In this paper, we study fully non-linear elliptic equations in nondivergence form which can be degenerate when “the gradient is small”. Typical examples are either equations involving the m-Laplace operator or BellmanIsaacs equations from stochastic control problems. We establish an AlexandroffBakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such degenerate e...

We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-λ method that does not use any size or smoothness estimates for the kernels.

We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-λ method that does not use any size or smoothness estimates for the kernels.

The subject of the paper is the mesh independent convergence of the preconditioned conjugate gradient method for nonsymmetric elliptic problems. The approach of equivalent operators is involved, in which one uses the discretization of another suitable elliptic operator as preconditioning matrix. By introducing the notion of compact-equivalent operators, it is proved that for a wide class of ell...

Rjk = R̄jk + ∇̄mD jk − ∇̄jD mk −Dn jkD nm −Dn mkD nj . (To see why this is true, just compute in local normal coordinates for ḡ!) From now on all components are tensorial (as opposed to local coordinate components.) The terms involving second covariant derivatives of the metric are: 1 2 g(−∇̄m∇̄lgjk +∇̄m∇̄kgjl +∇̄m∇̄jglk +∇̄j∇̄lgmk−∇̄j∇̄kgml−∇̄j∇̄mglk). The first term is a kind of Laplacian. The third and six...

Finite element or finite difference approximations to linear partial differential equations of elliptic type lead to algebraic systems, normally of very large size. However, since the matrices are sparse, even extremely large-sized systems can be handled. To save computer memory and elapsed time, such equations are normally solved by iteration, most commonly using a preconditioned conjugate gra...

Stephan Dahlke a, Massimo Fornasier b and Thorsten Raasch a a Philipps-Universität Marburg, FB 12 Mathematik und Informatik, Hans-Meerwein Straße, Lahnberge, 35032 Marburg, Germany E-mail: {dahlke;raasch}@mathematik.uni-marburg.de b Università “La Sapienza” in Roma, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Via Antonio Scarpa, 16/B, 00161 Roma, Italy E-mail: mfornasi...