Analysis and Geometry in Metric Spaces (Sep 2024)
Schauder estimates on bounded domains for KFP operators with coefficients measurable in time and Hölder continuous in space
Abstract
We consider degenerate Kolmogorov-Fokker-Planck operators ℒu=∑i,j=1qaij(x,t)uxixj+∑k,j=1Nbjkxkuxj−ut,{\mathcal{ {\mathcal L} }}u=\mathop{\sum }\limits_{i,j=1}^{q}{a}_{ij}\left(x,t){u}_{{x}_{i}{x}_{j}}+\mathop{\sum }\limits_{k,j=1}^{N}{b}_{jk}{x}_{k}{u}_{{x}_{j}}-{u}_{t}, such that the corresponding model operator having constant aij{a}_{ij} is hypoelliptic, translation invariant w.r.t. a Lie group operation in RN+1{{\mathbb{R}}}^{N+1} and 2-homogeneous w.r.t. a family of nonisotropic dilations. We assume that the aij{a}_{ij}’s are globally bounded and Hölder continuous in xx (w.r.t. some intrinsic distance induced by ℒ{\mathcal{ {\mathcal L} }}); the matrix {aij}i,j=1q{\{{a}_{ij}\}}_{i,j=1}^{q} is symmetric and uniformly positive on Rq{{\mathbb{R}}}^{q}. We prove “partial Schauder a priori estimates” on a bounded open set Ω⊂RN+1\Omega \subset {{\mathbb{R}}}^{N+1}, of the kind ∑i,j=1q([uxixj]∗,α,2x+[uxixj]∗,2)≤c{[ℒu]∗,α,2x+[ℒu]∗,2+supΩ∣u∣}\mathop{\sum }\limits_{i,j=1}^{q}({\left[{u}_{{x}_{i}{x}_{j}}]}_{\ast ,\alpha ,2}^{x}+{\left[{u}_{{x}_{i}{x}_{j}}]}_{\ast ,2})\le c\left\{{\left[{\mathcal{ {\mathcal L} }}u]}_{\ast ,\alpha ,2}^{x}+{\left[{\mathcal{ {\mathcal L} }}u]}_{\ast ,2}+\mathop{\sup }\limits_{\Omega }| u| \right\} for suitable functions uu, where [u]∗,α,2x=sup(x,t)≠(y,t)∈Ωd(x,t),(y,t)2+α∣u(x,t)−u(y,t)∣‖x−y‖α[u]∗,2=supξ∈Ωdξ2∣u(ξ)∣.\begin{array}{rcl}{\left[u]}_{\ast ,\alpha ,2}^{x}& =& \mathop{\sup }\limits_{\left(x,t)\ne (y,t)\in \Omega }{d}_{\left(x,t),(y,t)}^{2+\alpha }\frac{| u\left(x,t)-u(y,t)| }{\Vert x-y{\Vert }^{\alpha }}\\ {{[}u]}_{\ast ,2}& =& \mathop{\sup }\limits_{\xi \in \Omega }{d}_{\xi }^{2}| u\left(\xi )| .\end{array} Here ‖⋅‖\Vert \cdot \Vert is a homogeneous norm in RN{{\mathbb{R}}}^{N}, while dξ=dist(ξ,∂Ω){d}_{\xi }={\rm{dist}}\left(\xi ,\partial \Omega ) and dξ,η=min{dξ,dη}{d}_{\xi ,\eta }=\min \left\{{d}_{\xi },{d}_{\eta }\right\}. We also prove that the derivatives uxixj{u}_{{x}_{i}{x}_{j}} are locally Hölder continuous in space and time.
Keywords
