Calderón–Zygmund theory for nonlinear elliptic problems with irregular obstacles (original) (raw)
2008
In this paper we study the questions of existence and uniqueness of solutions for equations of type −div a(x, Du) + γ(u) φ, posed in an open bounded subset Ω of R N , with nonlinear boundary conditions of the form a(x, Du) • η + β(u) ψ. The nonlinear elliptic operator div a(x, Du) modeled on the p-Laplacian operator ∆p(u) = div(|Du| p−2 Du), with p > 1, γ and β maximal monotone graphs in R 2 such that 0 ∈ γ(0) ∩ β(0), R = D(γ) ⊂ D(β) and the data φ ∈ L 1 (Ω) and ψ ∈ L 1 (∂Ω). Since D(γ) = R, we are dealing with obstacle problems. For this kind of problems the existence of weak solution, in the usual sense, fails to be true for nonhomogeneous boundary conditions, so a new concept of solution has to be introduced.
On nonlinear elliptic problems with discontinuities
In this paper we examine nonlinear elliptic equations driven by the p-Laplacian and with a discontinuous forcing term. To develop an existence theory we pass to an elliptic inclusion by filling in the gaps at the discontinuity points of the forcing term. We prove three existence theorems. The first is a multiplicity result and proves the existence of two bounded solutions one strictly positive and the other strictly negative. The other two theorems deal with problems at resonance and prove the existence of solutions using Landesman-Lazer type conditions.
Boundary regularity for quasilinear elliptic equations with general Dirichlet boundary data
arXiv: Analysis of PDEs, 2018
We study global regularity for solutions of quasilinear elliptic equations of the form divA(x,u,nablau)=divF\div \A(x,u,\nabla u) = \div \F divA(x,u,nablau)=divF in rough domains Omega\OmegaOmega in Rn\R^nRn with nonhomogeneous Dirichlet boundary condition. The vector field A\AA is assumed to be continuous in uuu, and its growth in nablau\nabla unablau is like that of the ppp-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions uuu to the equation under the Reifenberg flat condition for Omega\OmegaOmega, a small BMO condition in xxx for A\AA, and an optimal condition for the Dirichlet boundary data.
Gradient estimates for elliptic systems in non-smooth domains
Mathematische Annalen, 2008
We obtain the global W 1, p , 1 < p < ∞, estimate for the weak solution of an elliptic system with discontinuous coefficients in non-smooth domains without using maximal function approach. It is assumed that the boundary of a bounded domain is well approximated by hyperplanes at every point and at every scale, and that the tensor coefficients belong to BMO space with their BMO semi-norms sufficiently small.
Some possibly degenerate elliptic problems with measure data and non linearity on the boundary
Annales de la faculté des sciences de Toulouse Mathématiques, 2011
The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a bounded domain with measure supported on the domain and on the boundary. A second one deals with the same type of data but involves a degenerate weight in the equation. In both cases, the nonlinearity under consideration lies on the boundary. For the first problem, we prove an optimal regularity result, whereas for the second one the optimality is not guaranteed but we provide however regularity estimates.
Degenerate problems with irregular obstacles
Journal für die reine und angewandte Mathematik (Crelles Journal), 2000
We establish the natural Calderó n and Zygmund theory for solutions of elliptic and parabolic obstacle problems involving possibly degenerate operators in divergence form of p-Laplacian type, and proving that the (spatial) gradient of solutions is as integrable as that of the assigned obstacles. We also include an existence and regularity theorem where obstacles are not necessarily considered to be non-increasing in time.
Gradient estimates for elliptic systems with measurable coefficients in nonsmooth domains
manuscripta mathematica, 2010
We consider an elliptic system in divergence form with measurable coefficients in a nonsmooth bounded domain to find a minimal regularity requirement on the coefficients and a lower level of geometric assumption on the boundary of the domain for a global W 1, p , 1 < p < ∞, regularity. It is proved that such a W 1, p regularity is still available under the assumption that the coefficients are merely measurable in one variable and have small BMO semi-norms in the other variables while the domain can be locally approximated by a hyperplane, a so called δ-Reifenberg domain, which is beyond the Lipschitz category. This regularity easily extends to a certain Orlicz-Sobolev space.
A priori estimates for solutions to elliptic equations on non-smooth domains
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2002
It is proved that elliptic boundary-value problems have a global smoothing property in Lebesgue spaces, provided the underlying space of weak solutions admits a Sobolev-type inequality. The results apply to all standard boundary conditions, and a wide range of non-smooth domains, even if the clas!3ical estimates fail. The dependence on the data is explicit. In particular, thi~ provides good control over the domain dependence, which is important for applications involving varying domains.