EIGENVALUE ESTIMATES FOR HIGHER ORDER ELLIPTIC EQUATIONS

Higher Order Derivative Estimates for Finite-difference Schemes for Linear Elliptic and Parabolic Equations

Singular Sets of Higher Order Elliptic Equations

The implicit function theorem implies that the zero set of a smooth function, the set where the function vanishes, is a smooth hypersurface away from the critical zero set. Hence to study zero sets it is important to understand the structure of the critical zero sets. For solutions of the second order elliptic equations the critical zero sets represent the singular parts of zero sets. They have...

Elliptic Equations of Higher Stochastic Order

This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Itô-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of theWiener Chaos (Cameron-Martin) expansions. The exist...

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumption of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma devel...

Global Estimates for Mixed Methods for Second Order Elliptic Equations

Global error estimates in L2(Q), L°°(Q), and H~S(Q), Q in R2 or R3, are derived for a mixed finite element method for the Dirichlet problem for the elliptic operator Lp = -div(a grad p + bp) + cp based on the Raviart-Thomas-Nedelec space V^ X Wh c H(div; Í2) X L2(ü). Optimal order estimates are obtained for the approximation of p and the associated velocity field u = -(a grad p + bp) in L2(fl) ...