On viscosity solutions for nontotally parabolic fully nonlinear equations

It is shown how to prove global unique solvability of the first initial-boundary value problem in the class of continuous viscosity solutions for some classes of equations −u_t+F(u_x,u_xx)=g(x, t, u_x), where F(p, A) is elliptic only on some nonlinear subsets of values of the arguments (p, A). For this purpose we use the techniques developed in the theory of viscosity solutions for degenerate elliptic equations. Bibliography: 12 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 112–130.     

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).     

Large viscosity solutions for some fully nonlinear equations

We study existence, uniqueness and asymptotic behavior near the boundary of solutions of the problem $$\left\{\begin{array}{ll}-F(D^{2} u) + \beta (u) = f \quad {\rm in} \, \Omega, \\ u = + \infty \quad \quad \quad \quad \quad \quad \,\,\,\, {\rm on}\, \partial \Omega, \end{array} \right.\quad \quad \quad \quad \quad {\rm (P)}$$ where Ω is a bounded smooth domain in ${{\mathbb R}^N, N >1 , F}$ is a fully nonlinear elliptic operator and β is a nondecreasing continuous function. Assuming that β satisfies the Keller–Osserman condition, we obtain existence results which apply to ${f \in L^\infty_{loc}(\Omega)}$ or f having only local integrability properties where viscosity solutions are well defined, i.e. ${f \in L^N_{loc}(\Omega)}$ . Besides, we find the asymptotic behavior near the boundary of solutions of (P) for a wide class of functions ${f \in \mathcal{C}(\Omega)}$ . Based in this behavior, we also prove uniqueness.     

Classical solutions for a class of fully nonlinear degenerate parabolic equations

This paper is concerned with the existence and comparison principle of classical solutions for a class of fully nonlinear degenerate parabolic 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.     

Pseudo-almost periodic viscosity solutions of second-order nonlinear parabolic equations

In this paper we generalize the comparison result of Bostan and Namah (2007) [8] to the second-order parabolic case and prove two properties of pseudo-almost periodic functions; then by using Perron’s method we prove the existence and uniqueness of time pseudo-almost periodic viscosity solutions of second-order parabolic equations under usual hypotheses.     

Strong solutions of a class of fully nonlinear degenerate parabolic equations

This paper is concerned with the Cauchy problem of a class of fully nonlinear degenerate parabolic equations with reaction sources. After establishing the necessary local existence theorems of strong solutions, we investigate the blow‐up and global existence profile. Copyright © 2015 John Wiley & Sons, Ltd.     

Good and viscosity solutions of fully nonlinear elliptic equations

We introduce the notion of a ``good" solution of a fully nonlinear uniformly elliptic equation. It is proven that ``good" solutions are equivalent to -viscosity solutions of such equations. The main contribution of the paper is an explicit construction of elliptic equations with strong solutions that approximate any given fully nonlinear uniformly elliptic equation and its -viscosity solution. The results also extend some results about ``good" solutions of linear equations.

     

A microscopic convexity principle for spacetime convex solutions of fully nonlinear parabolic equations

We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations. Under certain general structure condition, we establish a constant rank theorem for the spacetime convex solutions of fully nonlinear parabolic equations. At last, we consider the parabolic convexity of solutions to parabolic equations and the convexity of the spacetime second fundamental form of geometric flows.     

Gradient estimates for viscosity solutions of singular fully nonlinear elliptic equations

We establish Lipschitz regularity for solutions to a family of non-isotropic fully nonlinear partial differential equations of elliptic type. In general such a regularity is optimal. No sign constraint is imposed on the solution, thus limiting free boundaries may have two-phases. Our estimates are then employed in combination with fine regularizing techniques to prove existence of viscosity solutions to singular nonlinear PDEs.     

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.     

Analyticity of solutions to fully nonlinear parabolic evolution equations on symmetric spaces

It is shown that solutions to fully nonlinear parabolic evolution equations on symmetric Riemannian manifolds are real analytic in space and time, provided the propagator is compatible with the underlying Lie structure. Applications to Bellman equations and to a class of mean curvature flows are also discussed.     

Estimates of solutions of initial-irregular oblique derivative problems for fully nonlinear parabolic equations

In this paper the initial-irregular oblique derivative problems for fully nonlinear parabolic equations of second order are proposed, and then some a priori estimates of solutions for the above problems are given.     

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 ${\mathbb{R}^{n}}$ .