non-convex and its boundary includes reentrant corners or edges, which may lead the solution of Maxwell's equations to be a non-$H^1$ very weak function and thus causes many numerical difficulties. The proposed method is formulated in the stabilized form by adding an additional mesh-dependent stabilization term to the mixed variational formulation. A general framework of stability and error analysis is established. Specifically, a pair of $H^1$-conforming finite elements, namely the $CP_2$-$P_1$ elements, for electric field and multiplier is studied and its stability and error bounds are also derived. Numerical experiments for source problems as well as eigenvalue problems on the $L$-shaped and cracked domains are presented to illustrate the high performance of the proposed mixed $H^1$-conforming finite element method.

In this paper, we derive the CR Reilly'’s formula and its applications to studying of the fi…rst eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the …first Dirichlet eigenvalue estimate in a compact pseudo-hermitian manifold with boundary and the fi…rst eigenvalue estimate of the tangential sublaplacian on closed oriented embedded p-minimal hypersurfaces in a closed pseudohermitian manifold of vanishing torsion.

In this paper, we consider the heat ‡flow for p-pseudoharmonic maps from a closed Sasakian manifold into a compact Riemannian manifold. We prove global existence and asymptotic convergence of the solution for the p-pseudoharmonic map heat ‡flow provided that the sectional curvature of the target manifold is nonpositive. Moreover, without the curvature assumption on the target manifold, we obtain global existence and asymptotic convergence of the p-pseudoharmonic map heat fl‡ow as well when its initial p-energy is sufficiently small.

In this paper, we generalize the Cao-Yau'’s gradient estimate for the sum of squares of vector …elds up to higher step under ssumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel. 1.

For every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field is topologically dense in the set of its points with coordinates in the topological closure of this subgroup in the product of the multiplicative group of those local completions of this function field over all but finitely many places.