On second-order periodic elliptic operators in divergence form

On the Spectrum of a Class of Second Order Periodic Elliptic Differential Operators

Periodic homogenization of Dirichlet problem for divergence type elliptic operators

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Optimal Sobolev Regularity for Linear Second-Order Divergence Elliptic Operators Occurring in Real-World Problems

On bounded domains Ω ⊂ R, we consider divergence-type operators −∇ · μ∇, including mixed homogeneous Dirichlet and Neumann boundary conditions on ∂Ω \ Γ and Γ ⊂ ∂Ω, respectively, and discontinuous coefficient functions μ. We develop a general geometric framework for Ω, Γ and μ in which it is possible to prove that −∇ · μ∇ + 1 provides an isomorphism from W 1,q Γ (Ω) to W Γ (Ω) for some q > 3. W...

Maximum Principles for a Class of Nonlinear Second-order Elliptic Boundary Value Problems in Divergence Form

For a class of nonlinear elliptic boundary value problems in divergence form, we construct some general elliptic inequalities for appropriate combinations of u(x) and |∇u|2, where u(x) are the solutions of our problems. From these inequalities, we derive, using Hopf ’s maximum principles, some maximum principles for the appropriate combinations of u(x) and |∇u|2, and we list a few examples of p...

Hardy and BMO spaces associated to divergence form elliptic operators

Consider a second order divergence form elliptic operator L with complex bounded coefficients. In general, operators related to it (such as the Riesz transform or square function) lie beyond the scope of the Calderón-Zygmund theory. They need not be bounded in the classical Hardy, BMO and even some Lp spaces. In this work we develop a theory of Hardy and BMO spaces associated to L, which includ...