Fully Nonlinear Parabolic Equations and the Dini Condition

Interior regularity results for viscosity solutions of fully nonlinear uniformly parabolic equations under the Dini condition, which improve and generalize a result due to Kovats, are obtained by the use of the approximation lemma. Received November 21, 2000, Accepted April 27, 2001     

Viscosity solutions of fully nonlinear parabolic systems

In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second-order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in Proc. London Math. Soc. 63 (1991) 212-240 and Comm. Partial Differential Equations 16 (1991) 1095-1128.     

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.     

On the definition of viscosity solutions for parabolic equations

In this short note we suggest a refinement for the definition of viscosity solutions for parabolic equations. The new version of the definition is equivalent to the usual one and it better adapts to the properties of parabolic equations. The basic idea is to determine the admissibility of a test function based on its behavior prior to the given moment of time and ignore what happens at times after that.

     

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.     

Characterization of balls by Riesz-potentials

For a bounded convex domain and consider the unit- density Riesz-potential . We show in this paper that u  =  const. on ∂G if and only if G is a ball. This result corresponds to a theorem of L.E. Fraenkel, where the ball is characterized by the Newtonian-potential (α = 2) of unit density being constant on ∂G. In the case α = N the kernel |x − y| ^α-N is replaced by  − log|x − y| and a similar characterization of balls is given. The proof relies on a recent variant of the moving plane method which is suitable for Green-function representations of solutions of (pseudo-)differential equations of higher-order.      

Isolated singularities for some types of curvature equations

We consider the removability of isolated singularities for the curvature equations of the form H_k[u]=0, which is determined by the kth elementary symmetric function, in an n-dimensional domain. We prove that, for 1?k?n−1, isolated singularities of any viscosity solutions to the curvature equations are always removable, provided the solution can be extended continuously at the singularities. We also consider the class of “generalized solutions” and prove the removability of isolated singularities.     

椭圆型方程的边界正则性与Dini条件

笔者利用Caffarelli扰动方法,证明了Poisson方程和完全非线性一致椭圆型方程在Dini条件下其粘性解的边界正则性,从而给出了文[8]定理4.11不用位势理论的简化证明.     

MIXED INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS FOR ONE-DIMENSIONAL FULLY NONLINEAR SECOND ORDER ELLIPTIC AND PARABOLIC EQUATIONS

This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations （PDEs）. In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin （IP-DG） methods for approximating viscosity solutions of these fully nonlinear PDEs. In order to capture discontinuities of the second order derivative uxx of the solution u, three independent functions p1,p2 and p3 are introduced to represent numerical derivatives using various one-sided limits. The proposed DG frame- work, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative uxz. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator which is consistent with the differential operator and satisfies certain monotonicity （called g-monotonicity） properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG frame- work. The g-monotonicity gives the DG methods the ability to select the mathematically ＂correct＂ solution （i.e., the viscosity solution） among all possible solutions. Moreover, the g-monotonicity allows for the possible development of more efficient nonlinear solvers as the special nonlinearity of the algebraic systems can be explored to decouple the equations. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage the accuracy and efficiency of the proposed DG methods.     

On the Existence of Smooth Solutions for Fully Nonlinear Parabolic Equations with Measurable “Coefficients” without Convexity Assumptions

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable “coefficients” and bounded “free” term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.     

The W1,0∞ Estimate for Strong Solutions of Fully Nonlinear Parabolic PDE's

In this paper, we deal with the following fully nonlinear partial differential equation of parabolic type F(x, t, u,Du,D²u) + u_t = 0 in Q_T with the third boundary value conditions. We prove that under some structure conditions on F, the W^{2,1}_p (p > 3N + 1) strong solutions for the problem have the ^{1,0}_∞ a priori estimates.     

Singular sets for curvature equation of order k

We consider the removability of singular sets for the curvature equations of the form Hk[u]=ψ, which is determined by the kth elementary symmetric function, in an n-dimensional domain Ω. We prove that, for 1?k?n−1 and a compact set K whose (n−k)-dimensional Hausdorff measure is zero, any generalized solution to the curvature equation on Ω?K is always extendable to a generalized solution on the whole domain Ω.     

On symmetry properties of parabolic equations in bounded domains

We study symmetry properties of nonnegative bounded solutions of fully nonlinear parabolic equations on bounded domains with Dirichlet boundary conditions. We propose sufficient conditions on the equation and domain, which guarantee asymptotic symmetry of solutions.     

Global Lipschitz regularizing effects for linear and nonlinear parabolic equations

In this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic equations in (0,T)×R^N$(0,T)\times {\mathbb{R}}^N$. Our assumptions include the case that the coefficients be both unbounded and with very mild local regularity (possibly weaker than the Dini continuity), the estimates only depending on the parabolicity constant and on the modulus of continuity of coefficients (but not on their L^∞$L^{\infty }$-norm). Our proof provides the analytic counterpart to the probabilistic proof used in Priola and Wang (2006) [35] (J. Funct. Anal. 2006) to get this type of gradient estimates in the linear case. We actually extend such estimates to the case of possibly unbounded data and solutions as well as to the case of nonlinear operators including Bellman–Isaacs equations. We investigate both the classical regularizing effect (at time t>0$t>0$) and the possible conservation of Lipschitz regularity from t=0$t=0$, and similarly we prove global Hölder estimates under weaker assumptions on the coefficients. The estimates we prove for unbounded data and solutions seem to be new even in the classical case of linear equations with bounded and Hölder continuous coefficients. Applications to Liouville type theorems are also given in the paper. Finally, we compare in an appendix the analytic and the probabilistic approach discussing the analogy between the doubling variables method of viscosity solutions and the probabilistic coupling method.