Introduction Uniform estimates on the $p$-Laplacian, limits as $p\to\infty$ The transport set and transport rays Differentiability and smoothness properties of the potential Generic properties of transport rays Behavior of the transport density along rays Vanishing of the transport density at the ends of rays Approximate mass transfer plans Passage to limits a.e. Optimality Appendix: Approximating semiconcave and semiconvex functions by $C^2$ functions Bibliography.

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Differential equations methods for the Monge-Kantorovich mass transfer problem

In this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values. Geometrically, this can be interpreted as finding a minimal Lipschitz extension. Due to the weak convexity of the functional associated to this problem, solutions are generally nonunique. By adopting G. Aronsson's notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation. In fact, we actually prove a weak maximum principle for this partial differential equation, which in some sense is the Euler equation for the minimization problem. This is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity. To overcome this difficulty we use the weak solutions of M. G. Crandall and P.-L. Lions, also known as viscosity solutions, in conjunction with some arguments using integration by parts.

Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient

Le t F be a compac t set in Euc l idean n-space RL Suppose t h a t the funct ion ~(P) is def ined on F and t h a t i t satisfies a Lipschi tz condit ion. 1 I t is known t h a t ~0 can be e x t e n d e d to R ~ in such a w a y t h a t the new funct ion / satisfies t he same Lipsehi tz condit ion. An expl ic i t cons t ruc t ion was given in [1], where even more genera l s i tuat ions were t r ea t ed . I t is ea sy to give uppe r and lower bounds for the solutions. W e shal l do this, and also discuss quest ions of uniqueness. I t t u rns out t h a t a l l solut ions agree on a set E which has a ve ry s imple s t ruc ture (Theorem 2). I n ana logy wi th the t r e a t m e n t in [2] a n d [3], we shal l consider a subclass of solut ions (called absolu te minimals) which have the add i t i ona l p r o p e r t y of being solut ions of a corresponding p rob lem on each subregion of R ~ F . The pa r t i a l d i f ferent ia l equa t ion ~ . j=l, tx~ r tx~xj = 0 is de r ived in a pu re ly fo rmal m a n n e r and i t t u rns out t h a t a smooth funct ion satisfies th is equa t ion if and on ly if i t is an absolu te min ima l (Theorem 8). W e shal l also give an exis tence proof for abso lu te minima]s. I n a l a t e r paper , the two-d imens iona l case, a n d in pa r t i cu l a r the d i f ferent ia l equat ion r = 0, more r162 r tzu +Ca tuu will be s tud ied closely.

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Extension of functions satisfying lipschitz conditions

We consider a class of zero-sum two-player stochastic games called tug-of-war and use them to prove that every bounded real-valued Lipschitz function F on a subset Y of a length space X admits a unique absolutely minimal (AM) extension to X, i.e., a unique Lipschitz extension u : X → ℝ for which Lip U u = Lip ∂u u for all open U ⊂ X \ Y.

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Tug-of-war and the infinity Laplacian

I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the Dark Side...

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Regularity of minima: an invitation to the Dark Side of the Calculus of Variations

A detailed analysis of the class of absolutely minimizing functions in Euclidean spaces and the relationship to the infinity Laplace equation

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A tour of the theory of absolutely minimizing functions

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Minimization problems for the functional supxF(x, f(x), f′(x))

Fast/Slow Diffusion and Growing Sandpiles

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Gunnar Aronsson

Papers

Extension of functions satisfying lipschitz conditions

A tour of the theory of absolutely minimizing functions

Minimization problems for the functional supxF(x, f(x), f′(x))

Fast/Slow Diffusion and Growing Sandpiles

On the partial differential equation u ${}_{x}^{2}$ uxx+2uxuyuxy+u ${}_{y}^{2}$ uyy=0