An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞

On the numerical approximation of $$\infty $$ ∞ -harmonic mappings

“…The proof can be found in [36] and is a minor vectorial extension of standard results on limits of p-harmonic functions as p → ∞. Since the proof utilises only arguments involving norms, the proof in the vectorial case is essentially the same as in the scalar case [19,36]. Let us also remind here the standard notion of weak solutions to the p-Laplacian: …”

“…This idea, which dates back to Aronsson, has been effectively put into action by applying the theory of viscosity solutions to the ∞-Laplacian (see e.g. [12,19] and references therein) which, in view of the equivalence of weak and viscosity solutions [18,31] for the p finite case, is very stable under limits. Further, in view of the uniqueness in the scalar case, all subsequential limits as p → ∞ give rise to a viscosity solution of the limit equation.…”

“…Indeed, Aronsson himself demonstrated the existence of singular solutions in [6,7] which are minimisers of the functional. The theory of viscosity solutions of Crandall-Ishii-Lions [12,14,19] proved to be the appropriate framework in which to seek "weak" solutions for scalar L ∞ variational problems.…”

A Lie symmetry analysis and explicit solutions of the two‐dimensional ∞‐Polylaplacian

“…In recent years, many partial differential equations (PDEs) that appear as Euler–Lagrange equations in L ∞ variational problems have drawn considerable attention, see Refs. and references therein. These equations are strongly nonlinear elliptic PDEs and appear in many important applications such as modes for traveling waves in suspension bridges, the modeling of granular matter, image processing, and game theory …”

“…There are many difficulties typically encountered in these variational problems and the study of the associated Euler-Lagrange equations obtained in this way is notoriously challenging. 4 Usually, solutions are nonclassical and need to be made sense of weakly. The correct notion of weak solutions in this context is that of viscosity solutions.…”