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We study a class of linear parabolic equations in divergence form with degenerate coefficients on the upper half space. Specifically, the equations are considered in $(-infty, T) times mathbb{R}^d_+$, where $mathbb{R}^d_+ = {x in mathbb{R}^d,:, x_d>0}$ and $Tin {(-infty, infty]}$ is given, and the diffusion matrices are the product of $x_d$ and bounded uniformly elliptic matrices, which are degenerate at ${x_d=0}$. As such, our class of equations resembles well the corresponding class of degenerate viscous Hamilton-Jacobi equations. We obtain wellposedness results and regularity type estimates in some appropriate weighted Sobolev spaces for the solutions.
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We study a class of second-order degenerate linear parabolic equations in divergence form in $(-infty, T) times mathbb R^d_+$ with homogeneous Dirichlet boundary condition on $(-infty, T) times partial mathbb R^d_+$, where $mathbb R^d_+ = {x in mathbb R^d,:, x_d>0}$ and $Tin {(-infty, infty]}$ is given. The coefficient matrices of the equations are the product of $mu(x_d)$ and bounded uniformly elliptic matrices, where $mu(x_d)$ behaves like $x_d^alpha$ for some given $alpha in (0,2)$, which are degenerate on the boundary ${x_d=0}$ of the domain. Under a partially VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.
In this paper, we study both elliptic and parabolic equations in non-divergence form with singular degenerate coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under some suitable partially weighted BMO regularity conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional Laplace equations. Our results are new even in this setting and in the unmixed case. For the proof, we establish both interior and boundary Lipschitz estimates for solutions and for higher order derivatives of solutions to homogeneous equations. We then employ the perturbation method by using the Fefferman-Stein sharp function theorem, the Hardy-Littlewood maximum function theorem, as well as a weighted Hardys inequality.
In this paper, we establish $L_p$ estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean oscillations with respect to the other variables. The corresponding results for equations on a half space are also derived.
We study both divergence and non-divergence form parabolic and elliptic equations in the half space ${x_d>0}$ whose coefficients are the product of $x_d^alpha$ and uniformly nondegenerate bounded measurable matrix-valued functions, where $alpha in (-1, infty)$. As such, the coefficients are singular or degenerate near the boundary of the half space. For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the $x_d$ direction and have small mean oscillation in the other directions in small cylinders. Our results are new even in the special case when the coefficients are constants, and they are reduced to the classical results when $alpha =0$
In this paper, we study parabolic equations in divergence form with coefficients that are singular degenerate as some Muckenhoupt weight functions in one spatial variable. Under certain conditions, weighted reverse H{o}lders inequalities are established. Lipschitz estimates for weak solutions are proved for homogeneous equations with singular degenerate coefficients depending only on one spatial variable. These estimates are then used to establish interior, boundary, and global weighted estimates of Calder{o}n-Zygmund type for weak solutions, assuming that the coefficients are partially VMO (vanishing mean oscillations) with respect to the considered weights. The solvability in weighted Sobolev spaces is also achieved. Our results are new even for elliptic equations, and non-trivially extend known results for uniformly elliptic and parabolic equations. The results are also useful in the study of fractional elliptic and parabolic equations with measurable coefficients.
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