Elliptic Equations In Divergence Form Research Articles

Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces

Aim of this paper is to prove regularity results, in some Modified Local Generalized Morrey Spaces, for the first derivatives of the solutions of a divergence elliptic second order equation of the form Lu:=∑i,j=1naij(x)uxixj=∇·f,for almost allx∈Ω\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathscr {L}u{:}{=}\\sum _{i,j=1}^{n}\\left( a_{ij}(x)u_{x_{i}}\\right) _{x_{j}}=\\nabla \\cdot f,\\qquad \\hbox {for almost all }x\\in \\Omega \\end{aligned}$$\\end{document}where the coefficients a_{ij} belong to the Central (that is, Local) Sarason class CVMO and f is assumed to be in some Modified Local Generalized Morrey Spaces widetilde{LM}_{{x_{0}}}^{p,varphi }. Heart of the paper is to use an explicit representation formula for the first derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the representation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results.

Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems

Abstract This paper extends the applicability of the combined use of the virtual element method (VEM) and the boundary element method (BEM), recently introduced to solve the coupling of linear elliptic equations in divergence form with the Laplace equation, to the case of acoustic scattering problems in 2D and 3D. The well-posedness of the continuous and discrete formulations are established, and then Cea-type estimates and consequent rates of convergence are derived.

Regularity results for nonlocal equations and applications

We introduce the concept of $$C^{m,\alpha }$$ -nonlocal operators, extending the notion of second order elliptic operator in divergence form with $$C^{m,\alpha }$$ -coefficients. We then derive the nonlocal analogue of the key existing results for elliptic equations in divergence form, notably the Holder continuity of the gradient of the solutions in the case of $$C^{0,\alpha }$$ -coefficients and the classical Schauder estimates for $$C^{m+1,\alpha }$$ -coefficients. We further apply the regularity results for $$C^{m,\alpha }$$ -nonlocal operators to derive optimal higher order regularity estimates of Lipschitz graphs with prescribed Nonlocal Mean Curvature. Applications to nonlocal equation on manifolds are also provided.

The ergodic limit for weak solutions of elliptic equations with Neumann boundary condition

We consider the so-called it ergodic problem for weak solutions of elliptic equations in divergence form, complemented with Neumann boundary conditions. The simplest example reads as the following boundary value problem in a bounded domain of $\mathbb{R}^N$: $ \left\{ \begin{array}{l} - {\rm{div}}(A(x)\nabla u) + \lambda = H(x,\nabla u)\;\;\;\;\;{\rm{in}}\;\Omega ,\\ A(x)\nabla u \cdot \vec n = 0\;\;\;\;\;\;\;{\rm{on}}\;\partial \Omega , \end{array} \right. $ where A(x) is a coercive matrix with bounded coefficients, and $H(x, \nabla u)$ has Lipschitz growth in the gradient and measurable $x$-dependence with suitable growth in some Lebesgue space (typically, $|H(x, \nabla u)|\leq b(x) |\nabla u|+ f(x)$ for functions b(x)∈ LN(Ω) and f (x) ∈ Lm(Ω), $m\geq 1$). We prove that there exists a unique real value $\lambda$ for which the problem is solvable in Sobolev spaces and the solution is unique up to addition of a constant. We also characterize the ergodic limit, say the singular limit obtained by adding a vanishing zero order term in the equation. Our results extend to weak solutions and to data in Lebesgue spaces LN(Ω) (or in the dual space (H1(Ω))'), previous results which were proved in the literature for bounded solutions and possibly classical or viscosity formulations.

On the regularity of very weak solutions for linear elliptic equations in divergence form

In this paper we consider a linear elliptic equation in divergence form 0.1∑i,jDj(aij(x)Diu)=0inΩ.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \\quad \\hbox {in } \\Omega . \\end{aligned}$$\\end{document}Assuming the coefficients a_{ij} in W^{1,n}(Omega ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution uin L^{n'}_mathrm{loc}(Omega ) of (0.1) is actually a weak solution in W^{1,2}_mathrm{loc}(Omega ).

Growth conditions and regularity for weak solutions to nonlinear elliptic pdes

We describe some aspects of the process/approach to interior regularity of weak solutions to a class of nonlinear elliptic equations in divergence form, as well as of minimizers of integrals of the calculus of variations. In particular with respect to the growth conditions we emphasize some differences between solving equations and minimizing energy integrals, and also between x−dependence or x−independence.

Hölder Continuity of Solutions to an Elliptic Equation with Drift Degenerating in a Part of the Domain

We consider a uniformly elliptic equation of divergence form with lower-order terms in a domain D divided by a hyperplane into two parts. The equation is uniformly degenerate with respect to a small parameter e in one of these parts. It is shown that any solution is Holder continuous in D with Holder exponent independent of e.

Global Lorentz estimates for nonuniformly nonlinear elliptic equations via fractional maximal operators

This article is a contribution to the study of regularity theory for nonlinear elliptic equations. The aim of this article is to establish some global estimates for nonuniformly elliptic equations in divergence form as−div(|∇u|p−2∇u+a(x)|∇u|q−2∇u)=−div(|F|p−2F+a(x)|F|q−2F), which arises from double-phase functional problems. In particular, the main results provide the regularity estimates for the distributional solutions in terms of maximal and fractional maximal operators. This work extends that of Colombo and Mingione (2016) [23] and Byun and Oh (2017) [9] by dealing with the global estimates in Lorentz spaces. It also extends our recent result Tran and Nguyen (2020) [66] concerning new estimates of divergence elliptic equations using cutoff fractional maximal operators. For future research, the approach developed in this article will allow global estimates of distributional solutions to nonuniformly nonlinear elliptic equations to be obtained in the framework of other spaces.

Global Hölder estimates for 2D linearized Monge–Ampère equations with right-hand side in divergence form

We establish global Hölder estimates for solutions to inhomogeneous linearized Monge–Ampère equations in two dimensions with the right hand side being the divergence of a bounded vector field. These equations arise in the semi-geostrophic equations in meteorology and in the approximation of convex functionals subject to a convexity constraint using fourth order Abreu type equations. Our estimates hold under natural assumptions on the domain, boundary data and Monge-Ampère measure being bounded away from zero and infinity. They are an affine invariant and degenerate version of global Hölder estimates by Murthy-Stampacchia and Trudinger for second order elliptic equations in divergence form.

Higher-Order Riesz Transforms of Hermite Operators on New Besov and Triebel–Lizorkin Spaces

Consider the Hermite operator $$H=-\Delta +|x|^2$$ on the Euclidean space $${\mathbb {R}}^n$$. The aim of this article is to prove the boundedness of higher-order Riesz trnsforms on appropriate Besov and Triebel–Lizorkin spaces. As an application, we prove certain regularity estimates of second-order elliptic equations in divergence form with the oscillator perturbations.

On conormal and oblique derivative problem for elliptic equations with Dini mean oscillation coefficients

We show that weak solutions to conormal derivative problem for elliptic equations in divergence form are continuously differentiable up to the boundary provided that the mean oscillations of the leading coefficients satisfy the Dini condition, the lower order coefficients satisfy certain suitable conditions, and the boundary is locally represented by a $C^1$ function whose derivatives are Dini continuous. We also prove that strong solutions to oblique derivative problem for elliptic equations in nondivergence form are twice continuously differentiable up to the boundary if the mean oscillations of coefficients satisfy the Dini condition and the boundary is locally represented by a $C^1$ function whose derivatives are double Dini continuous. This in particular extends a result of M. V. Safonov (Comm. Partial Differential Equations 20:1349--1367, 1995)

Formula omitted] regularity theory for a class of nonlocal elliptic equations

In this paper, we study the regularity of weak solutions to a class of nonlocal elliptic equations in Bessel potential spaces Hs,p. Our main results can be seen as an extension of the well-known W1,p regularity theory for local second-order elliptic equations in divergence form to the nonlocal setting.

Uniform estimate of an iterative method for elliptic problems with rapidly oscillating coefficients

We study the iterative algorithm proposed by S. Armstrong, A. Hannukainen, T. Kuusi, J.-C. Mourrat to solve elliptic equations in divergence form with stochastic stationary coefficients. Such equations display rapidly oscillating coefficients and thus usually require very expensive numerical calculations, while this iterative method is comparatively easy to compute. In this article, we strengthen the estimate for the contraction factor achieved by one iteration of the algorithm. We obtain an estimate that holds uniformly over the initial function in the iteration, and which grows only logarithmically with the size of the domain.

Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients

In this paper we derive a propagation of smallness result for a scalar second elliptic equation in divergence form whose leading order coefficients are Lipschitz continuous on two sides of a C2 hypersurface that crosses the domain, but may have jumps across this hypersurface. Our propagation of smallness result is in the most general form regarding the locations of domains, which may intersect the interface of discontinuity. At the end, we also list some consequences of the propagation of smallness result, including stability results for the associated Cauchy problem, a propagation of smallness result from sets of positive measure, and a quantitative Runge approximation property.

Higher-order pathwise theory of fluctuations in stochastic homogenization

We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order $$\frac{d}{2}$$ , where d is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise proximity of the macroscopic fluctuations of a general solution to those of the (higher-order) correctors, via a (higher-order) two-scale expansion injected into the “homogenization commutator”, thus confirming the scope of this notion. This higher-order generalization sheds a clearer light on the algebraic structure of the higher-order versions of correctors, flux correctors, two-scale expansions, and homogenization commutators. It reveals that in the same way as this algebra provides a higher-order theory for microscopic spatial oscillations, it also provides a higher-order theory for macroscopic random fluctuations, although both phenomena are not directly related. We focus on the model framework of an underlying Gaussian ensemble, which allows for an efficient use of (second-order) Malliavin calculus for stochastic estimates. On the technical side, we introduce annealed Calderon–Zygmund estimates for the elliptic operator with random coefficients, which conveniently upgrade the known quenched large-scale estimates.

Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains

Let n≥2 and Ω be a bounded Lipschitz domain in Rn. In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in Ω. More precisely, for any given p∈(2,∞), two necessary and sufficient conditions for W1,p estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W1,q estimates of solutions with q∈[2,p] and some Muckenhoupt weights, are obtained. As applications, for any given p∈(1,∞) and ω∈Ap(Rn) (the class of Muckenhoupt weights), the authors establish weighted Wω1,p estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces.

Analysis of a Quadratic Finite Element Method for Second-Order Linear Elliptic PDE, With Low Regularity Data

In the present work, we propose extend an approximation for the second order linear elliptic equation in divergence form with coefficients in and L1-Data, based on the usual quadratic finite element techniques. We study the convergence with low-regularity solutions only belonging to with and where the class of renormalized solution is considered as limit. Statements and proofs of linear finite elements approximation case in [1]; remain valid in our case, and when the Data is a bounded Radon measure, a weaker convergence is obtained. An error estimate in is then deduced under suitable regularity assumptions on the solution, the coefficients and the L1-Data f.

New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data

This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions:{div(A(x,∇u))=div(|F|p−2F)inΩ,u=σon∂Ω, where Ω⊂Rn (n≥2), the nonlinearity A is a monotone Carathéodory vector valued function defined on W01,p(Ω) for p>1 and the p-capacity uniform thickness condition is imposed on the complement of our bounded domain Ω. Moreover, for given data F∈Lp(Ω;Rn), the problem is set up with general Dirichlet boundary data σ∈W1,p(Ω). In this paper, the optimal good-λ type bounds technique is applied to prove some results of fractional maximal estimates for gradient of solutions. And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works [45,52,53] and references therein.

A New Boundary Harnack Principle (Equations with Right Hand Side)

We introduce a new boundary Harnack principle in Lipschitz domains for equations with a right hand side. Our approach, which uses comparisons and blow-ups, will adapt to more general domains as well as other types of operators. We prove the principle for divergence form elliptic equations with lower order terms including zero order terms. The inclusion of a zero order term appears to be new even in the absence of a right hand side.

Reconstruction for the coefficients of a quasilinear elliptic partial differential equation

In this paper we consider an inverse coefficients problem for a quasilinear elliptic equation of divergence form ∇⋅C→(x,∇u(x))=0, in a bounded smooth domain Ω. We assume that C⃗(x,p→)=γ(x)p→+b→(x)|p→|2+O(|p→|3), by expanding C⃗(x,p→) around p→=0. We give a reconstruction method for γ and b→ from the Dirichlet to Neumann map defined on ∂Ω.