On $L_p$-estimates for a class of non-local elliptic equations

2011

90 citations recorded by the source corpus.

We consider non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$, where $0<\sigma<2$ is a constant and $a$ is a bounded measurable function. By using a purely analytic method, we prove the continuity of the non-local operator $L$ from the Bessel potential space $H^\sigma_p$ to $L_p$, and the unique strong solvability of the corresponding non-local elliptic equations in $L_p$ spaces. As a byproduct, we also obtain interior $L_p$-estimates. The novelty of our results is that the function $a$ is not necessarily to be homogeneous, regular, or symmetric. An application of our result is the uniqueness for the martingale problem associated to the operator $L$.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b2b3c5572422939cd434a84b9996a85c46f8e21d

Citations in the corpus

• A relatively short proof of Itô’s formula for SPDEs and its applications (2012, unknown): “unded measurable m(y),y ∈Rd,and α∈(0,2), set for v ∈ S(Rd),x∈Rd, Lv(x) = Lαv(x) = Z ∇αv(x)m(y) dy |y|d+α . We will need the following continuity estimate (see [2] for a symmetric case, Theorem 2.1 in [5] for a general case using H¨older estimates, and Lemma 10 in [14] for a direct proof). Lemma 18 (”
• Interior regularity of solutions of non-local equations in Sobolev and Nikol’skii spaces (2016, unknown): “Concerning regularity results in Sobolev spaces, H2s estimates are proved in [DK12] for entire translation invariant equations.”
• DENSITY PROPERTIES FOR FRACTIONAL SOBOLEV SPACES (2015, unknown): “…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.”
• WEAK AND VISCOSITY SOLUTIONS OF THE FRACTIONAL LAPLACE EQUATION (2014, unknown): “Proof: The case of smooth solutions was considered in [7], so it is enough to reduce to this case by a standard convolution argument.”
• Schauder estimates for a class of non-local elliptic equations (2011, unknown): “Furthermore, while in [10] we dealt with K independent of x, we prove the same estimates for operators with x-dependent kernels under reasonable assumptions on a(x, y).”