A note on the strong maximum principle for elliptic differential inequalities

AbstractWe consider the strong maximum principle and the compact support principle for quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. This allows us to give necessary and sufficient conditions for the validity of both principles

The strong maximum principle revisited

AbstractIn this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf's paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf's principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here.In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered.The results have unexpected ramifications for other problems, as will develop from the exposition, e.g. (i)two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3 and 4);(ii)the exterior Dirichlet boundary value problem (Section 5);(iii)the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7);(iv)Euler–Lagrange inequalities on a Riemannian manifold (Section 9);(v)comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10). The case of p-regular elliptic inequalities is briefly considered in Section 11

Global Nonexistence Theorems for Quasilinear Evolution Equations with Dissipation

The final publication is available at Springer via https://doi.org/10.1007/s002050050032. Levine, Howard A., and James Serrin. "Global Nonexistence Theorems for Quasilinear Evolution Equations with Dissipation." Archive for Rational Mechanics and Analysis 137, no. 4 (1997): 341-361. Posted with permission.</p

A new critical curve for a class of quasilinear elliptic systems

We study a class of systems of quasilinear differential inequalities associated to weakly coercive differential operators and power reaction terms. The main model cases are given by the $p$-Laplacian operator as well as the mean curvature operator in non parametric form. We prove that if the exponents lie under a certain curve, then the system has only the trivial solution. These results hold without any restriction provided the possible solutions are more regular. The underlying framework is the classical Euclidean case as well as the Carnot groups setting.Comment: 28 page

Singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities

We consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities. The dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect. We investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function. Necessary conditions to locate the possible concentration points are also given

The Three Rs and Beyond : Public Perceptions on the Role of Australian Local Government Today

Despite the growing consensus among local government scholars and practitioners that the sector has now moved beyond the ‘Three Rs’, there remains a trenchant perception in public debate that when local councils do more than provide the narrow range of local services to property they are overreaching. But to what extent are these views actually reflective of Australian public opinion? This article reports on the findings of a new national survey and analyses public perceptions on the changing role of local government in Australia. It reaches three key findings. The first is that Australians have now largely outgrown the three longstanding ideological underpinnings of Australian urban politics. The second is that Australians increasingly have an appetite for local government to address contentious cultural and political issues. Finally, the third is that local council category had little effect in determining how residents conceived of the role of local government

Nonlinear Hodge maps

We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models certain kinds of compressible flow. Conditions are found under which singular sets of prescribed dimension cannot occur. Various degrees of smoothness are proven for the sonic limit, high-dimensional flow, and flow having nonzero vorticity. The gradient flow of solutions is estimated. Implications for other quasilinear field theories are suggested.Comment: Slightly modified and updated version; tcilatex, 32 page

Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

We study the problem of the existence and nonexistence of positive solutions to a superlinear second-order divergence type elliptic equation with measurable coefficients $(*)$: $-\nabla\cdot a\cdot\nabla u=u^p$ in an unbounded cone--like domain $G\subset\bf R^N$ $(N\ge 3)$. We prove that the critical exponent $p^*(a,G)=\inf\{p>1 : (*) \hbox{has a positive supersolution in} G\}$ for a nontrivial cone-like domain is always in $(1,N/(N-2))$ and in contrast with exterior domains depends both on the geometry of the domain $G$ and the coefficients $a$ of the equation.Comment: 20 page