We show the existence of a Lipschitz viscosity solution $u$ in $Ω$ to a system of fully nonlinear equations involving Pucci-type operators. We study the regularity of the interface $\partial \{ u> 0 \}\cap\Om$ and we show that the viscosity inequalities of the system imply, in the weak sense, the free boundary condition $u^{+}_{ν_{+}} = u^{-}_{ν_{-}}$, and hence $u$ is a solution to a two-phase free boundary problem. We show that we can apply the classical method of sup-convolutions developed by the first author in \cite{caffarelli_harnack_1987,caffarelli_harnack_1989}, and generalized by Wang \cite{wang_regularity_2000,wang_regularity_2002} and Feldman \cite{Fel} to fully nonlinear operators, to conclude that the regular points in $\partial \{ u> 0 \}\cap\Om$ form an open set of class $C^{1,α}$. A novelty in our problem is that we have different operators, $\puccip$ and $\puccin$, on each side of the free boundary. In the particular case when these operators are the Pucci's extremal operators $\ppuccip$ and $\ppuccin$, our results provide an alternative approach to obtain the stationary limit %proof of existence to the one obtained from of a segregation model of populations with nonlinear diffusion in \cite{quitalo_free_2013}.
We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, α}$-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator.
We extend the Caffarelli-Świech-Winter $C^{1,α}$ regularity estimates to $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form with superlinear growth in the gradient and unbounded coefficients. As an application, in addition to the usual $W^{2,p}$ results, we prove the existence of positive eigenvalues for proper operators with nonnegative unbounded weight, in particular for Pucci's operators with unbounded coefficients.
We study the two membranes problem for two different fully nonlinear operators. We give a viscosity formulation for the problem and prove existence of solutions. Then we prove a general regularity result and the optimal $C^{1,1}$ regularity when the operators are the Pucci extremal operators. We also give an example that shows that no regularity for the free boundary is to be expected to hold in general.
The existence of global nonnegative martingale solutions to a cross-diffusion system of Shigesada-Kawasaki-Teramoto type with multiplicative noise is proven. The model describes the segregation dynamics of populations with an arbitrary number of species. The diffusion matrix is generally neither symmetric nor positive semidefinite, which excludes standard methods. Instead, the existence proof is based on the entropy structure of the model, approximated by a Wong-Zakai argument, and on suitable higher moment estimates and fractional time regularity. In the case without self-diffusion, the lack …
