Related papers: Viscosity solutions and the minimal surface system

First, a new sufficient condition for uniqueness of weak solutions is proved for the system of 2D viscous Primitive Equations. Second, global existence and uniqueness are established for several classes of weak solutions with partial…

We solve the Cauchy-Dirichlet problem for the minimal surface system in arbitrary dimension and codimension assuming a condition on the variation of the initial submanifold .

In this paper, we investigate the moduli of continuity for viscosity solutions of a wide class of nonsingular quasilinear evolution equations and also for the level set mean curvature flow, which is an example of singular degenerate…

We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local…

This is an announcement of a series of upcoming works on boundary regularity for area minimizing currents, one of which is in collaboration with Reinaldo Resende. The setting we consider is that of an area minimizing current with a smooth…

In an extended mean field game the vector field governing the flow of the population can be different from that of the individual player at some mean field equilibrium. This new class strictly includes the standard mean field games. It is…

This paper proves the existence of viscosity solutions of path dependent semilinear PDEs via Perron's method, i.e. via showing that the supremum of viscosity subsolutions is a viscosity solution. We use the notion of viscosity solutions…

We consider almost minimizers to the thin-one phase energy functional and we prove optimal regularity of the solution and partial regularity of the free boundary. We thus recover the theory for energy minimizers. Our methods are based on a…

In this article, a notion of viscosity solutions is introduced for fully nonlinear second order path-dependent partial differential equations in the spirit of [Zhou, Ann. Appl. Probab., 33 (2023), 5564-5612]. We prove the existence,…

The response of Newtonian liquids to small perturbations is usually considered to be fully described by homogeneous transport coefficients like shear and dilatational viscosity. However, the presence of strong density gradients at the…

In this paper we study a non strictly system of conservation law when viscosity is present and viscosity is zero, which is studied in [10]. We show the existence and uniqueness of the solution in the space of generalized functions of…

In this paper, we shall study the Dirichlet problem for the minimal surfaces equation. We prove some results about the boundary behaviour of a solution of this problem. We describe the behaviour of a non-converging sequence of solutions in…

Recently, it was realised that liquid viscosity has a lower bound which is nearly constant for all liquids and is governed by fundamental physical constants. This was supported by experimental data in noble and molecular liquids. Here, we…

We employ the viscosity solution technique to analyze optimal stopping problems with regime switching. Specifically, we obtain the viscosity property of value functions, the uniqueness of viscosity solutions, the regularity of value…

In this paper, we shall extend the definition of $\mathcal{C}$-subsolution condition and adapt the argument of Guo-Phong-Tong[18] to replace Alexandroff-Bakelman-Pucci estimate in complex cases. As an application, we shall define and study…

We introduce a notion of viscosity solutions for a general class of elliptic-parabolic phase transition problems. These include the Richards equation, which is a classical model in filtration theory. Existence and uniqueness results are…

We prove existence of weak solutions to a diffuse interface model describing the flow of a fluid through a deformable porous medium consisting of two phases. The system non-linearly couples Biot's equations for poroelasticity, including…

We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solutions are considered and compared. We establish existence and regularity results. In particular…

A system of linear equations is said underdetermined when there are more unknowns than equations. Such systems may have infinitely many solutions. In this case, it is important to single out solutions possessing special features. A well…

This is a survey paper for the recent results on and beyond propagation of singularities of viscosity solutions. We also collect some open problems in this topic.