Gradient estimates for higher order elliptic equations on nonsmooth domains

We establish optimal gradient estimates in Orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain. Our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane. As a consequence we prove the classical Wm,p, m=1,2,…, 1<p<∞, estimates for such a higher order equation. Our results easily extend to higher order elliptic and parabolic systems.     

Global weighted estimates for quasilinear elliptic equations with non-standard growth

In this paper we obtain a new global gradient estimates in weighted Lorentz spaces for weak solutions of p(x)$p(x)$-Laplacian type equation with small BMO coefficients in a δ-Reifenberg flat domain. The modified Vitali covering lemma, the maximal function technique and the appropriate localization method are the main analytical tools. Our results improve the known results for such equations.     

Global regularity for divergence form elliptic equations on quasiconvex domains

In this paper, we introduce a notion of quasiconvex domain, and show that the global W1,p regularity holds on such domains for a wide class of divergence form elliptic equations. The modified Vitali covering lemma, compactness method and the maximal function technique are the main analytical tools.     

Optimal <Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript> regularity theory for parabolic equations in divergence form

This work treats L^p regularity theory for weak solutions of parabolic equations in divergence form with discontinuous coefficients on nonsmooth domains. We essentially obtain an optimal condition on the coefficients under which the global W^1,p regularity theory holds. This work was supported by SNU foundation in 2005.     

Boundary estimates for elliptic systems with L<Superscript>1</Superscript>–data

We obtain boundary estimates for the gradient of solutions to elliptic systems with Dirichlet or Neumann boundary conditions and L ¹–data, under some condition on the divergence of the data. Similar boundary estimates are obtained for div–curl and Hodge systems.     

Fine regularity for elliptic systems with discontinuous ingredients

We propose results on interior Morrey, BMO and H?lder regularity for the strong solutions to linear elliptic systems of order 2b with discontinuous coefficients and right-hand sides belonging to the Morrey space L^p,λ. Received: 20 October 2004     

Hessian estimates in Orlicz spaces for fourth-order parabolic systems in non-smooth domains

We establish the global Hessian estimate in Orlicz spaces for a fourth-order parabolic system with discontinuous tensor coefficients in a non-smooth domain under the assumptions that the coefficients have small weak BMO semi-norms, the boundary of a domain is δ-Reifenberg flat for δ>0 small and the given Young function satisfies some moderate growth condition. As a corollary we obtain an optimal global W2,p regularity for such a system.     

Parabolic equations with BMO coefficients in Lipschitz domains

In this paper, we are concerned with certain natural Sobolev-type estimates for weak solutions of inhomogeneous problems for second-order parabolic equations in divergence form. The geometric setting is that of time-independent cylinders having a space intersection assumed to be locally given by graphs with small Lipschitz coefficients, the constants of the operator being uniformly parabolic. We prove the relevant L^p estimates, assuming that the coefficients are in parabolic bounded mean oscillation (BMO) and that their parabolic BMO semi-norms are small enough.     

Existence of bounded solutions for nonlinear elliptic equations in unbounded domains

In this paper we study the existence of bounded weak solutions for some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the p-laplacian operator, and the lower order terms, which depend on the solution u and its gradient u, have a power growth of order p–1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity.     

Gradient estimates for Dirichlet parabolic problems in unbounded domains

We consider autonomous parabolic Dirichlet problems in a regular unbounded open set Ω⊂R^N involving second-order operator A with (possibly) unbounded coefficients. We determine new conditions on the coefficients of A yielding global gradient estimates for the bounded classical solution.     

The Green function estimates for strongly elliptic systems of second order

We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain $\Omega \subseteq {\mathbb{R}}^n, n \geq 3We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain , under the assumption that solutions of the system satisfy De Giorgi-Nash type local H?lder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation.     

Parabolic equations in time dependent Reifenberg domains

In this paper we define time dependent parabolic Reifenberg domains and study Lp estimates for weak solutions of uniformly parabolic equations in divergence form on these domains. The basic assumption is that the principal coefficients are of parabolic BMO space with small parabolic BMO seminorms. It is shown that Lp estimates hold for time dependent parabolic δ-Reifenberg domains.     

THE SINGULARLY PERTURBED NONLINEAR ELLIPTIC SYSTEMS IN UNBOUNDED DOMAINS

Abstract. The singularly perturbed problems for elliptic systems in unbounded domains are considered. Under suitable conditions and by using the comparison theorem the existence and asymptotic behavior of solution for the boundary value problems studied,     

Elliptic equations with BMO nonlinearity in Reifenberg domains

Given p∈[2,+∞), we obtain the global W1,p estimate for the weak solution of a boundary-value problem for an elliptic equation with BMO nonlinearity in a Reifenberg domain, assuming that the nonlinearity has sufficiently small BMO seminorm and that the boundary of the domain is sufficiently flat.     

Fourth-order parabolic equations with weak BMO coefficients in Reifenberg domains

We study optimal W2,p-regularity for fourth-order parabolic equations with discontinuous coefficients in general domains. We obtain the global W2,p-regularity for each 1<p<∞ under the assumption that the coefficients have suitably small BMO semi-norm of weak type and the boundary of the domain is δ-Reifenberg flat. The situation of our main theorem arises when the conductivity on fractals is controlled by a random variable in the time direction.     

Comparison principle for non-cooperative elliptic systems

This paper presents some sufficient conditions for the validity of the comparison principle for weak and classical solutions of non-cooperative weakly-coupled systems of elliptic second-order PDE.     

Schauder estimates for solutions to boundary value problems for second order elliptic systems in polyhedral domains

Boundary value problems for second order elliptic differential equations and systems in a polyhedral domain are considered. The authors prove Schauder estimates and obtain regularity assertions for the solutions.     

Uniform asymptotic formulae for Green's functions in singularly perturbed domains

Asymptotic formulae for Green's functions for the operator -Δ$-\Delta$ in domains with small holes are obtained. A new feature of these formulae is their uniformity with respect to the independent variables. The cases of multi-dimensional and planar domains are considered.