Related papers: Regularity theory for parabolic equations with sin…

This paper studies the Sobolev regularity estimates for weak solutions of a class of degenerate, and singular quasi-linear elliptic problems of the form $\text{div}[\mathbf{A}(x,u, \nabla u)]= \text{div}[\mathbf{F}]$ with non-homogeneous…

This paper studies the Sobolev regularity of weak solution of degenerate elliptic equations in divergence form $\text{div}[\mathbf{A}(X) \nabla u] = \text{div}[\mathbf{F}(X)]$, where $X = (x,y) \in \mathbb{R}^{n} \times \mathbb{R}$ . The…

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…

This paper studies a class of linear parabolic equations with measurable coefficients in divergence form whose volumetric heat capacity coefficients are assumed to be in some Muckenhoupt class of weights. As such, the coefficients can be…

This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - \mbox{div}[\mathbb{A}(x,t,u,\nabla u)]= \mbox{div}[{\mathbf F}]$ with homogeneous Dirichlet…

We investigate the inhomogeneous boundary value problem for elliptic and parabolic equations in divergence form in the half space $\{x_d > 0\}$, where the coefficients are measurable, singular or degenerate, and depend only on $x_d$. The…

We investigate weighted Sobolev regularity of weak solutions of non-homogeneous parabolic equations with singular divergence-free drifts. Assuming that the drifts satisfy some mild regularity conditions, we establish local weighted…

We study a class of degenerate parabolic and elliptic equations in divergence form in the upper half space $\{x_d>0\}$. The leading coefficients are of the form $x_d^2a_{ij}$, where $a_{ij}$ are bounded, uniformly elliptic, and measurable…

We study a class of parabolic equations in non-divergence form with measurable coefficients that exhibit singular and/or degenerate behavior governed by weights in the $A_{1+\frac{1}{n}}$-Muckenhoupt class. Under a smallness assumption on a…

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…

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…

This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate as a weight belonging to the $A_{1+\frac{1}{n}}$ class of Muckenhoupt…

Global weighted $L^{p}$-estimates are obtained for the gradient of solutions to a class of linear singular, degenerate elliptic Dirichlet boundary value problems over a bounded non-smooth domain. The coefficient matrix is symmetric,…

We study a class of non-divergence form elliptic and parabolic equations with singular first-order coefficients in an upper half space with the homogeneous Dirichlet boundary condition. In the simplest setting, the operators in the…

We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb{R}^d_+$, where the coefficients are the product of $x_d^\alpha, \alpha \in (-\infty, 1),$ and a bounded uniformly elliptic…

We prove the well-posedness and regularity of solutions in mixed-norm weighted Sobolev spaces for a class of second-order parabolic and elliptic systems in divergence form in the half-space $\mathbb{R}^d_+ = \{x_d > 0\}$ subject to the…

We present recent advances in the regularity theory for weak solutions to some classes of elliptic and parabolic equations with strongly singular or degenerate structure. The equations under consideration satisfy standard $p$-growth and…

This paper studies the regularity of weak solutions to a class of parabolic perturbed fractional $1$-Laplace equations. Our analysis combines finite difference quotients, energy estimates, and iterative arguments, with a key step being the…

In this paper, we study the global H\"older regularity of solutions to uniformly degenerate parabolic equations. We also study the convergence of solutions as time goes to infinity under extra assumptions on the characteristic exponents of…

We establish some $C^{0,\alpha}$ and $C^{1,\alpha}$ regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane $\Sigma$ as a…