2020/02/23 by Hao, Zimo, Wang, Zhen, Wu, Mingyan
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2002.09887
In this paper, we establish Schauder's estimates for the following non-local equations in \mRd : ∂tu=\mathscr L(α)κ,σ u+b⋅∇ u+f, u(0)=0, where α∈(1/2,2) and b:\mathbb R+×\mathbb Rd→\mathbb R is an unbounded local β-order Hölder function in x uniformly in t , and \mathscr L(α)κ,σ is a non-local α-stable-like operator with form: \mathscr L(α)κ,σu(t,x):=∫\mathbb Rd(u(t,x+σ(t,x)z)-u(t,x)-σ(t,x)z(α)⋅∇ u(t,x))κ(t,x,z)ν(α)(\mathord\rm d z), where z(α)=z1α∈(1,2)+z1|z|≤ 11α=1, κ:\mathbb R+×\mathbb R2d→\mathbb R+ is bounded from above and below, σ:\mathbb R+×\mathbb Rd→ \mathbb Rd⊗ \mathbb Rd is a γ-order Hölder continuous function in x uniformly in t , and ν(α) is a singular non-degenerate α-stable Lévy measure.