Regularity estimates for nonlocal space-time master equations in bounded domains
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Journal of Evolution Equations
Regularity estimates for nonlocal space-time master equations in bounded domains Animesh Biswas and Pablo Raúl Stinga
Abstract. We obtain sharp parabolic interior and global Schauder estimates for solutions to nonlocal spacetime master equations (∂t + L)s u = f in R×, where L is an elliptic operator in divergence form, subject to homogeneous Dirichlet and Neumann boundary conditions. In particular, we establish the precise behavior of solutions near the boundary. Along the way, we prove a characterization of the correct intermediate parabolic Hölder spaces in the spirit of Sergio Campanato.
1. Introduction We study interior and global Schauder estimates for solutions u to nonlocal spacetime master equations H s u ≡ (∂t − div(A(x)∇x ))s u = f
in R ×
(1.1)
where f = f (t, x) : R × → R is a given datum and ⊂ Rn , n ≥ 1, is a bounded Lipschitz domain. Here, H s = (∂t − div(A(x)∇x ))s is the fractional power of order 0 < s < 1 of the parabolic operator H = ∂t − div(A(x)∇x ). The coefficients in (1.1) are symmetric A(x) = (Ai j (x)) = (A ji (x)) for i, j = 1, . . . , n, bounded and measurable and satisfy the uniform ellipticity condition 1 |ξ |2 ≤ A(x)ξ ·ξ ≤ 2 |ξ |2 , for all ξ ∈ Rn and almost every x ∈ , for some ellipticity constants 0 < 1 ≤ 2 . The problem is subject to either homogeneous Dirichlet or Neumann boundary conditions, that is, u=0
or
∂ A u ≡ A(x)∇x u · ν = 0
on R × ∂
where ν is the exterior unit normal to ∂. Master equations as in (1.1) arise in several different physical applications such as the phenomenon of osmosis in semipermeable membranes, in diffusion models for Mathematics Subject Classification: Primary 35R11, 35B65, 35K65; Secondary 35R09, 46E35, 58J35 Keywords: Master equations, Hölder regularity, Extension problem, Compactness, Parabolic Hölder spaces. The authors were partially supported by Simons Foundation Grant 580911 and by Grant MTM2015-66157C2-1-P (MINECO/FEDER) from Government of Spain.
J. Evol. Equ.
A. Biswas and P. R. Stinga
biological invasions, in financial mathematics, in the Signorini problem of elasticity in heterogeneous materials and also in probability, among others, see, for instance, [1,3,5,6,9,18,27] and references therein. All these phenomena are governed by a master equation given in generalized form as ∞ (u(t − τ, z) − u(t, x))K (t, x, τ, z) dτ dz = f (t, x) (1.2) Rn
0
for t ∈ R and x ∈ Rn , for some kernel K . In terms of regularity, Caffarelli and Silvestre proved Hölder estimates of viscosity solutions to (1.2) with bounded right-hand side, see [9]. They assumed conditions on the kernel K that ensure that (1.2) is an equation of fractional order s in time and 2s in space. On the other hand, in [27], Stinga and Torrea studied the problem (∂t −)s u = f , for 0 < s < 1, which is the most basic form of a master equation. The systematic study of weak solutions to master equations as in (1.1) was initiated in [7], where a precise definition of the fractional power operator H s is given. In particular, in [7],
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