Regularity theory for general stable operators: parabolic equations
By Xavier Fernández-Real, Xavier Ros-Oton. 2015
89 citations recorded by the source corpus.
We establish sharp interior and boundary regularity estimates for solutions to $\partial_t u - L u = f(t, x)$ in $I\times \Omega$, with $I \subset \mathbb{R}$ and $\Omega \subset\mathbb{R}^n$. The operators $L$ we consider are infinitessimal generators of stable L\'evy processes. These are linear nonlocal operators with kernels that may be very singular. On the one hand, we establish interior estimates, obtaining that $u$ is $C^{2s+\alpha}$ in $x$ and $C^{1+\frac{\alpha}{2s}}$ in $t$, whenever $f$ is $C^{\alpha}$ in $x$ and $C^{\frac{\alpha}{2s}}$ in $t$. In the case $f\in L^\infty$, we prove that $u$ is $C^{2s-\epsilon}$ in $x$ and $C^{1-\epsilon}$ in $t$, for any $\epsilon > 0$. On the other hand, we study the boundary regularity of solutions in $C^{1,1}$ domains. We prove that for solutions $u$ to the Dirichlet problem the quotient $u/d^s$ is H\"older continuous in space and time up to the boundary $\partial\Omega$, where $d$ is the distance to $\partial\Omega$. This is new even when $L$ is the fractional Laplacian.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/5130494f3d2ebeaee51c51f9fe9c84b128c2f250
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In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Asymptotic method of moving planes for fractional parabolic equations (2020, background): “By standard parabolic regularity estimates [23], we infer that there is a subsequence of wk(x, t) (still denoted by wk(x, t)) which converges uniformly to a function w∞(x, t) in Σλ × [0, 2] and ∂wk ∂t (x, t) + (−∆)wk(x, t) → ∂w∞ ∂t (x, t) + (−∆)w∞(x, t), ck(x, t) → c∞(x, t), as k → ∞.”
- Liouville Theorems for Fractional Parabolic Equations (2021, unknown): “7) Since u(x, t) is uniformly bounded, by the regularity estimates for the fractional parabolic equations ([17]), up to a subsequence (still denoted by u), as k → +∞, we have u(x, t) → ũ(x, t), (−∆)u(x, t) → (−∆)ũ(x, t), and w λk(x, t) → w̃λ0(x, t) = ũλ0(x, t)− ũ(x, t) ≥ 0, (x, t) ∈ Σλ0 × R.”