ﻻ يوجد ملخص باللغة العربية
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 matrix of coefficients. Thus, the coefficients are singular or degenerate near the boundary ${x_d =0}$ and they may not locally integrable. The novelty of the work is that we find proper weights under which the existence, uniqueness, and regularity of solutions in Sobolev spaces are established. These results appear to be the first of their kind and are new even if the coefficients are constant. They are also readily extended to systems of equations.
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 (-
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 establis
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
Let $ngeq 3$, $0le m<frac{n-2}{n}$, $rho_1>0$, $beta>beta_0^{(m)}=frac{mrho_1}{n-2-nm}$, $alpha_m=frac{2beta+rho_1}{1-m}$ and $alpha=2beta+rho_1$. For any $lambda>0$, we prove the uniqueness of radially symmetric solution $v^{(m)}$ of $La(v^m/m)+alph
We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $Mtimes mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solu