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.     

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.     

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.     

Convexity of solutions and estimates for fully nonlinear elliptic equations

The starting point of this work is a paper by Alvarez, Lasry and Lions (1997) concerning the convexity and the partial convexity of solutions of fully nonlinear degenerate elliptic equations. We extend their results in two directions. First, we deal with possibly sublinear (but epi-pointed) solutions instead of 1-coercive ones; secondly, the partial convexity of C² solutions is extended to the class of continuous viscosity solutions. A third contribution of this paper concerns C^1,1 estimates for convex viscosity solutions of strictly elliptic nonlinear equations. To finish with, all the tools and techniques introduced here permit us to give a new proof of the Alexandroff estimate obtained by Trudinger (1988) and Caffarelli (1989).     

On viscosity solutions of fully nonlinear equations with measurable ingredients

We study fully nonlinear, uniformly elliptic equations with measurable ingredients. Caffarelli's recent work on W^2,p estimates for viscosity solutions has led to significant progress in this area. Here we present a unified treatment of this theory based on an appropriate notion of viscosity solution. For instance, it is shown that strong solutions are viscosity solutions, that viscosity solutions are twice differentiable a.e., and that the pointwise derivatives satisfy the equation a.e. An important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations. This extends results of this type due to Evans and Krylov. Our work is to some extent expository, the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients. © 1996 John Wiley & Sons, Inc.     

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.     

Maximum principle and existence of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms

We study L^p-viscosity solutions of fully nonlinear, second-order, uniformly elliptic partial differential equations (PDE) with measurable terms and quadratic nonlinearity. We present a sufficient condition under which the maximum principle holds for L^p-viscosity solution. We also prove stability and existence results for the equations under consideration.     

Existence of Multiple Positive Solutions of Inhomogeneous Semi–Linear Elliptic Problems Involving Critical Exponents ∗

We consider the obstacle problem for the degenerate Monge-Ampére equation. We prove the existence of the greatest viscosity sub-solution u(x) below a given obstacle φ(x), and its C ^{1, 1}-regularity which is optimal. Then the solution satisfies the concave uniformly elliptic equation if it doesn't touch the obstacle. We use the author's previous work to show the C ^{1, α}-regularity of the free boundary, ?{u(x) = φ(x)}. Finally, we discuss the stability of this free boundary.     

Regularity of free boundaries of two‐phase problems for fully nonlinear elliptic equations of second order I. Lipschitz free boundaries are C1,α

In this paper, we study an extension of a C^1,α regularity theory developed by L. Caffarelli in [2] to some fully nonlinear elliptic equations of second order. In fact, we investigate a two‐phase free boundary problem in which a fully nonlinear elliptic equation of second order is verified by the solution in the positive and the negative domains. Assuming the free boundary is locally a Lipschitz graph, we have established the C^1,α regularity of the free boundary. © 2000 John Wiley & Sons, Inc.     

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

We prove weak and strong maximum principles, including a Hopf lemma, for C ² subsolutions to equations defined by linear, second-order, linear, elliptic partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the C ² subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol's vanishing locus. We obtain uniqueness and a priori maximum principle estimates for C ² solutions to boundary value and obstacle problems defined by these boundary-degenerate elliptic operators with partial Dirichlet or Neumann boundary conditions. We also prove weak maximum principles and uniqueness for W ^{1, 2} solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions.     

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.     

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.     

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 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.     

Existence of strong solutions of Pucci extremal equations with superlinear growth in Du

We prove existence of strong solutions of Pucci extremal equations with superlinear growth in Du and unbounded coefficients. We apply this result to establish the weak Harnack inequality for L^p-viscosity supersolutions of fully nonlinear uniformly elliptic PDEs with superlinear growth terms with respect to Du.      

Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds

We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold M with the sectional curvature bounded from below by −κ   for κ≥0$\kappa \ge 0$. In the elliptic case, Wang and Zhang [24] recently extended the results of [5] to nonlinear elliptic equations in nondivergence form on such M, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on M. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved.     

Concentrated terms and varying domains in elliptic equations: Lipschitz case

In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior, which is uniformly Lipschitz and nonlinear terms, are concentrated in a region, which neighbors the boundary of domain. We prove that this family of solutions converges to the solutions of a limit problem in H¹an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.Copyright © 2015 John Wiley & Sons, Ltd.     

On the Wl q-Regularity of Solutions to Systems of Differential Equations in the Case When the Equations Are Constructed from Discontinuous Functions

Some solution, final in a sense from the standpoint of the theory of Sobolev spaces, is obtained to the problem of regularity of solutions to a system of (generally) nonlinear partial differential equations in the case when the system is locally close to elliptic systems of linear equations with constant coefficients. The main consequences of this result are Theorems 5 and 8. According to the first of them, the higher derivatives of an elliptic C ^l -smooth solution to a system of lth-order nonlinear partial differential equations constructed from C ^l -smooth functions meet the local Hoelder condition with every exponent , 0<<1. Theorem 8 claims that if a system of linear partial differential equations of order l with measurable coefficients and right-hand sides is uniformly elliptic then, under the hypothesis of a (sufficiently) slow variation of its leading coefficients, the degree of local integrability of lth-order partial derivatives of every W ^l _q,loc-solution, q>1, to the system coincides with the degree of local integrability of lower coefficients and right-hand sides.     

Multi-valued solutions to fully nonlinear uniformly elliptic equations

In this paper, we discuss the existence and regularity of multi-valued viscosity solutions to fully nonlinear uniformly elliptic equations. We use the Perron method to prove the existence of bounded multi-valued viscosity solutions.     

A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR, UNIFORMLY ELLIPTIC EQUATIONS

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear,uniformly elliptic equations under Dirichlet boundary conditions. When ...