Mixed local and nonlocal elliptic operators: regularity and maximum principles

Nonlinear Maximum Principles for Dissipative Linear Nonlocal Operators and Applications

We obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative...

Pseudodifferential Operators and Elliptic Regularity

Theorem 1. Let M be a compact manifold, E and F be two (smooth) vector bundles over M , and P : C∞(M ;E) → C∞(M ;F ) be a differential operator (with smooth coefficients) of order n. Assume that P is elliptic, in the sense defined in the next section. Then: 1. The operator P (or rather its relevant extension) is Fredholm W n 2 (M ;E)→ L(M ;F ). Here W n 2 is a Sobolev space (introduced below). ...

Generalized Local Maximum Principles for Finite-Difference Operators

The generalized local maximum principle for a difference operator L» asserts that if Lam(jc) > 0 then Vu cannot attain its positive maximum at the net-point x. Here r is a local net-operator such that Tu = u + 0(/i) for any smooth function u. This principle, with simple forms of V, is proved for some quite general classes of second-order elliptic operators Lh, whose associated global matrices a...

Liouville Principles and a Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space

We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the regularity at the boundary: We consider problems posed on the halfspace with homogeneous Dirichlet boundary conditions and derive an associated C1,α-type large-scale ...

Elliptic operators and maximal regularity on periodic little-Hölder spaces