Related papers: Regularity and monotonicity for solutions to a con…

We study the regularity of entropy solutions for quasilinear parabolic equations with anisotropic degeneracy and stochastic forcing. Building on previous works, we establish space-time regularity under a non-degeneracy condition that does…

In this paper, we study the evolution of smooth, closed planar curves under a fourth order biharmonic flow with an external forcing term. Such flows arise naturally in the theory of biharmonic maps and geometric variational problems…

We study unstable epitaxy on singular surfaces using continuum equations with a prescribed slope-dependent surface current. We derive scaling relations for the late stage of growth, where power law coarsening of the mound morphology is…

This is a study of a class of nonlocal nonlinear diffusion equations. We present a strong maximum principle for nonlocal time-dependent Dirichlet problems. Results are for bounded functions of space, rather than (semi)-continuous functions.…

We study regularity properties for solutions to elliptic equations that are degenerate or singular along orthogonal hyperplanes. The degenerate ellipticity is carried out by a weight term which is the monomial product of different powers of…

In this manuscript, we study a nonlinear model of tumor growth, described by a coupled hyperbolic-elliptic system of partial differential equations. In this model, the compressible flow of tumor cells is modeled by a transport equation for…

In this work, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive…

In this paper, we develop systematically the pointwise regularity for viscosity solutions of fully nonlinear elliptic equations in general forms. In particular, the equations with quadratic growth (called natural growth) in the gradient are…

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…

In this work, we study nonlocal differential equations with particular focus on those with reflection in their argument and piecewise constant dependence. The approach entails deriving the explicit expression of the solution to the linear…

The nonlocal models of peridynamics have successfully predicted fractures and deformations for a variety of materials. In contrast to local mechanics, peridynamic boundary conditions must be defined on a finite volume region outside the…

This is the first of a series of papers on the interior regularity of fully nonlinear degenerate elliptic equations. We consider a stochastic optimal control problem in which the diffusion coefficients, drift coefficients and discount…

In this paper we consider the non-cutoff Boltzmann equation in spatially inhomogeneous case. We prove the propagation of Gevrey regularity for the so-called smooth Maxwellian decay solutions to the Cauchy problem of spatially inhomogeneous…

We consider a nonlocal functional equation that is a generalization of the mathematical model used in behavioral sciences. The equation is built upon an operator that introduces a convex combination and a nonlinear mixing of the function…

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…

During the life of animals, epithelial tissues undergo extensive deformations--first to form organs during embryogensis and later to preserve integrity and function in adulthood. To what extent these deformations resemble that of non-living…

In this paper we study the regularity of the local minima of integral functionals: in particular, not convexity (quasi-convexity, policonvexity or rank one convexity) hypothesis will be made on the density, neither structure hypothesis nor…

This work studies a variational formulation and numerical solution of a regularized morphoelasticity problem of shape evolution. The foundation of our analysis is based on the governing equations of linear elasticity, extended to account…

In this paper we study the regularity properties of solutions to the Davey-Stewartson system. It is shown that for initial data in a Sobolev space, the nonlinear part of the solution flow resides in a smoother space than the initial data…

We study an initial value problem for two-dimensional needle crystal growth with anisotropic surface tension. The initial value problem is derived from the so called one-sided model based on complex variables method. We then obtain the…