Related papers: The Schauder estimates for higher-order parabolic …

We prove $L_p$ estimates of solutions to a conormal derivative problem for divergence form complex-valued higher-order elliptic systems on a half space and on a Reifenberg flat domain. The leading coefficients are assumed to be merely…

Li-Vogelius and Li-Nirenberg gave a gradient estimate for solutions of strongly elliptic equations and systems of divergence forms with piecewise smooth coefficients, respectively. The discontinuities of the coefficients are assumed to be…

We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which…

We prove Schauder type estimates for solutions of stationary and evolution equations driven by weak generators of transition semigroups associated to a semilinear stochastic partial differential equations with values in a separable Hilbert…

In this paper, we establish Schauder's estimates for the following non-local equations in \mR^d : $$ \partial_tu=\mathscr L^{(\alpha)}_{\kappa,\sigma} u+b\cdot\nabla u+f,\ u(0)=0, $$ where $\alpha\in(1/2,2)$ and $ b:\mathbb R_+\times\mathbb…

We obtain Schauder estimates for a general class of linear integro-differential equations. The estimates are applied to a scalar non-local Burgers equation and complete the global well-posedness results obtained in \cite{ISV}.

We consider a family of second-order parabolic operators $\partial_t+\mathcal{L}_\varepsilon$ in divergence form with rapidly oscillating, time-dependent and almost-periodic coefficients. We establish uniform interior and boundary H\"older…

We study parabolic equations in variable H\"older spaces on domains of Euclidean spaces. The existence and uniqueness of solutions is proved.

This paper discusses the local Calder\'on-Zygmund type estimate for the singular parabolic double-phase system. The proof covers the counterpart $p<2$ of the result in [23]. Phase analysis is employed to determine an appropriate intrinsic…

Schauder estimates hold in nonuniformly elliptic problems under optimal assumptions on the growth of the ellipticity ratio.

We consider a class of possibly degenerate second order elliptic operators $\cal A$ on $\R^n$. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We…

We consider parabolic systems in divergence form with piecewise $C^{(s+\delta)/2,s+\delta}$ coefficients and data in a bounded domain consisting of a finite number of cylindrical subdomains with interfacial boundaries in $C^{s+1+\mu}$,…

We consider backward problems for semilinear coupled parabolic systems in bounded domains. We prove conditional stability estimates for linear and semilinear systems of strongly coupled parabolic equations involving general semilinearities.…

For a family of second-order parabolic systems with bounded measurable, rapidly oscillating and time-dependent periodic coefficients, we investigate the sharp convergence rates of weak solutions in $L^2$. Both initial-Dirichlet and…

In this paper, we examine regularity estimates for solutions to fully nonlinear, degenerated elliptic equations, at interior vanishing source points. At these points, we obtain Schauder-type regularity estimates, which depend on the…

We study boundary gradient estimates for second-order divergence type parabolic and elliptic systems in $C^{1,\alpha}$ domains. The coefficients and data are assumed to be H\"older in the time variable and all but one spatial variables.…

We consider an implicit finite difference scheme on uniform grids in time and space for the Cauchy problem for a second order parabolic stochastic partial differential equation where the parabolicity condition is allowed to degenerate. Such…

We study arbitrary order symmetry operators for the linear Schr\"odinger equations with arbitrary number of spatial variables. We deduce determining equations for coefficient functions of such operators and consider in detail some cases…

We prove a sparse bound in the context of Schauder theory for divergence form elliptic partial differential equations. In addition, we show how an iteration argument inspired by sparse domination bounds can be used to deduce gradient…

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…