Lawrence C. Evans

Motion of level sets by mean curvature. I

We continue our investigation of the “level-set” technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature. The principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by K. Brakke: we prove that almost every level set of the solution to the mean curvature evolution PDE is in fact aunit-density varifold moving according to its mean curvature. In particular, a.e. level set is endowed with a kind of “geometric structure.” The proof utilizes compensated compactness methods to pass to limits in various geometric expressions.

Motion of level sets by mean curvature. II

We continue our investigation of the “level-set” technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature. The principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by K. Brakke: we prove that almost every level set of the solution to the mean curvature evolution PDE is in fact aunit-density varifold moving according to its mean curvature. In particular, a.e. level set is endowed with a kind of “geometric structure.” The proof utilizes compensated compactness methods to pass to limits in various geometric expressions.

The perturbed test function method for viscosity solutions of nonlinear PDE

The method of viscosity solutions for nonlinear partial differential equations (PDEs) justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions. We describe in this paper a “perturbed test function” device, which entails various modifications of the test functions by lower order correctors. Applications include homogenisation for quasilinear elliptic PDEs and approximation of quasilinear parabolic PDEs by systems of Hamilton-Jacobi equations.

Partial Regularity for Stationary Harmonic Maps into Spheres

In an interesting recent paper [12], F. HI~LEIN has shown that any weakly harmonic mapping from a two-dimensional surface into a sphere is smooth. I present here a kind of generalization to higher dimensions, asserting in effect that a stationary harmonic mapping from an open subset of Nn(n __> 3) into a sphere is smooth, except possibly for a closed singular set of (n 2 ) dimensional Hausdorff measure zero. My proof crucially depends upon several of H~L~INs observations (as streamlined by P.-L. LIoNs). To state the result precisely let us suppose that m, n => 2, U is a smooth open subset of Nn, and S m-1 denotes the unit sphere in Nm. A function u in the Sobolev space HI (U;Rm) , u = (u 1 . . . . ,urn), belongs to H I ( U ; S ml ) provided t u[ = 1 a.e. in U

A new proof of local C1,α regularity for solutions of certain degenerate elliptic p.d.e.

Abstract We prove Cloc1,α estimates for solutions u ϵ W1,p + 2 of the degenerate elliptic p.d.e. div (¦Du¦ p Du) = 0 (p > 0).

Optimal Switching for Ordinary Differential Equations

We consider the problem of controlling an ordinary differential equation, subject to positive switching costs, and show in particular that the value functions form the “viscosity solution” (cf. [6], [7]) of the dynamic programming quasi-variational inequalities. This interpretation allows for a rigorous application of various dynamic programming techniques.

Nonlinear evolution equations in an arbitrary Banach space

This paper proves the existence of an evolution operatorU(t, s)x0 corresponding to a weak or generalized solution of the differential equation:du (t)/dt +A (t)u(t) ∋ f(t), u(s) =x0,t ≧ s; the operatorsA(t) are eachm-accretive in a Banach spaceX and, loosely speaking, have an “L1 modulus of continuity” int. The continuity and differentiability properties ofU(t, s)x0 are investigated, and some simple examples are presented.

Effective Hamiltonians and Averaging¶for Hamiltonian Dynamics II

Abstract We extend to time-dependent Hamiltonians some of the PDE methods from our previous paper [E-G1], and in particular the theory of “effective Hamiltonians” introduced by Lions, Papanicolaou & Varadhan [L-P-V]. These PDE techniques augment the variational approach of Mather [Mt1,Mt2,Mt3,Mt4,M-F] and the weak KAM methods of Fathi [F1,F2,F3,F4,F5].We also provide a weak interpretation of adiabatic invariance of the action and suggest a formula for the Berry-Hannay geometric phase in terms of an effective Hamiltonian.

The infinity Laplacian, Aronsson’s equation and their generalizations

The infinity Laplace equation Δ ∞ u = 0 arose originally as a sort of Euler-Lagrange equation governing the absolute minimizer for the L ∞ variational problem of minimizing the functional ess-sup U |Du|. The more general functional ess-sup U F(x, u, Du) leads similarly to the so-called Aronsson equation A F [u] = 0. In this paper we show that these PDE operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to L ∞ variational problems, including two-person game theory with random order of play, rapid switching of states in control problems, etc. The resulting equations can be parabolic and inhomogeneous, equation types precluded in conventional L ∞ variational problems.

Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators

We prove under appropriate hypotheses that the Hamilton-JacobiBellman dynamic programming equation with uniformly elliptic operators, max¡^k<m{Lku —/*} = 0, has a classical solution u E C2,ß, for some (small) Holder exponent ß > 0.