Interior W
1,p
Regularity and Hölder Continuity of Weak Solutions to a Class of Divergence Kolmogorov Equations with Discontinuous Coefficients |
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Authors: | Maochun Zhu Pengcheng Niu |
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Institution: | 1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi, 710129, China
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Abstract: | We consider a class of Kolmogorov equation $$Lu={\sum^{p_0}_{i,j=1}{\partial_{x_i}}(a_{ij}(z){\partial_{x_j}}u)}+{\sum^{N}_{i,j=1}b_{ij}x_{i}{\partial_{x_j}}u-{\partial_t}u}={\sum^{p_0}_{j=1}{\partial_{x_j}}F_{j}(z)}$$ in a bounded open domain ${\Omega \subset \mathbb{R}^{N+1}}$ , where the coefficients matrix (a ij (z)) is symmetric uniformly positive definite on ${\mathbb{R}^{p_0} (1 \leq p_0 < N)}$ . We obtain interior W 1,p (1 < p < ∞) regularity and Hölder continuity of weak solutions to the equation under the assumption that coefficients a ij (z) belong to the ${VMO_L\cap L^\infty}$ and ${({b_{ij}})_{N \times N}}$ is a constant matrix such that the frozen operator ${L_{z_0}}$ is hypoelliptic. |
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