A deterministic‐control‐based approach motion by curvature

The level‐set formulation of motion by mean curvature is a degenerate parabolic equation. We show that its solution can be interpreted as the value function of a deterministic two‐person game. More precisely, we give a family of discrete‐time, two‐person games whose value functions converge in the continuous‐time limit to the solution of the motion‐by‐curvature PDE. For a convex domain, the boundary's “first arrival time” solves a degenerate elliptic equation; this corresponds, in our game‐theoretic setting, to a minimum‐exit‐time problem. For a nonconvex domain the two‐person game still makes sense; we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the “positive part of the curvature.” These results are unexpected, because the value function of a deterministic control problem is normally the solution of a first‐order Hamilton‐Jacobi equation. Our situation is different because the usual first‐order calculation is singular. © 2005 Wiley Periodicals, Inc.     

A global approach to fully nonlinear parabolic problems

We consider the general initial-boundary value problem

(1)        
(2)        
(3)        
where is a bounded open set in with sufficiently smooth boundary.  The problem (1)-(3) is first reduced to the analogous problem in the space with zero initial condition and

The resulting problem is then reduced to the problem where the operator satisfies Condition  This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces.  The local and global solvability of the operator equation are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.

     

Homogeneous solutions to fully nonlinear elliptic equations

We classify homogeneous degree solutions to fully nonlinear elliptic equations.

     

Singular viscosity solutions to fully nonlinear elliptic equations

We prove the existence of a viscosity solution of a fully nonlinear elliptic equation in 24 dimensions with blowing up second derivative.     

Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs

For a d‐dimensional diffusion of the form dX_t = μ(X_t)dt + σ(X_t)dW_t and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, Γ, and A solving the second‐order backward stochastic differential equation (2BSDE) If the associated PDE has a sufficiently regular solution, then it follows directly from Itô's formula that the processes solve the 2BSDE, where ?? is the Dynkin operator of X without the drift term. The main result of the paper shows that if f is Lipschitz in Y as well as decreasing in Γ and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then the existence of a solution (Y, Z,Γ, A) to the 2BSDE implies that the associated PDE has a unique continuous viscosity solution v and the process Y is of the form Y_t = v(t, X_t), t ∈ [0, T]. In particular, the 2BSDE has at most one solution. This provides a stochastic representation for solutions of fully nonlinear parabolic PDEs. As a consequence, the numerical treatment of such PDEs can now be approached by Monte Carlo methods. © 2006 Wiley Periodicals, Inc.     

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.     

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).     

Near operators theory and fully nonlinear elliptic equations

We give a short survey of the Campanato near operators theory and of its applications to fully nonlinear elliptic equations.     

Homogenization of fully nonlinear,uniformly elliptic and parabolic partial differential equations in stationary ergodic media

We study the homogenization of fully nonlinear, uniformly elliptic and parabolic second‐order partial differential equations in stationary ergodic media. © 2004 Wiley Periodicals, Inc.     

Long-time asymptotics for fully nonlinear homogeneous parabolic equations

We study the long-time asymptotics of solutions of the uniformly parabolic equation $$ u_t + F(D^2u) = 0 \quad{\rm in}\, {\mathbb{R}^{n}}\times \mathbb{R}_{+},$$ for a positively homogeneous operator F, subject to the initial condition u(x, 0) =  g(x), under the assumption that g does not change sign and possesses sufficient decay at infinity. We prove the existence of a unique positive solution Φ⁺ and negative solution Φ^?, which satisfy the self-similarity relations $$\Phi^\pm (x,t) = \lambda^{\alpha^\pm}\Phi^\pm ( \lambda^{1/2} x,\lambda t ).$$ We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to ${\Phi^+}$ ( ${\Phi^-}$ ) locally uniformly in ${\mathbb{R}^{n} \times \mathbb{R}_{+}}$ . The anomalous exponents α⁺ and α^? are identified as the principal half-eigenvalues of a certain elliptic operator associated to F in ${\mathbb{R}^{n}}$ .     

A maximum rank problem for degenerate elliptic fully nonlinear equations

The solutions to the Dirichlet problem for two degenerate elliptic fully nonlinear equations in n + 1 dimensions, namely the real Monge–Ampère equation and the Donaldson equation, are shown to have maximum rank in the space variables when n ≤ 2. A constant rank property is also established for the Donaldson equation when n = 3.     

Quasilinear parabolic and elliptic equations with nonlinear boundary conditions

This paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both Dirichlet and nonlinear Neumann boundary conditions. The equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient. The consideration of the class of equations is motivated by some heat-transfer problems where the heat capacity and thermal conductivity are both temperature dependent. The aim of the paper is to show the existence and uniqueness of a global time-dependent solution of the parabolic problem, existence of maximal and minimal steady-state solutions of the elliptic problem, including conditions for the uniqueness of a solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Applications are given to some heat-transfer problems and an extended logistic reaction–diffusion equation.     

Well-posedness of fully nonlinear and nonlocal critical parabolic equations

In this paper, we prove the existence of smooth solutions in Sobolev spaces to fully nonlinear and nonlocal parabolic equations with critical index. Our argument is to transform the fully nonlinear equation into a quasi-linear nonlocal parabolic equation.     

Dini estimates for nonlocal fully nonlinear elliptic equations

We obtain Dini type estimates for a class of concave fully nonlinear nonlocal elliptic equations of order $\sigma \in (0,2)$ with rough and non-symmetric kernels. The proof is based on a novel application of Campanato's approach and a refined $C^{\sigma +\alpha }$ estimate in [9].     

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.     

Fundamental solutions of homogeneous fully nonlinear elliptic equations

We prove the existence of two fundamental solutions Φ and of the PDE \input amssym $F(D^2\Phi) = 0 \quad {\rm in} \ {\Bbb{R}}^n \setminus \{ 0 \}$ for any positively homogeneous, uniformly elliptic operator F. Corresponding to F are two unique scaling exponents α*, > −1 that describe the homogeneity of Φ and . We give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation F(D²u) = 0, which is bounded on one side. A Liouville‐type result demonstrates that the two fundamental solutions are the unique nontrivial solutions of F(D²u) = 0 in \input amssym ${\Bbb{R}}^n \setminus \{ 0 \}$ that are bounded on one side in both a neighborhood of the origin as well as at infinity. Finally, we show that the sign of each scaling exponent is related to the recurrence or transience of a stochastic process for a two‐player differential game. © 2010 Wiley Periodicals, Inc.