Nonlinear differential inequalities and functions of compact support

Abstract:

This paper is concerned with strongly nonlinear (and possibly degenerate) elliptic partial differential equations in unbounded regions. To broaden the class of problems for which solutions exist, the equation and boundary conditions are expressed by use of set-valued functions; this involves no technical complications. The concept of “solution” is so formulated that existence is needed only in bounded regions. Uniform boundedness is first established, and compactness of support is then deduced by a comparison argument, similar to that in recent work of Brezis, but simpler in detail. The central problems here are not associated with the comparison argument, but with the nonlinearities. Our hypotheses are given only when $|{\text {grad}}\;u|$ is small, so that the minimal surface operator (for example) is just as tractable as the Laplacian. Further nonlinearity is allowed by the use of the Bernstein-Serrin condition on the quadratic form, and by a suitably generalized version of the Meyers-Serrin concept of essential dimension. Although the boundary can have corners, we allow nonlinear boundary conditions of mixed type. Counterexamples show that certain seemingly ad hoc distinctions are in fact necessary to the truth of the theorems.