Schauder estimates for a class of non-local elliptic equations

Abstract

We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$ and either Dini or H\"older continuous data. Here $0 < \sigma < 2$ is a constant and $a$ is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz--Zygmund spaces $\Lambda^{\alpha+\sigma}$ and $\Lambda^\alpha$ for any $\alpha>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.Comment: final submitted version, 32 page

Weak and viscosity solutions of the fractional Laplace equation

“…As a consequence of this, we may apply [7,Corollary 4.3] and obtain that u ε ∈ C α (B 1/4 ) for any 0 < α < min{2s, 1} and…”

“…Proof: The case of smooth solutions was considered in [7], so it is enough to reduce to this case by a standard convolution argument. Namely, we take ρ ∈ C ∞ 0 (B 1 ; [0, 1]) and we consider the mollifier ρ ε (x) := ε −n ρ(x/ε), ε > 0.…”

Density properties for fractional Sobolev spaces

“…Motivations. Recently, in the literature a deep interest was shown for nonlocal operators, thanks to their intriguing analytical structure and in view of several applications in a wide range of contexts, such as the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science and water waves: see for instance [3,4,5,6,8,9,10,11,14,15,16,20,21,22,23,24,27,29,30,31,32,33] and references therein. One of the typical models considered is the equation may be defined as (1.2) −(−∆) s u(x) =ˆR n u(x + y) + u(x − y) − 2u(x) |y| n+2s dy for x ∈ R n (see [12,28] and references therein for further details on the fractional Laplacian) and the right-hand side f is a function satisfying suitable regularity and growth conditions.…”

On $$L_{p}$$ - theory for stochastic parabolic integro-differential equations

“…Theorem 5 below shows that the estimates of the main Theorem 3.6 in [8] are not sharp and the assumptions can be relaxed. Contrary to the case of a partial differential equation, in order to handle an equation with A (α) , it is not sufficient to consider an equation with fractional Laplacian like (5). Since only measurability of…”

“…We will need the following continuity estimate (see [2] for a symmetric case, Theorem 2.1 in [5] for a general case using Hö lder estimates, and Lemma 10 in [13] for a direct proof).…”