Second order elliptic equations with mixed boundary conditions

Fractional Order Diierentiability of the Gradient of Solutions to Elliptic Equations with Mixed Boundary Conditions

Boundary Estimates for Positive Solutions to Second Order Elliptic Equations

Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which guarantee the Hopf-Oleinik type estimates and the boundary Lipschitz estimates for solutions. These conditions are sharp even for harmonic functions.

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) ...

On Second Order Elliptic and Parabolic Equations of Mixed Type

It is known that solutions to second order uniformly elliptic and parabolic equations, either in divergence or nondivergence (general) form, are Hölder continuous and satisfy the interior Harnack inequality. We show that even in the one-dimensional case (x ∈ R1), these properties are not preserved for equations of mixed divergence-nondivergence structure: for elliptic equations Di(a 1 ijDju) + ...

Eigenfunction Expansions for Second-Order Boundary Value Problems with Separated Boundary Conditions

In this paper, we investigate some properties of eigenvalues and eigenfunctions of boundary value problems with separated boundary conditions. Also, we obtain formal series solutions for some partial differential equations associated with the second order differential equation, and study necessary and sufficient conditions for the negative and positive eigenvalues of the boundary value problem....