Higher gradients estimates in Morrey spaces for weak solutions to linear ultraparabolic equations

2014/04/25 by Yan Dong, Dong, Yan, Pengcheng Niu +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.1404.6428

Abstract

The aim of this paper is to consider the linear ultraparabolic equation with bounded and VMO coefficients a_ij (z). Assume that the operator L₀ obtained by freezing the coefficients a_ij(z) at any point z₀ ∈ ℝ^{N + 1} is hypoelliptic. We first establish a Caccioppoli type inequality by choosing a cutoff function, a Sobolev type inequality by prosperities of the fundamental solution to L₀, and a Poincaré type inequality with a new cutoff function. Then L^p estimate for weak solutions is derived by using the reverse Hölder inequality on homogeneous spaces. Finally, higher Morrey estimates for weak solutions to the above equation are shown by investigating a homogeneous ultraparabolic equation of variable coefficients with a nonhomogeneous boundary value condition, and a nonhomogeneous ultraparabolic equation of variable coefficients with homogeneous boundary value condition.

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