Elliptic Partial Differential Equations of Second Order

The principal objective of this work is the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process. This means we shall be concerned with the solvability of boundary value problems (primarily the Dirichlet problem) and related general properties of solutions of linear equations and of quasilinear equations . Here Du = (D ₁u,..., D _nu), where D _iu = ∂u/∂x _i, D _iju = ∂ ²u/∂x _i ∂x _j, etc., and the summation convention is understood. The ellipticity of these equations is expressed by the fact that the coefficient matrix [a ^ij] is (in each case) positive definite in the domain of the respective arguments. We refer to an equation as uniformly elliptic if the ratio γ of maximum to minimum eigenvalue of the matrix [a ^ij] is bounded. We shall be concerned with both non-uniformly and uniformly elliptic equations.

Let Ω be a domain in ℝⁿ and u a C ²(Ω) function. The Laplacian of u, denoted Δu, is defined by . The function u is called harmonic (subharmonic, superharmonic) in Ω if it satisfies there . In this chapter we develop some basic properties of harmonic, subharmonic and superharmonic functions which we use to study the solvability of the classical Dirichlet problem for Laplace’s equation, Δu = 0. As mentioned in Chapter 1, Laplace’s equation and its inhomogeneous form, Poisson’s equation, are basic models of linear elliptic equations.

The purpose of this chapter is to extend the classical maximum principles for the Laplace operator, derived in Chapter 2, to linear elliptic differential operators of the form , where x = (x ₁,..., x _n) lies in a domain Ω of ℝⁿ, n≥2. It will be assumed, unless otherwise stated, that u belongs to C ²(Ω). The summation convention that repeated indices indicate summation from 1 to n is followed here as it will be throughout. L will always denote the operator (3.1).

In Chapter 2 we introduced the fundamental solution Г of Laplace’s equation given by . For an integrable function f on a domain Ω, the Newtonian potential of f is the function w defined on ℝⁿ by . From Green’s representation formula (2.16), we see that when ∂Ω is sufficiently smooth a C ²(ΩΩ) function may be expressed as the sum of a harmonic function and the Newtonian potential of its Laplacian. It is not surprising therefore that the study of Poisson’s equation Δu = f can largely be effected through the study of the Newtonian potential of f. This chapter is primarily devoted to the estimation of derivatives of the Newtonian potential. As well as leading to existence theorems for the classical Dirichlet problem for Poisson’s equation, these estimates form the basis for the Schauder or potential theoretic approach to linear elliptic equations treated in Chapter 6.

This chapter supplies the functional analytic material required for our study of existence of solutions of linear elliptic equations in Chapters 6 and 8. This material will be familiar to a reader already versed in basic functional analysis but we shall assume some acquaintance with elementary linear algebra and the theory of metric spaces. Unless otherwise indicated, all linear spaces used in this book are assumed to be defined over the real number field. The theory of this chapter, however, carries over almost unchanged if the real numbers are replaced by the complex numbers.

This chapter develops a theory of second order linear elliptic equations that is essentially an extension of potential theory. It is based on the fundamental observation that equations with Holder continuous coefficients can be treated locally as a perturbation of constant coefficient equations. From this fact Schauder [SC 4, 5] was able to construct a global theory, an extension of which is presented here. Basic to this approach are apriori estimates of solutions, extending those of potential theory to equations with Hölder continuous coefficients. These estimates provide compactness results that are essential for the existence and regularity theory, and since they apply to classical solutions under relatively weak hypotheses on the coefficients, they play an important part in the subsequent nonlinear theory.

To motivate the theory of this chapter we now consider a different approach to Poisson’s equation from that of Chapter 4. By the divergence theorem (equation (2.3)) a C2(Q) solution of Δu=f satisfies the integral identity for all φ ∈ C ₀ ¹ ∈ (Ω). The bilinear form is an inner product on the space C ₀ ¹ (Ω) and the completion of C ₀ ¹ (Ω) under the metric induced by (7.2) is consequently a Hubert space, which we call W ₀ ^1,2 (Ω).

This chapter treats linear elliptic operators having principal part in divergence form under relatively weak smoothness assumptions on the coefficients.

Until now in this work we have concentrated on either weak or classical solutions of second-order elliptic equations; a weak solution need only be once weakly differentiable while a classical solution must be at least twice continuously differ-entiable. The formulation of the weak solution concept depended on the operator L under consideration having a “divergence form” while the concept of classical solution made sense for operators with completely arbitrary coefficients. In this chapter our concern is with the intermediate situation of strong solutions.

The purpose of this chapter is to provide various maximum and comparison principles for quasilinear equations which extend corresponding results in Chapter 3. We consider second order, quasilinear operators Q of the form (10.1) Qu = a ^ij(x, u, Du)D _iju + b(x, u, Du), a ^ij = a ^ji, where x = (x ₁..., x _n) is contained in a domain Ω of ℝⁿ, n ≥ 2, and, unless other-wise stated, the function u belongs to C ²(Ω). The coefficients of Q, namely the functions a ^ij(x, z, p), i, j= 1,..., n, b(x, z, p) are assumed to be defined for all values of (x, z, p) in the set Q × ℝ × ℝⁿ. Two operators of the form (10.1) will be called equivalent if one is a multiple of the other by a fixed positive function in Q × ℝ × ℝⁿ. Equations Qu = 0 corresponding to equivalent operators Q will also be called equivalent.

In this chapter the solvability of the classical Dirichlet problem for quasilinear equations is reduced to the establishment of certain apriori estimates for solutions. This reduction is achieved through the application of topological fixed point theorems in appropriate function spaces. We shall first formulate a general criterion for solvability and illustrate its application in a situation where the required apriori estimates are readily derived from our previous results. The derivation of these apriori estimates under more general hypotheses will be the major concern of the ensuing chapters.

In this chapter we derive interior and global Hölder estimates for the derivatives of solutions of quasilinear elliptic equations of the form in a bounded domain Ω. From the global results we shall see that Step IV of the existence procedure described in Chapter 11 can be carried out if, in addition to the hypotheses of Theorem 11.4, we assume that either the coefficients a ^ij are in C ^l(ΩΩ × ℝ × ℝⁿ) or that Q is of divergence form or that n = 2. The estimates of this chapter will be established through a reduction to the results of Chapter 8, in particular to Theorems 8.18, 8.24, 8.26 and 8.29.

An examination of the proof of Theorem 11.5 shows that for elliptic operators of the forms (11.7) or (11.8) the solvability of the classical Dirichlet problem with smooth data depends only upon the fulfillment of Step II of the existence procedure, that is, upon the existence of a boundary gradient estimate. In this chapter we provide a variety of hypotheses for the general equation, in Ω ⊂ ℝⁿ, that guarantee a boundary gradient estimate for solutions. These hypotheses are combinations of structural conditions on Q and geometric conditions on the domain Ω. It will be seen that the gradient bound aspect of the theory of quasilinear elliptic equations is not as profound as other aspects such as the Holder estimates of Chapters 6 and 13. The boundary gradient estimates are tied through the classical maximum principle to judicious and generally natural choices of barrier functions. Nevertheless these estimates are of considerable importance since they seem to be the principal factor in determining the solvability character of the Dirichlet problem. This will be evidenced by the non-existence results at the end of the chapter.

In this chapter we are mainly concerned with the derivation of apriori estimates for the gradients of C ²(Ω) solutions of quasilinear elliptic equations of the form in terms of the gradients on the boundary ∂Ω and the magnitudes of the solutions. The resulting estimates facilitate the establishment of Step III of the existence procedure described in Section 11.3. On combination with the estimates of Chapters 10,13 and 14, they yield existence theorems for large classes of quasilinear elliptic equations including both uniformly elliptic equations and equations of form similar to the prescribed mean curvature equation (10.7). Since the methods of this chapter involve the differentiation of equation (15.1), our hypotheses will generally require structural conditions to be satisfied by the derivatives of the coefficients a ^ij,b. In Section 15.4 we shall see that these derivative conditions can be relaxed somewhat for equations in divergence form, where different types of arguments are appropriate.

In this chapter we focus attention on both the prescribed mean curvature equation, and a related family of equations in two variables. Our main concern is with interior derivative estimates for solutions. We shall see that not only can interior gradient bounds be established for solutions of these equations but that also their non-linearity leads to strong second derivative estimates which distinguish them from uniformly elliptic equations such as Laplace’s equation. In particular we shall derive an extension of the classical result of Bernstein that a C ²(ℝ²) solution of the minimal surface equation in ℝ² must be a linear function (Theorem 16.12).