James Serrin

A symmetry problem in potential theory

The proof of this result is given in Section 1 ; in Section 3 we give various generalizations to elliptic differential equations other than (1). Before turning to the detailed arguments it will be of interest to discuss the physical motivation for the problem itself. Consider a viscous incompressible fluid moving in straight parallel streamlines through a straight pipe of given cross sectional form f2. If we fix rectangular coordinates in space with the z axis directed along the pipe, it is well known that the flow velocity u is then a function of x, y alone satisfying the Poisson differential equation (for n = 2) A u = A i n f2

Local behavior of solutions of quasi-linear equations

This paper deals with the local behavior of solutions of quasi-linear partial differential equations of second order in n/> 2 independent variables. ~re shall be concerned specifically with the a priori majorization of solutions, the nature of removable singularities, and the behavior of a positive solution in the neighborhood of an isolated singularity. Corresponding results are for the most par t well known for the case of the Laplace equation; roughly speaking, our work constitutes an extension of these results to a wide class of non-linear equations. Throughout the paper we are concerned with real quasi-linear equations of the general form div .,4(x, u, uz) = ~(x, u, ux). (1)

Mathematical Principles of Classical Fluid Mechanics

Classical fluid mechanics is a branch of continuum mechanics; that is, it proceeds on the assumption that a fluid is practically continuous and homogeneous in structure. The fundamental property which distinguishes a fluid from other continuous media is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the common surface. Though this property is the basis of hydrostatics and hydrodynamics, it is by itself insufficient for the description of fluid motion. In order to characterize the physical behavior of a fluid the property must be extended, given suitable analytical form, and introduced into the equations of motion of a general continuous medium, this leading ultimately to a system of differential equations which are to be satisfied by the, velocity, density, pressure, etc. of an arbitrary fluid motion. In this article we shall consider these differential equations, their derivation from fundamental axioms, and the various forms which they take when more or less special assumptions concerning the fluid or the fluid motion are made.

The Problem of Dirichlet for Quasilinear Elliptic Differential Equations with Many Independent Variables

This paper is concerned with the existence of solutions of the Dirichlet problem for quasilinear elliptic partial differential equations of second order, the conclusions being in the form of necessary conditions and sufficient conditions for this problem to be solvable in a given domain with arbitrarily assigned smooth boundary data. A central position in the discussion is played by the concept of global barrier functions and by certain fundamental invariants of the equation. With the help of these invariants we are able to distinguish an important class of ‘ regularly elliptic5 equations which, as far as the Dirichlet problem is concerned, behave comparably to uniformly elliptic equations. For equations which are not regularly elliptic it is necessary to impose significant restrictions on the curvatures of the boundaries of the underlying domains in order for the Dirichlet problem to be generally solvable; the determination of the precise form of these restrictions constitutes a second primary aim of the paper. By maintaining a high level of generality throughout, we are able to treat as special examples the minimal surface equation, the equation for surfaces having prescribed mean curvature, and a number of other non-uniformly elliptic equations of classical interest.

Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities

1. In t roduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2. Liouville theorems I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3. Liouville theorems II . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4. The Harnack inequal i ty . . . . . . . . . . . . . . . . . . . . . . . . . 104 5. Universal a priori es t imates . . . . . . . . . . . . . . . . . . . . . . . 110 Chapte r II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6. A general integral inequali ty I . . . . . . . . . . . . . . . . . . . . . 114 7. A general integral inequali ty II . . . . . . . . . . . . . . . . . . . . . 124 8. Regular i ty theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 9. Historical note: Canchy and Liouville, a quest ion of priori ty . . 138 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

A mountain pass theorem

143 In other words, the mountain separating 0 and e may even be assumed of “zero” altitude and the conclusion is still true; moreover (as we shall show) if c = a the critical point can be chosen with llx,,ll = R. It is interesting to ask whether this extension of the Ambrosetti- Rabinowitz theorem remains true in the infinite-dimensional case. We prove here that if (1) is strengthened a little, to the form (1’) there exist real numbers u, r, R such that 0 a for every x E A := {X E X: Y a. Moreover, if c = a the critical point can be chosen with r < I/x0/I < R. Roughly speaking, in brief, the mountain pass theorem continues to hold for a mountain of zero altitude, provided it also has non-zero thickness; in addition, if c = a, the “pass” itself occurs precisely on the mountain-i.e., satisfying r < llroll < R. There are two interesting and immediate corollaries. The first one says that a C’ function which has IWO local minimum points also has a third critical point. The second states that a u-periodic C’ function with a local minimum e has a critical point x,, # e + ku, k = 0, + 1, +2 ,.... The precise statements of the above results will be given in the next sec- tions. The proof depends upon a lemma of Clark [3], formulated here in a version suitable to our purpose. In Section 3 some applications are presen- ted for the forced pendulum equation. 2.

On the definition and properties of certain variational integrals

Here R denotes an open set in the number space En and u = u(x) is a realvalued function defined on R. The integrand f(x, it, p) is assumed to be nonnegative and continuous. Moreover, we suppose throughout the paper that f is convex in p so that I[u] is the integral of a regular variational problem. For the purposes of the calculus of variations, and also for aesthetic reasons, it is natural to want the class of admissible functions u to be as large as possible. Now I[u], as it stands, is certainly well-defined for continuously differentiable functions, but once we go beyond this class there is some question as to the meaning of the integral. If measurable partial derivatives can be associated with u, then one can define I[u] simply as the Lebesgue integral of f(x, u, ut;). This procedure cannot be used indiscriminately, however, for it assigns the absurd value ff(x, u, O)dx to any nonconstant function u whose partial derivatives are zero almost everywhere. As an alternate definition of the integral, we have introduced in [13] a certain lower semicontinuous functional which in general agrees with I[u] whenever u is continuously differentiable, but which at the same time is defined for a much larger class of functions. For convenience in discussing these two integrals the former will be denoted simply by I[u] and the latter by g[u], (a formal definition of these quantities will be given in ?1). Both functionals I[i] and 4 [u] are of interest in the calculus of variations; it is the purpose of this paper to clarify the relation between them. An important illustration of the present situation may be found in the theory of area of a nonparametric surface. Indeed, let us denote by a [u] the Lebesgue area of a surface z=u(x, y) over a region R in the ordinary (x, y) plane, and set

Isolated singularities of solutions of quasi-linear equations

Here ~ is a given vector function of the variables x, u, ux, B is a given scalar function of the same variables, and u x = ( ~ u / ~ x l . . . . . ~u /Dxn) denotes the gradient of the dependent variable u = u(x), where x = (x 1 . . . . . x=). The final chapter of [1] t reated a var iety of problems concerning the behavior of solutions at an isolated singularity. In tha t chapter I found it necessary to impose the condition B-~0, a condition which had not been required in the preceding parts of [1]. The main purpose of the present paper is to show tha t this additional condition can be removed, and thus to complete, in an important way, the theory of the earlier paper. We assume as always that the functions Jd and B are defined for all points x in some connected open set (domain) ~ of the Euclidean number space E n, and for all va lues of u and ux. Furthermore, they are to satisfy inequalities of the general form

Extensions of the mountain pass theorem

On considere une fonctionnelle C 1 :I:X→R ou X est un espace de Banach reel et ou I satisfait la condition de compacite de Palais-Smale. On presente une serie de theoremes qui generalisent le theoreme du col

A strong maximum principle and a compact support principle for singular elliptic inequalities

Abstract Vazquez in 1984 established a strong maximum principle for the classical m-Laplace differential inequality Δ m u−f(u)≤0, where Δ m u =div(| Du | m −2 Du ) and f ( u ) is a non-decreasing continuous function with f (0)=0. We extend this principle to a wide class of singular inequalities involving quasilinear divergence structure elliptic operators, and also consider the converse problem of compact support solutions in exterior domains.