相关论文: Viscosity solutions and the minimal surface system
This is a very brief report on recent developments on the Dirichlet problem for the minimal surface system and minimal cones in Euclidean spaces. We shall mainly focus on two directions: (1) Further systematic developments after…
We consider the Cauchy problem for a class of nonlinear degenerate parabolic equa- tion with forcing. By using the vanishing viscosity method we obtain generalized solutions. We prove some regularity results about this generalized…
In this short note, we present new observations and examples concerning the existence and rigidity of solutions to the Allen-Cahn equation with degenerate minimal hypersurfaces as their limit interfaces.
A new concept of viscosity solutions, namely, the Hausdorff continuous viscosity solution for the Hamilton-Jacobi equation is defined and investigated. It is shown that the main ideas within the classical theory of continuous viscosity…
In this paper we introduce a notion of viscosity solutions for Eikonal equations defined on topological networks. Existence of a solution for the Dirichlet problem is obtained via representation formulas involving a distance function…
We introduce a new notion of viscosity solutions for a class of very singular nonlinear parabolic problems of non-divergence form in a periodic domain of arbitrary dimension, whose diffusion on flat parts with zero slope is so strong that…
Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this…
In this paper, we prove the existence of viscosity solutions to complex Hessian equations on compact Hermitian manifolds, assuming the existence of a strict subsolution in the viscosity sense. The results cover the complex Hessian quotient…
In this paper, we study a hydrodynamic system modeling the deformation of vesicle membranes in incompressible viscous fluids. The system consists of the Navier-Stokes equations coupled with a fourth order phase-field equation. In the three…
The goal of the present work is twofold. First we prove the existence of an Hilbert Manifold structure on the space of immersed oriented closed surfaces with three derivatives in $L^2$ in an arbitrary sub-manifold $M^m$ of an euclidian…
We study certain obstacle type problems involving standard and nonlocal minimal surfaces. We obtain optimal regularity of the solution and a characterization of the free boundary.
By making use of the approximation method, we obtain the existence and regularity of the viscosity solutions for the generalized mean curvature flow. The asymptotic behavior of the flow is also considered. In particular, the Dirichlet…
Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds,…
The fact that minimal surfaces in the four-dimensional Euclidean space admit natural parameters implies that any minimal surface is determined uniquely up to a motion by two curvature functions, satisfying a system of two PDE's (the system…
We show the existence and uniqueness of a continuous viscosity solution of a system of partial differential equations (PDEs for short) without assuming the usual monotonicity conditions on the driver function as in Hamad\`ene and Morlais's…
We consider free surface dynamics of a two-dimensional incompressible fluid with odd viscosity. The odd viscosity is a peculiar part of the viscosity tensor which does not result in dissipation and is allowed when parity symmetry is broken.…
We study the Navier-Stokes equations with an extra eddy viscosity term in the whole space in three dimensions. We introduce a suitable regularized system for which we prove the existence of a regular solution defined for all time. We prove…
We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to…
In this paper, a regularity result for the Willmore flow is presented. It is established by means of a truncated translation technique in conjunction with the Implicit Function Theorem.
We address the global persistence of analyticity and Gevrey-class regularity of solutions to the two and three-dimensional visco-elastic second-grade fluid equations. We obtain an explicit novel lower bound on the radius of analyticity of…