Reifenberg Domains Research Articles

$$W^{1,p}$$ estimates for nonlinear Schrödinger equations with partially BMO coefficients in Reifenberg domains

$$W^{1,p}$$ estimates for nonlinear Schrödinger equations with partially BMO coefficients in Reifenberg domains

Pointwise boundary differentiability on Reifenberg domains for fully nonlinear elliptic equations

Pointwise boundary differentiability on Reifenberg domains for fully nonlinear elliptic equations

Boundary Lipschitz regularity and the Hopf lemma on Reifenberg domains for fully nonlinear elliptic equations

In this paper, we prove the boundary Lipschitz regularity and the Hopf Lemma by a unified method on Reifenberg domains for fully nonlinear elliptic equations. Precisely, if the domain $$\Omega $$ satisfies the exterior Reifenberg $$C^{1,\mathrm {Dini}}$$ condition at $$x_0\in \partial \Omega $$ (see Definition 1.3), the solution is Lipschitz continuous at $$x_0$$ ; if $$\Omega $$ satisfies the interior Reifenberg $$C^{1,\mathrm {Dini}}$$ condition at $$x_0$$ (see Definition 1.4), the Hopf lemma holds at $$x_0$$ . Our paper extends the results under the usual $$C^{1,\mathrm {Dini}}$$ condition.

Lorentz estimates to nonlinear elliptic obstacle problems of p(x)-growth in Reifenberg domains

We prove a global estimate in the scale of Lorentz spaces for the gradient of weak solution to a general nonlinear obstacle problem of p(x)-growth. It is mainly assumed that the nonlinearity satisfies small BMO semi-norm in the spatial variable and the boundary of domain is flat in the Reifenberg sense, while the variable exponent p(x) is a strongly log-Hölder continuous function. Here, an essential ingredient of our main proof is to make use of a combination of the so-called large-M-inequality principle and the geometric argument.

Orlicz estimates for general parabolic obstacle problems with p(t,x)-growth in Reifenberg domains

This article shows a global gradient estimate in the framework of Orlicz spaces for general parabolic obstacle problems with p(t,x)-Laplacian in a bounded rough domain. It is assumed that the variable exponent p(t,x) satisfies a strong log-Holder continuity, that the associated nonlinearity is measurable in the time variable and have small BMO semi-norms in the space variables, and that the boundary of the domain has Reifenberg flatness.

Pointwise estimates for Stokes systems with BMO coefficients in Reifenberg domains

Pointwise estimates of weak solution pairs to a stationary Stokes system with small [Formula: see text] semi-norm coefficients are established in Reifenberg flat domains by using the restricted sharp maximal function. These pointwise estimates provide a unified treatment of the Calderón–Zygmund estimates for the solution pair to Stokes systems in [Formula: see text] and [Formula: see text] spaces.

Potential estimates of superquadratic elliptic systems with VMO coefficients in Reifenberg domains

The purpose of this paper is to study an inhomogeneous Dirichlet problem of a superquadratic nonlinear elliptic system with VMO coefficients in Reifenberg flat domains. We derive pointwise potential estimates for weak solution to the Dirichlet problem. As a consequence, we derive global integrability estimates for solutions with respect to Lorentz and Morrey norms.

Global gradient estimates for nonlinear equations of p-Laplace type from composite materials

We study global gradient estimates for the weak solution to divergence form nonlinear elliptic equations with p -growth nonlinearities from composite materials. We assume that the domain is bounded and is composed of disjoint Reifenberg flat domains and the nonlinearities have small BMO seminorms in each subdomain. Under these assumptions, based on our new geometric observation for disjoint Reifenberg domains and gradient estimates for nonlinear equations with measurable p -growth nonlinearities, we establish global W 1 , q estimates for p ≤ q < ∞ .

Gradient estimates for solutions of elliptic systems with measurable coefficients from composite material

We study global gradient estimates for the weak solution to elliptic systems with measurable coefficients from composite materials in a nonsmooth bounded domain. The principal coefficients are assumed to be merely measurable in one variable and have small bounded mean oscillation (BMO) seminorms in the other variables on each subdomain whose boundary satisfies the so‐called δ‐Reifenberg flat condition. Under these assumptions and based on our new geometric observation for disjoint Reifenberg domains in a previous study, we obtain global W1,p estimates.

Weighted Lorentz estimates for nonlinear elliptic obstacle problems with partially regular nonlinearities

A global estimate in the framework of weighted Lorentz spaces is reached for the gradient of weak solution to a class of nonlinear elliptic obstacle problems with partially regular nonlinearities in a bounded Reifenberg flat domain. We mainly assume that the nonlinearities are merely measurable in one spatial variable and have small BMO seminorms in the remaining variables and that the underlying domain is flat in the sense of Reifenberg. As an application, we also present a global Lorentz estimate for the gradients of weak solutions to Dirichlet problems of relevant nonlinear elliptic equations under controlled growth in Reifenberg domains based on the bootstrap argument.

Weighted variable exponent Sobolev estimates for elliptic equations with non-standard growth and measure data

Consider the following nonlinear elliptic equation of p(x)-Laplacian type with nonstandard growth $$\begin{aligned} {\left\{ \begin{array}{ll} \mathrm{div}a(Du, x)=\mu &{}\quad \text {in} \quad \Omega ,\\ u=0 &{} \quad \text {on} \quad \partial \Omega , \end{array}\right. } \end{aligned}$$where \(\Omega \) is a Reifenberg domain in \(\mathbb {R}^n\), \(\mu \) is a Radon measure defined on \(\Omega \) with finite total mass and the nonlinearity \(a: \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}^n\) is modeled upon the \(p(\cdot )\)-Laplacian. We prove the estimates on weighted variable exponent Lebesgue spaces for gradients of solutions to this equation in terms of Muckenhoupt–Wheeden type estimates. As a consequence, we obtain some new results such as the weighted \(L^q-L^r\) regularity (with constants \(q < r\)) and estimates on Morrey spaces for gradients of the solutions to this non-linear equation.

Homogenization of the conormal derivative problem for elliptic systems in Reifenberg domains

We study homogenization of the conormal derivative problem for an elliptic system with discontinuous coefficients in a bounded domain. A uniform global [Formula: see text] estimate for [Formula: see text] is obtained under optimal assumptions that the coefficients have a small bounded mean oscillation (BMO) seminorm and the domain is a [Formula: see text]-Reifenberg flat domain whose boundary might be fractal.

W1,p(⋅) regularity for quasilinear problems with irregular obstacles on Reifenberg domains

In this paper, we prove the global gradient estimates on the generalized Lebesgue spaces for weak solutions to elliptic quasilinear obstacle problems. It is worth noticing that the coefficients related to the obstacle problems are merely measurable with small BMO norms and the underlying domain does not satisfy any smoothness conditions.

Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

Consider the nonlinear parabolic equation in the form $$\begin{aligned} u_t-\mathrm{div}{\mathbf {a}}(D u,x,t)=\mathrm{div}\,(|F|^{p-2}F) \quad \text {in} \quad \Omega \times (0,T), \end{aligned}$$ where $$T>0$$ and $$\Omega $$ is a Reifenberg domain. We suppose that the nonlinearity $${\mathbf {a}}(\xi ,x,t)$$ has a small BMO norm with respect to x and is merely measurable and bounded with respect to the time variable t. In this paper, we prove the global Calderon-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calderon-Zygmund estimates extend certain previous results to equations with less regularity assumptions on the nonlinearity $${\mathbf {a}}(\xi ,x,t)$$ and to more general setting of Lorentz spaces.

Global W 1,p(·) estimate for renormalized solutions of quasilinear equations with measure data on Reifenberg domains

Abstract In this paper, we prove the gradient estimate for renormalized solutions to quasilinear elliptic equations with measure data on variable exponent Lebesgue spaces with BMO coefficients in a Reifenberg flat domain.

Gradient estimates for [formula omitted]-Laplacian equation in composite Reifenberg domains

We obtain a local gradient estimate for weak solutions of p-Laplacian type equation in a non-smooth domain composed of a finite number of disjoint subdomains with (δ0,R)-Reifenberg flat boundaries. This kind of domain was called (δ0,R) composite-Reifenberg domain, which allows certain subdomain to exist with highly non-smooth boundaries worse than Reifenberg domain. Our results improve the known results for such equations.

Hessian estimates for fourth order elliptic systems with singular BMO coefficients in composite Reifenberg domains

We consider fourth order elliptic systems in divergence form with greatly discontinuous coefficients in a nonsmooth domain. Our domain is composed of a finite number of disjoint subdomains with (δ,R)-Reifenberg flat boundaries, and the coefficients, which are allowed to have great discontinuity across the boundaries of subdomains, have small BMO semi-norms on each subdomain. Our main result is deriving the global W2,p (1<p<∞) regularity, with the estimates independent of the distance between the subdomains.

Elliptic equations with singular BMO coefficients in Reifenberg domains

W1,p estimate for the solutions of elliptic equations whose coefficient matrix can have large jump along the boundary of subdomains is obtained. The principal coefficients are supposed to be in the John–Nirenberg space with small BMO seminorms, and the domain and subdomains are Reifenberg flat domains. Moreover, it has been shown that the estimates are uniform with respect to the distance between the subdomains.

Global weighted estimates for the gradient of solutions to nonlinear elliptic equations

We consider nonlinear elliptic equations of p-Laplacian type that are not necessarily of variation form when the nonlinearity is allowed to be discontinuous and the boundary of the domain can go beyond the Lipschitz category. Under smallness in the BMO nonlinearity and sufficient flatness of the Reifenberg domain, we obtain the global weighted Lq estimates with q∈(p,∞) for the gradient of weak solutions.

Parabolic Systems with Measurable Coefficients in Reifenberg Domains

Parabolic Systems with Measurable Coefficients in Reifenberg Domains