1. Minimax problems for nonlinear differential operators and related questions; on the line, in the plane or in $R^n$

A) On the line

The first problem is basically clear from the three titles

• Minimization problems for the functional $sup_x F(x,f(x),f'(x))$}, Ark. Mat.6(1965), 33-53;
• Minimization problems for the functional $sup_x F(x,f(x),f'(x))$ (II)}, Ark. Mat.6(1966), 409-431;
• Minimization problems for the functional $sup_x F(x,f(x),f'(x))$ (III)}, Ark. Mat.7(1968), 509-512.

Discussion: the functional $sup_x F(...)$ is considered here for absolutely continuous scalar functions on a finite interval, taking prescribed values at the endpoints. Questions of existence, uniqueness and properties of a minimizing function are considered. The cost function $F(x,y,z)$ must satisfy some structure conditions. The concept of an absolutely minimizing function (= minimizing on sub-intervals) is introduced and studied. A kind of Euler-type equation is derived. It is proved that any absolutely minimizing function in fact satisfies this equation in a weakened sense.

The above problem has also been considered for vector-valued functions:

• Pontryagin's maximum principle and a minimax problem, Math. Scand. 29(1971), 55-71;
• On certain minimax problems and Pontryagin's maximum Principle, Calculus of variations and Partial Differential Equations, 2009-2010.

Discussion: The situation here is more difficult, but it turned out possible to adapt the Pontryagin principle to the present situation. Also here, an absolutely minimizing function is necessarily in $C^1$ and satisfies an Euler-type equation. The case of higher derivatives is also considered.

B) in $R^n$, $n\ge 2$.

It is well known that a scalar Lipschitz function, defined on the boundary of a bounded domain $\Omega \subset R^n$ can be extended (interpolated) into $\Omega$ without violating the Lipschitz condition. A "good extension" into $\Omega$ is one which does not violate the particular Lipschitz condition (the strongest) which is defined from the boundary values. An absolutely minimizing Lipschitz extension, called AMLE, is a Lipschitz function in $\Omega$ which, for each subdomain $\omega$, is a "good" extension into $\omega$ of its own values on $\partial \omega$. The AMLE concept turns out to be central for developing a more ambitious theory. A nonlinear Euler-type PDE is formally derived by approaching the problem by means of $L^p-norms$ of grad(u) for big p. Here is a main result: a function in $C^2$ is an AMLE if and only if it satisfies that particular PDE. These results and many others were derived in the paper

• Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6(1967), 551-561.

Important results for the non-smooth case were proved by R. Jensen in 1993.

C) In the plane

The above-mentioned Euler-type PDE here takes the nice form $$ u_x^2u_{xx}+ 2u_x u_y u_{xy}+u_y^2 u_{yy} =0. $$

It is often called the infinity Laplace equation. The theory for classical solutions of this equation is developed in some detail in the paper

• On the partial differential equation $$ u_x^2u_{xx}+ 2u_x u_y u_{xy}+u_y^2 u_{yy} =0, $$ Ark. Mat. 7(1968), 395-425.

Here is a central result: the gradient can never vanish for a non-constant $C^2$ solution of this equation. It is even bounded away from zero on any smooth, bounded domain.
Non-smooth, "singular" solutions have also been considered:

• On certain singular solutions of the partial differential equation $$ u_x^2u_{xx}+ 2u_x u_y u_{xy}+u_y^2 u_{yy} =0. $$ Manuscripta Mathematica 47(1984), 133-151;
• Construction of singular solutions to the p-harmonic equation and its limit equation for $p = \infty$.
  Manuscripta Mathematica 56(1986), 135-158.
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• Examples of infinity harmonic functions having singular lines, LiTH-Mat-R-2006-3.

Remarks: A well known singular solution is $u(x,y)= |x|^{4/3}- |y|^{4/3}$. It has been conjectured by several people that the "optimal smoothness" for viscosity solutions of the infinity Laplace equation in the plane should be $C^{1,1/3}$.

A comprehensive survey of the above area is given in

• A Tour of the Theory of Absolutely Minimizing Functions, by G.A., M.G. Crandall and P. Juutinen, Bull. Amer. Math. Soc. 41:4(2004), 439-505.

D) Other aspects. The paper

• Interpolation under a gradient bound, J. Austral. Math. Soc., 2009,

discusses the relationship (also local) between regularity and uniqueness for interpolation under the condition $|grad(u)|\le g(x),$ where $g(x)$ is a prescribed bound.