Boundedness of non-local operators with spatially dependent coefficients and $$L_p$$-estimates for non-local equations

Novel Inverse Estimates for Non-Local Operators

Local inverse estimates for non-local boundary integral operators

We prove local inverse-type estimates for the four non-local boundary integral operators associated with the Laplace operator on a bounded Lipschitz domain Ω in R for d ≥ 2 with piecewise smooth boundary. For piecewise polynomial ansatz spaces and d ∈ {2, 3}, the inverse estimates are explicit in both the local mesh width and the approximation order. An application to efficiency estimates in a ...

Non-local Gradient Dependent Operators

In this paper we study a general class of “quasilinear non-local equations” depending on the gradient which arise from tug-of-war games. We establish a C/C/C regularity theory for these equations (the kind of regularity depending on the assumptions on the kernel), and we construct different non-local approximations of the p-Laplacian.

Universal estimates for parabolic equations and applications for non-linear and non-local problems

We obtain some ”universal” estimates for L2-norm of the solution of a parabolic equation via a weighted version of H-norm of the free term. More precisely, we found the limit upper estimate that can be achieved by transformation of the equation by adding a constant to the zero order coefficient. The inverse matrix of the higher order coefficients of the parabolic equation is included into the w...

Perturbation by Non-Local Operators

Suppose that d ≥ 1 and 0 < β < α < 2. We establish the existence and uniqueness of the fundamental solution q(t, x, y) to a class of (possibly nonsymmetric) non-local operators L = ∆ + S, where Sf(x) := A(d,−β) ∫ R ( f(x+ z)− f(x)−∇f(x) · z1{|z|≤1} ) b(x, z) |z|d+β dz and b(x, z) is a bounded measurable function on R×R with b(x, z) = b(x,−z) for x, z ∈ R. Here A(d,−β) is a normalizing constant ...