相关论文: Existence of solution to modified Gursky-Streets e…

The Gursky-Streets equation are introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of $\sigma_2$ Yamabe problem in dimension…

Gursky-Streets introduced a formal Riemannian metric on the space of conformal metrics in a fixed conformal class of a compact Riemannian four-manifold in the context of the $\sigma_2$-Yamabe problem. The geodesic equation of…

This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity $g$ such that $g'(0)>0$. We show the existence of oscillating solutions, namely with an…

We establish that when n >= 2 and H is a C^1 Hamiltonian such that some level set contains a line segment, the Aronsson equation admits explicit entire viscosity solutions. They are superpositions of a linear part plus a Lipschitz…

For a class of fully nonlinear equations having second order operators which may be singular or degenerate when the gradient of the solutions vanishes, and having first order terms with power growth, we prove the existence and uniqueness of…

We investigate the Cauchy problem for a two-component generalization of the Novikov equation with cubic nonlinearity -- an integrable system whose solutions may develop strong nonlinear phenomena such as gradient blow-up and interactions…

In this article, we adapt the definition of viscosity solutions to the obstacle problem for fully nonlinear path-dependent PDEs with data uniformly continuous in $(t,\omega)$, and generator Lipschitz continuous in $(y,z,\gamma)$. We prove…

We illustrate an alternative derivation of the viscous regularization of the diffusion equation which was studied in [A. Novick-Cohen and R. L. Pego. {\em Trans. Amer. Math. Soc.}, 324:331--351]. We provide an alternative proof of existence…

We prove a global well-posedness and regularity result of strong solutions to a slightly modified Michelson-Sivashinsky equation in any spatial dimension and in the absence of physical boundaries. Local-in-time well-posedness (and…

A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz…

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for…

We show how a theorem about solvability in $C^{1,1}$ of special Isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate Isaacs equations. We apply it also to establish the…

We show that a viscosity solution of a uniformly elliptic, fully nonlinear equation which vanishes on an open set must be identically zero, provided that the equation is $C^{1,1}$. We do not assume that the nonlinearity is convex or…

In bounded domains, without any geometric conditions, we study the existence and uniqueness of globally Lipschitz and interior strong C^{1,1}, (and classical C^2), solutions of general semilinear oblique boundary value problems for…

We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $\sigma_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab}…

This paper introduces a convenient solution space for the uniformly elliptic fully nonlinear path dependent PDEs. It provides a wellposedness result under standard Lipschitz-type assumptions on the nonlinearity and an additional assumption…

In this paper we consider a class of fully nonlinear equations which cover the equation introduced by S. Donaldson a decade ago and the equation introduced by Gursky-Streets recently. We solve the equation with uniform weak $C^2$ estimates,…

We study the Dirichlet problem for a class of curvature equations arising from conformal geometry on Riemannian manifolds $(M^n, g)$ with boundary where $n \geq 3$. We prove there exists a unique solution using the continuity method which…

We prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the product of the maximal and minimal slopes is strictly less than 1. The curvature of these solutions solutions decays to 0 as $t$…

In this manuscript we present an approach to analyze the discontinuous Galerkin solution for general quasilinear elliptic problems. This approach is sufficiently general to extend most of the well-known discretization schemes, including…