Schauder estimates for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and Hölder continuous in space

2022/05/20 by Stefano Biagi, Biagi, Stefano, Marco Bramanti +1
Mathematics · #Spectral Theory in Mathematical Physics #advanced mathematical theories #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2205.10270

Abstract

We consider degenerate Kolmogorov-Fokker-Planck operators Lu=∑_i,j=1^qa_ij(x,t)∂_x_ix_j²u+∑_k,j=1^Nb_jkx_k∂_x_ju-∂_tu, (x,t)∈ℝ^N+1,N≥ q≥1 such that the corresponding model operator having constant a_ij is hypoelliptic, translation invariant w.r.t. a Lie group operation in ℝ^N+1 and 2-homogeneous w.r.t. a family of nonisotropic dilations. The coefficients a_ij are bounded and Hölder continuous in space (w.r.t. some distance induced by L in ℝ^N) and only bounded measurable in time; the matrix \ a_ij\_i,j=1^q is symmetric and uniformly positive on ℝ^q. We prove "partial Schauder a priori estimates" the kind ∑_i,j=1^q\Vert∂_x_ix_j²u\Vert_C_x^α(S_T)+\Vert Yu\Vert_C_x^α(S_T)≤ c\ \VertLu\Vert _C_x^α(S_T)+\Vert u\Vert_C⁰(S_T)\ for suitable functions u, where \Vert f\Vert_C_x^α(S_T)=sup_{t≤ T}sup_x₁,x₂∈ℝ^N,x₁≠ x₂\frac\vert f( x₁,t) -f( x₂,t) \vert \Vert x₁-x₂\Vert ^α. We also prove that the derivatives ∂_x_ix_j²u are locally Hölder continuous in space and time while ∂_x_iu and u are globally Hölder continuous in space and time.

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