Calculus of Variations and Geometric Measure Theory

S. Biagi - M. Bramanti - B. Stroffolini

KFP operators with coefficients measurable in time and Dini continuous in space

created by stroffolini on 25 May 2023



Inserted: 25 may 2023
Last Updated: 25 may 2023

Year: 2023

ArXiv: 2305.11641 PDF


We consider degenerate KFP operators \[ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)u_{x_{i}x_{j}}+\sum_{k,j=1}^{N} b_{jk}x_{k}u_{x_{j}}-u_{t}, \] $(x,t)\in\mathbb{R}^{N+1}$, $1\leq q\leq N$, such that the model operator having constant $a_{ij}$ is hypoelliptic, translation invariant w.r.t. a Lie group in $\mathbb{R}^{N+1}$ and $2$-homogeneous w.r.t. a family of nonisotropic dilations. The matrix $(a_{ij})_{i,j=1}^{q}$ is symmetric and uniformly positive on $\mathbb{R}^{q} $. The $a_{ij}$ are bounded and Dini continuous in space, bounded measurable in time, i.e., letting $S_{T}=\mathbb{R}^{N}\times\left( -\infty,T\right) $, \[ \omega_{f,S_{T}}\left( r\right) =\sup_{\left( x,t\right) ,\left( y,t\right) \in S_{T}}\left\vert f\left( x,t\right) -f\left( y,t\right) \right\vert \] \[ \left\Vert f\right\Vert _{D\left( S_{T}\right) }=\int_{0}^{1}\frac {\omega_{f,S_{T}}\left( r\right) }{r}dr+\left\Vert f\right\Vert _{L^{\infty }\left( S_{T}\right) }% \] we require the finiteness of $\left\Vert a_{ij}\right\Vert _{D\left( S_{T}\right) }$. We bound $\omega_{u_{x_{i}x_{j}},S_{T}}$, $\left\Vert u_{x_{i}x_{j}}\right\Vert _{L^{\infty}\left( S_{T}\right) }$, $\omega _{Yu,S_{T}}$, $\left\Vert Yu\right\Vert _{L^{\infty}\left( S_{T}\right) }$ in terms of $\omega_{\mathcal{L}u,S_{T}}$, $\Vert\mathcal{L}u\Vert_{L^{\infty }\left( S_{T}\right) }$ and $\Vert u\Vert_{L^{\infty}\left( S_{T}\right) }$, getting a control on the uniform continuity in space of $u_{x_{i}x_{j}}$ and $Yu$ if $\mathcal{L}u$ is partially Dini-continuous. Moreover, if both $a_{ij}$ and $\mathcal{L}u$ are log-Dini continuous, we prove that $u_{x_{i}x_{j}}$ and $Yu$ are Dini continuous; moreover, in this case, the derivatives $u_{x_{i}x_{j}}$ are locally uniformly continuous in space and time.