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In 1982, J. Ball formulated a pioneering theory on the existence and uniqueness of weak radial equilibria to the pure displacement boundary value problem associated with isotropic, frame-invariant strain-energy functions in nonlinear hyperelasticity. In the theory [Bal82], he posed the following question: "Does strong ellipticity ('of the stored energy') imply that all solutions to the equilibrium equations which pass through the origin and have finite energy are trivial?" J. Ball's work depended critically on the number of elasticity dimensions.In this chapter, we will present models in n-dimensional elasticity that establish that the answer to J. Ball's question is negative. This work extends to higher dimensional elasticity the approach and results we presented, for the first time, on this question in [Ha07]. These models also provide further namely, that of regularity of weak equilibria, which would be hard to gain by other methods such as the common, but delicate, phase plane analysis.

All undergraduate heat transfer textbooks available today give an overriding role to the method of separation of variables to the point of excluding other frameworks and views for solving the underlying governing PDEs. However, the transition from a linear to a nonlinear heat transfer model not only makes the method of separation of variables inapplicable, but also introduces additional mathematical and computational difficulties that must be studied further and overcome. Yet, none of these textbooks discuss integral methods for solving the governing PDEs in heat transfer which are at least as good as the common separation of variables and finite difference techniques taught in the classroom. In this paper, we extend our new methodology from the linear to the nonlinear heat conduction problems by bringing such powerful methods to the undergraduate heat transfer classroom with no prior student experience with PDEs [1]. Integral methods of Von Kármán together with Ritz and Kantorovich methods are used to show our students in the undergraduate heat transfer course how to find approximate analytic solutions to nonlinear multidimensional steady and unsteady conduction problems involving surface radiation and temperature-dependent thermo-physical properties under distinct temperature profiles. The approach has a certain elegance in the sense that it expresses the complete physical effect of the system in terms of a single integral representing the first law of thermodynamics; moreover, the implications of using integral methods in this undergraduate course show the value of mathematical simplification in reducing the order of the governing PDEs and/or the number of associated independent variables. No knowledge of separation of variables or transform methods is needed to obtain an approximate analytic solution to such nonlinear multidimensional steady or unsteady problems with accuracy acceptable by most engineering standards.