Optimal Regularity of Lower-Dimensional Obstacle Problems

In this paper, we prove that solutions to the “boundary obstacle problem” have the optimal regularity, C^1,1/2, in any space dimension. This bound depends only on the local L²-norm of the solution. Main ingredients in the proof are the quasiconvexity of the solution and a monotonicity formula for an appropriate weighted average of the local energy of the normal derivative of the solution. Bibliography: 8 titles. Dedicated to Nina Nikolaevna Uraltseva on the occasion of her 70th birthday __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 49–66.     

Existence and regularity properties of non-isotropic singular elliptic equations

We establish existence and sharp regularity results for solutions to singular elliptic equations of the order u ^−β , 0 < β < 1, with gradient dependence and involving a forcing term λ f(x, u). Our approach is based on a singularly perturbed technique. We show that if the forcing parameter λ > 0 is large enough, our solution is positive. For λ small solutions vanish on a nontrivial set and therefore they exhibit free boundaries. We also establish regularity results for the free boundary and study the asymptotic behavior of the problem as b\searrow 0{\beta\searrow 0} and b\nearrow 1{\beta\nearrow 1}. In the former, we show that our solutions u _β converge to a C ^1,1 function which is a solution to an obstacle type problem. When b\nearrow 1{\beta\nearrow 1} we recover the Alt-Caffarelli theory.     

Classical solutions of fully nonlinear parabolic equations

In this paper, we study the classical solutions of the fully nonlinear parabolic equation u_t-F(D_x²u)=0,{u_{t}-F(D_{x}^2u)=0,} where the nonlinear operator F is locally C ^1,β almost everywhere with 0 < β < 1. The interior C ^2,α regularity of the classical solutions will be shown without the assumption that F is convex (or concave).     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two

We prove C ¹ regularity of c-convex weak Alexandrov solutions of a Monge–Ampère type equation in dimension two, assuming only a bound from above on the Monge–Ampère measure. The Monge–Ampère equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely non-negative cost-sectional curvature, a condition introduced in Ma et al. (Arch Ration Mech Anal 177(2):151–183, 2005), that was shown in Loeper (Acta Math, to appear) to be necessary for C ¹ regularity. Such a condition holds in particular for the case “cost = distance squared” which leads to the usual Monge–Ampère equation det D ² u = f. Our result is in some sense optimal, both for the assumptions on the density [thanks to the regularity counterexamples of Wang (Proc Am Math Soc 123(3):841–845, 1995)] and for the assumptions on the cost-function [thanks to the results of Loeper (Acta Math, to appear)].     

<Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Bold">1,</Emphasis><Emphasis Type="Italic">α</Emphasis></Superscript> regularity for infinity harmonic functions in two dimensions

We propose a new method for showing C ^{1, α} regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE. LCE is supported in part by NSF Grant DMS-0500452. OS was supported in part by the Miller Institute for Basic Research in Science, Berkeley.     

Variation on a theme of caffarelli and vasseur

Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur have shown that a certain class of weak solutions to the drift diffusion equation with initial data in L² gain H¨older continuity, provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on the BMO norm of a smooth velocity implies a uniform bound on the C^β norm of the solution for some β > 0. We apply elementary tools involving the control of H¨older norms by using test functions. In particular, our approach offers a third proof of the global regularity for the critical surface quasigeostrophic (SQG) equation in addition to the two proofs obtained earlier. Bibliography: 6 titles.     

Orlicz regularity for higher order parabolic equations in divergence form with coefficients in weak BMO

We consider higher order parabolic equations in divergence form with measurable coefficients to find optimal regularity in Orlicz spaces of the maximum order derivatives of the weak solutions. The relevant minimal regularity requirement on the tensor matrix coefficients is of small BMO in the spatial variable and is measurable in the time variable. As a consequence we prove the classical W ^m,p regularity, m = 1, 2, . . . , 1 < p < ∞, for such higher order equations. In the same spirit the results easily extend to higher order parabolic systems as well as up to the boundary.     

Optimal Control of the Obstacle for an Elliptic Variational Inequality

An optimal control problem for an elliptic obstacle variational inequality is considered. The obstacle is taken to be the control and the solution to the obstacle problem is taken to be the state. The goal is to find the optimal obstacle from H ¹ ₀ (Ω) so that the state is close to the desired profile while the H ¹ (Ω) norm of the obstacle is not too large. Existence, uniqueness, and regularity as well as some characterizations of the optimal pairs are established. Accepted 11 September 1996     

Direct Epiperimetric Inequalities for the Thin Obstacle Problem and Applications

We introduce a new logarithmic epiperimetric inequality for the 2m -Weiss energy in any dimension, and we recover with a simple direct approach the usual epiperimetric inequality for the 3/2-Weiss energy. In particular, even in the latter case, unlike the classical statements, we do not assume any a priori closeness to a special class of homogeneous functions. In dimension 2 , we also prove for the first time the classical epiperimetric inequality for the (2m − 1/2)-Weiss energy, thus covering all the admissible energies. As a first application, we classify the global λ -homogeneous minimizers of the thin obstacle problem, with , showing as a consequence that the frequencies 3/2 and 2m are isolated and thus improving on the previously known results. Moreover, we give an example of a new family of (2m − 1/2) -homogeneous minimizers in dimension higher than 2 . Second, we give a short and self-contained proof of the regularity of the free boundary of the thin obstacle problem, previously obtained by Athanasopoulos, Caffarelli, and Salsa (2008) for regular points and Garofalo and Petrosyan (2009) for singular points. In particular, we improve the C¹ regularity of the singular set with frequency 2m by an explicit logarithmic modulus of continuity. © 2019 Wiley Periodicals, Inc.     

C ∞ regularity of solutions of an equation of Levi’s type in R 2 n +1

We study the regularity of the solutions u of a class of P.D.E., whose prototype is the prescribed Levi curvature equation in ℝ^{2 n +1}. It is a second-order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every function u∈C ². If the Levi curvature never vanishes, we represent the operator ℒ associated with the Levi equation as a sum of squares of non-linear vector fields which are linearly independent at every point. By using a freezing method we first study the regularity properties of the solutions of a linear operator, which has the same structure as ℒ. Then we apply these results to the classical solutions of the equation, and prove their C ^∞ regularity. Received: October 10, 1998; in final form: March 5, 1999?Published online: May 10, 2001     

On the smoothness of conjugation of circle diffeomorphisms with rigid rotations

We prove that any C^3+β-smooth diffeomorphism preserving the orientation of a circle with rotation number from the Diophantine class D_δ, 0 < β < δ < 1, is C^2+β−δ-smoothly conjugate to a rigid rotation of the circle by a certain angle. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 268–282, February, 2008.     

Stokes and Navier–Stokes problems in a half-space: the existence and uniqueness of solutions a priori nonconvergent to a limit at infinity

The Cauchy problem and the initial boundary value problem in the half-space of the Stokes and Navier–Stokes equations are studied. The existence and uniqueness of classical solutions (u, π) (considered at least C ² × C ¹ smooth with respect to the space variable and C ¹ × C ⁰ smooth with respect to the time variable) without requiring convergence at infinity are proved. A priori the fields u and π are nondecreasing at infinity. In the case of the Stokes problem, the existence, for any t > 0, and the uniqueness of solutions with kinetic field and pressure field are established for some β ∈ (0, 1) and γ ∈ (0, 1 − β). In the case of Navier–Stokes equations, the existence (local in time) and the uniqueness of classical solutions to the Navier–Stokes equations are shown under the assumption that the initial data are only continuous and bounded, by proving that, for any t ∈ (0, T), the kinetic field u(x, t) is bounded and, for any γ ∈ (0, 1), the pressure field π(x, t) is O(1 + |x| ^γ ). Bibliography: 20 titles. To V. A. Solonnikov on his 75th birthday Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 176–240.     

Rigidity of critical circle mappings I

We prove that two C ³ critical circle maps with the same rotation number in a special set ? are C ^1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C ⁰ sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ^∞ critical circle maps with the same rotation number that are not C ^1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers. Received November 1, 1998 / final version received July 7, 1999     

Compact and subelliptic estimates for the $${\overline{\partial}}$$-Neumann operator on C2 pseudoconvex domains

In this paper, we will show that Catlin’s property (P) implies regularity for the -Neumann operator in W ¹ on C ² domains. We will also look at the subelliptic case and derive the natural generalization to C ^k domains when k ≥  2.     

Hölder regularity of optimal mappings in optimal transportation

It is known that optimal mappings in optimal transportation problems are uniquely determined by corresponding potential functions. In this paper we prove various local properties of potential functions. In particular we obtain the C ^1,α regularity of potential functions with optimal exponent α, which improves previous regularity results of Loeper.     

Nonlinear PDE's for stochastic optimal control with switchings and impulses

We show existence, uniqueness, andW ^2, -regularity of the system of nonlinear partial differential equations, associated with stochastic optimal control problems involving costly switchings and impulses. The impulse obstacle is approximated by a sequence of switching obstacles in the proof of regularity. The semiconcavity of the impulse obstacle is exploited.     

A local regularity of the complex Monge–Ampère equation

We prove a local regularity (and a corresponding a priori estimate) for plurisubharmonic solutions of the nondegenerate complex Monge–Ampère equation assuming that their W ^{2, p} -norm is under control for some p > n(n − 1). This condition is optimal. We use in particular some methods developed by Trudinger and an estimate for the complex Monge–Ampère equation due to Kołodziej.     

Sous ellipticite dans le cadre du calcul S(m,g)

Abstract

We study the regularity of the free boundary in the two membranes problem. We prove that around any point the free boundary is either a C ^{1, α} surface or a cusp, as in the obstacle problem. We also prove C ^{1, 1} regularity for the pair of functions solving the problem.     

Global existence and uniform decay for the coupled Klein-Gordon-Schr?dinger equations

This paper is concerned to the existence, uniqueness and uniform decay for the solutions of the coupled Klein-Gordon-Schr?dinger damped equations where ω is a bounded domain of R ⁿ , n≤ 3, F : R ²→R is a C ¹-function; γ, β; θ are constants such that γ, β > 0 and 1 ≤ 2θ≤ 2. Received January 1999 – Accepted October 1999     

<Emphasis Type="Italic">C</Emphasis><Superscript>1+<Emphasis Type="Italic">α</Emphasis></Superscript>-Regularity for Two-Dimensional Almost-Minimal Sets in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We give a new proof and a partial generalization of Jean Taylor’s result (Ann. Math. (2) 103(3), 489–539, 1976) that says that Almgren almost-minimal sets of dimension 2 in ℝ³ are locally C ^1+α -equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in Ann. Fac. Sci. Toulouse 18(1), 65–246, 2009 and an extension of Reifenberg’s parameterization theorem (David et al. in Geom. Funct. Anal. 18, 1168–1235, 2008). The key idea is still that if X is the cone over an arc of small Lipschitz graph in the unit sphere, but X is not contained in a disk, we can use the graph of a harmonic function to deform X and substantially diminish its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in ℝ ⁿ , but in this setting our final regularity result on E may depend on the list of minimal cones obtained as blow-up limits of E at a point.