The Scalar Case

The result of Ladyzenskaya on global bounds for weak solutions of elliptic equations is carried over to certain quasilinear elliptic systems in diagonal form. A generalization of De Giorgi's example shows that there are unbounded weak solutions for systems with only “small” deviation from the diagonal form.

In this paper we show how a second order scalar uniformly elliptic equation in divergence form with measurable coefficients and Dirichlet boundary conditions can be transformed into a first order elliptic system with half-Dirichlet boundary condition. This first order system involves Hodge-Dirac operators and can be seen as a natural generalization of the Beltrami equation in the plane and we develop a theory for this equation, extending results from the plane to higher dimension. The reduction to a first order system applies both to linear as well as quasilinear second order equations and we believe this to be of independent interest. Using the first order system, we give a new representation formula of the solution of the Dirichlet problem both on simply and finitely connected domains. This representation formula involves only singular integral operators of convolution type and Neumann series there of, for which classical Calderón-Zygmund theory is applicable. Moreover, no use is made of any fundamental solution or Green's function beside fundamental solutions of constant coefficient operators. Remarkably, this representation formula applies also for solutions of the fully non-linear first order system. We hope that the representation formula could be used for numerically solving the equations. Using these tools we give a new short proof of Meyers' higher integrability theorem. Furthermore, we show that the solutions of the first order system are Hölder continuous with the same Hölder coefficient as the solutions of the second order equations. Finally, factorization identities and representation formulas for the higher dimensional Beurling-Ahlfors operator are proven using Clifford algebras, and certain integral estimates for the Cauchy transform is extended to higher dimensions.

In [GG], Giaquinta & Giusti introduced the notion of quasiminima, a concept which is very useful in unifying the C 0,α-regularity theory of elliptic equations in divergence form. In particular, their treatment provides regularity results for minima of nondifferentiable variational integrals. They also show that weak solutions of nonlinear elliptic equations in divergence form, under natural and general structural conditions, are quasiminima as are solutions to a wide class of obstacle problems. Because of the great interplay between quasiminima and the Holder theory of partial differential equations, it is important to develop the subject fully. Indeed, significant progress has already been made. Recently, for example, Harnack inequalities have been established in [DBT] for sub and super-quasiminima which run parallel to the results obtained by Trudinger [T] for weak sub and supersolutions of quasilinear equations in divergence form. In particular, the results in [DBT] imply that quasiminima are Holder continuous, a result first established by Giaquinta & Giusti, [GG].

In [1, 2, 3, 4, 6], the authors posed and discussed some boundary‐value problems of second order elliptic equations with parabolic degeneracy. In [5], the authors posed and discussed some boundary‐value problems of second order mixed equations with degenerate curve on the sides of an angle, but the coefficients of the equations possess strong restrictions. The present paper deals with the mixed problem for elliptic equations with degenerate curve on the sides of an angle, where the coefficients satisfy general conditions. We first give the formulation of the problem and estimates of solutions of the problem for the equations, and then prove the existence of solutions for the above problem by the Leray–Schauder theorem. In this paper, we use the complex analytic method, namely we first introduce the new notation (2.1) below and reduce the degenerate elliptic equations of second order to the corresponding problem for degenerate elliptic complex equations of first order, afterwards the above problem of second order degenerate elliptic equations can be solved. The results in this paper will be used in a subsequent paper to handle the Tricomi problem of second order equations of mixed type with degenerate curve on the sides of an angle. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Let Ω be an open set in high dimension Euclidean spaces R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , and let L denote the second order elliptic operator in the general form, which will characterize many of the original phenomenon of nature. As a very useful space, Orlicz space L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Φ</sup> includes many classical spaces such as L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> space, where 1< p < ∞ and Φ is a convex increasing function. The purpose of this paper is to use the stopping time technique and establish the regularity estimates in Orlicz spaces for the second order derivatives of the solutions to the general second order elliptic equations and their Dirichlet problems. We will obtain that, if u is the solution of the Dirichlet problem to the equation Lu = f for any given data f, then the weak solution u as well as its gradient ∇u and second order derivatives ∇ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> u can be controlled by the data f even in the Orlicz spaces.

For the weak solutions of second-order equations in a plane bounded domain, one obtains the asymptotics and estimates of the solutions near the interior angular points of the line of discontinuity of the coefficients. It is established that the number of terms of the asymptotics, not in the Sobolev classW 2 2 , can be arbitrarily large.

The aim of this chapter is to prove a Sobolev space regularity result for weak solutions of elliptic equations. This result will be used for the construction of \({L^p}\)-strong Feller processes in the next chapter. It will be also used for the construction of the boundary local time.

Gradient regularity for elliptic equations in the Heisenberg group

First, we will apply the method to study W 1,p regularity for a nonlinear elliptic operator in divergence form. We would like to point out that in the particular case of a linear elliptic equation, this method gives an alternative proof to the classical one, which uses the general theory of singular integrals. The utility of the method that we describe below is that hypotheses (A) and (B) are obtained directly by studying the deviation of the “coefficients” of A from the “coefficients” of A0, and this is usually not a difficult task. A more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give W 1,p estimates with a weak hypothesis of regularity in the coefficients. For instance, we are able to study the following cases:

This paper studies eigenvalues of elliptic operators on a bounded domain in a Euclidean space.We obtain lower bounds for the eigenvalues of elliptic operators of higher orders with Navier boundary condition.We also prove lower bounds and universal inequalities of Payne-Pólya-Weinberger-Yang type for the eigenvalues of second order elliptic equations in divergence form with Dirichlet boundary condition.

The following situation occurs in many problems of local regularity: Knowing a certain estimate on a certain ball Br(x0), deduce a better estimate on a smaller ball, say B r /2(x0). In the fifties, this point of view was put to a high degree of sophistication by De Giorgi in his famous proof of Holder estimates for elliptic second-order partial differential equations in divergence form; and it also plays a role in the alternative solutions found at the same time by Nash, and later by Moser. When fine analysis on metric spaces started to develop, it became an important issue to understand what were the key ingredients lying at the core of the methods of De Giorgi, Nash and Moser. It is now accepted by many that the two key inequalities are: a doubling inequality for the reference volume measure; a local Poincare inequality, controlling the deviation of a function on a smaller ball by the integral of its gradient on a larger ball.

We consider a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened ellipticity condition involving two weighted functions. Existence of solutions of Dirichlet problem for equations of the given class in the case where these equations have L 1-right-hand sides depending on unknown function has been established in our recent article. In the present article we prove some results which show the improvement of summability of solutions of the same problem under additional conditions on right-hand sides of the equations under consideration.

We prove the unique solvability of second order elliptic equations in nondivergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain the weak uniqueness of the Martingale problem associated with the elliptic equations.

In this paper we study the uniform approximation of continuous functions on compact subsets of R n by solutions of second order elliptic equations. Also we discuss the problem of existence of a continuous best approximation and the connection of the approximation problem to some isoperimetric inequalities.KeywordsMaximum PrincipleElliptic EquationDirichlet ProblemUniform ApproximationIsoperimetric InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions d \geq 2 . In previous works we studied the model problem of a discrete elliptic equation on \mathbb Z^d . Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions d &gt; 2 and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages – the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions.