Optimal Regularity of Viscosity Solutions of Fully Nonlinear Singular Equations and Their Limiting Free Boundary Problems
نویسنده
چکیده
In the present paper, we start the journey of investigation into fully nonlinear elliptic singular equations of the form F (D2u, x) = βε(uε), where βε(uε) converges to the Dirac delta measure δ0. We show optimal regularity, uniform in ε, as well as H1 compactness for Bellman’s singular equations. We also provide a complete picture of limiting one-dimensional profiles. The study of further geometric-measure properties as well as smoothness of the general limiting free boundary problem is currently in progress [16].
منابع مشابه
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 C1,α on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is C2,α on the b...
متن کاملLocal C Estimates for Viscosity Solutions of Neumann-type Boundary Value Problems
In this article, we prove the local C0,α regularity and provide C0,α estimates for viscosity solutions of fully nonlinear, possibly degenerate, elliptic equations associated to linear or nonlinear Neumann type boundary conditions. The interest of these results comes from the fact that they are indeed regularity results (and not only a priori estimates), from the generality of the equations and ...
متن کاملA General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations
In this paper we study the fully nonlinear free boundary problem { F (Du) = 1 a.e. in B1 ∩ Ω |Du| ≤ K a.e. in B1 \ Ω, where K > 0, and Ω is an unknown open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W 2,n solutions are locally C inside B1. Under the extra condition that Ω ⊃ {Du 6= 0} and a uniform thickness assumption on the coincidence s...
متن کاملFree Boundary Problems for Tumor Growth: a Viscosity Solutions Approach
The mathematical modeling of tumor growth leads to singular “stiff pressure law” limits for porous medium equations with a source term. Such asymptotic problems give rise to free boundaries, which, in the absence of active motion, are generalized Hele-Shaw flows. In this note we use viscosity solutions methods to study limits for porous medium-type equations with active motion. We prove the uni...
متن کاملNonlinear oblique derivative problems for singular degenerate parabolic equations on a general domain
We establish comparison and existence theorems of viscosity solutions of the initial-boundary value problem for some singular degenerate parabolic partial di/erential equations with nonlinear oblique derivative boundary conditions. The theorems cover the capillary problem for the mean curvature 1ow equation and apply to more general Neumann-type boundary problems for parabolic equations in the ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007