On $C^{1/2,1}$, $C^{1,2}$, and $C^{0,0}$ estimates for linear parabolic operators

Abstract

We show that weak solutions to parabolic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain conditions. Similar results are obtained for non-divergence form parabolic operators and their adjoint operators. Under similar conditions, we also establish a Harnack inequality for nonnegative adjoint solutions, together with upper and lower Gaussian bounds for the global fundamental solution.

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Cited by in corpus (3)

• Estimates for fundamental solutions of parabolic equations in non-divergence form
• Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients
• Two-sided Gaussian estimates for fundamental solutions of second-order parabolic equations in non-divergence form