Regularity for Fully Nonlinear Elliptic Equations with Neumann Boundary Data

We obtain local C ^α, C ^1,α, and C ^2,α regularity results up to the boundary for viscosity solutions of fully nonlinear uniformly elliptic second order equations with Neumann boundary conditions.     

REGULARITY OF FREE BOUNDARIES OF TWO-PHASE PROBLEMS FOR FULLY NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER. II. FLAT FREE BOUNDARIES ARE LIPSCHITZ

ABSTRACT

In this second paper, we continue our study on the regularity of free boundaries for some fully nonlinear elliptic equations. Our result is if the free boundary is trapped in a sufficiently narrow strip formed by two Lipschitz graphs, then it is also a Lipschitz graph. Combining with the results in Part 1 (see Ref. [Wang]), the free boundary is C ^1,α.     

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

We establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C^1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C^1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C^1,α, provided that the hyperbolic phase is close in C^1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C^2,α, the free boundary is C^2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.     

REGULARITY OF THE FREE BOUNDARY FOR SOME ELLIPTIC AND PARABOLIC PROBLEMS. I

We study the free boundary of solutions to some obstacle problems in the elliptic and parabolic cases. In the one-phase Stefan problem, the parabolic case, we prove that the points where the zero set has no density lie in a Lipschitz surface in space and time.

For some fully nonlinear elliptic equations of second order, we get similar results.

Furthermore, we prove the C ¹ regularity for singular points with some (n ? 1)-dimensional density.     

Existence of solutions of Two-Phase free boundary problems for fully nonlinear elliptic equations of second order

In this article, we have established existence of a solution to the 2 -phase free boundary problem for some fully nonlinear elliptic equations and also shown the free boundary has finite Hⁿ⁻¹ Hausdorff measure and a normal in a measuretheoretic sense Hⁿ⁻¹ almost everywhere. The regularity theory developed in [9] and [10] for this free boundary problem then leads to the fact that the free boundary is locally a C^1,α surface near Hⁿ⁻¹-a.e. point.     

Boundary Regularity for Viscosity Solutions of Fully Nonlinear Elliptic Equations

We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is C ^{1, α} on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is C ^{2, α} on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

Regularity of solutions to regular shock reflection for potential flow

The shock reflection problem is one of the most important problems in mathematical fluid dynamics, since this problem not only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional conservation laws that is still largely incomplete. However, most of the fundamental issues for shock reflection have not been understood, including the regularity and transition of different patterns of shock reflection configurations. Therefore, it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, for a regular reflection configuration, the potential flow governs the exact behavior of the solution in C ^1,1 across the pseudo-sonic circle even starting from the full Euler flow, that is, both of the nonlinear systems are actually the same in a physically significant region near the pseudo-sonic circle; thus, it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo-sonic circle (the transonic boundary from the elliptic to hyperbolic region) and at the point where the pseudo-sonic circle (the degenerate elliptic curve) meets the reflected shock (a free boundary connecting the elliptic to hyperbolic region). In this paper, we study the regularity of solutions to regular shock reflection for potential flow. In particular, we prove that the C ^1,1-regularity is optimal for the solution across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock. We also obtain the C ^2,α regularity of the solution up to the pseudo-sonic circle in the pseudo-subsonic region. The problem involves two types of transonic flow: one is a continuous transition through the pseudo-sonic circle from the pseudo-supersonic region to the pseudo-subsonic region; the other a jump transition through the transonic shock as a free boundary from another pseudo-supersonic region to the pseudo-subsonic region. The techniques and ideas developed in this paper will be useful to other regularity problems for nonlinear degenerate equations involving similar difficulties.     

Principal Eigenvalues for Isaacs Operators with Neumann Boundary Conditions

In this paper we show the existence of two principal eigenvalues associated to general non-convex fully nonlinear elliptic operators with Neumann boundary conditions in a bounded C ² domain. We study these objects and we establish some of their basic properties. Finally, Lipschitz regularity, uniqueness and existence results for the solution of the Neumann problem are given.      

Interior C1, γ‐regularity for weak solutions of nonlinear second orderelliptic systems

The interior C^{0, γ}‐regularity for the first gradient of a weak solution to a class of nonlinear second order elliptic systems is proved under the assumption that oscillations of coefficients are controlled by the ellipticity constant. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C^2,α$C^{2,\alpha }$-estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.     

Fully nonlinear singularly perturbed equations and asymptotic free boundaries

In this paper we study one-phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials, ζε(u) with ζε→δ₀⋅∫ζ. We establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non-degenerated. For problems governed by concave equations, we establish uniform weak geometric properties of approximating level surfaces. We also provide a thorough analysis of the free boundary problem obtained as a limit as the ε-parameter term goes to zero. We find the precise jumping condition of limiting solutions through the phase transition, which involves a subtle homogenization process of the governing fully nonlinear operator. In particular, for rotational invariant operators, F(D²u), we show the normal derivative of limiting function is constant along the interface. Smoothness properties of the free boundary are also addressed.     

C 1, α Interior Regularity for Nonlinear Nonlocal Elliptic Equations with Rough Kernels

We prove a C ^{1, α} interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.     

Partial regularity of solutions of fully nonlinear,uniformly elliptic equations

We prove that a viscosity solution of a uniformly elliptic, fully nonlinear equation is C^2,α on the complement of a closed set of Hausdorff dimension at most ? less than the dimension. The equation is assumed to be C¹, and the constant ? > 0 depends only on the dimension and the ellipticity constants. The argument combines the W^2,? estimates of Lin with a result of Savin on the C^2,α regularity of viscosity solutions that are close to quadratic polynomials. © 2012 Wiley Periodicals, Inc.     

Regularity in a one-phase free boundary problem for the fractional Laplacian

For a one-phase free boundary problem involving a fractional Laplacian, we prove that “flat free boundaries” are C^1,α$C^{1,\alpha }$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.     

Lipschitz Estimates in Almost‐Periodic Homogenization

We establish uniform Lipschitz estimates for second‐order elliptic systems in divergence form with rapidly oscillating, almost‐periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C^1,α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat nonsymmetric coefficients. We also obtain uniform W^1,p estimates.© 2016 Wiley Periodicals, Inc.     

Regularity theory for fully nonlinear integro‐differential equations

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C^{1, α} regularity for general fully nonlinear integro‐differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. © 2008 Wiley Periodicals, Inc.     

Regularity for the nonlinear Signorini problem

We study the regularity of the solution to a fully nonlinear version of the thin obstacle problem. In particular we prove that the solution is C1,α for some small α>0. This extends a result of Luis Caffarelli of 1979. Our proof relies on new estimates up to the boundary for fully nonlinear equations with Neumann boundary data, developed recently by the authors.     

A note on the regularity of the free boundaries in the optimal partial transport problem

In this work we study the global regularity of the free boundaries arising in the optimal partial transport problem. Assuming the supports of both the source and the target measure to be convex, we show that the free boundaries of the active regions are globally C ^0,1/2.      

Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C1,γ-smooth for some γ∈(0,1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.