Related papers: N-barrier maximum principle for degenerate ellipti…
We study the diffusive limit approximation for a nonlinear radiative heat transfer system that arises in the modeling of glass cooling, greenhouse effects and in astrophysics. The model is considered with the reflective radiative boundary…
In this paper we consider a one-dimensional fully parabolic quasilinear Keller-Segel system with critical nonlinear diffusion. We show uniform-in-time boundedness of solutions, which means, that unlike in higher dimensions, there is no…
In this paper, we consider anisotropic diffusion with decay, and the diffusivity coefficient to be a second-order symmetric and positive definite tensor. It is well-known that this particular equation is a second-order elliptic equation,…
We prove the non-degeneracy for the critical Lane--Emden system $$ -\Delta U = V^p,\quad -\Delta V = U^q,\quad U, V > 0 \quad \text{in } \mathbb{R}^N $$ for all $N \ge 3$ and $p,q > 0$ such that $\frac{1}{p+1} + \frac{1}{q+1} =…
The finite element solution of two-dimensional anisotropic diffusion problems is considered. A Delaunay-type mesh condition is developed for linear finite element approximations to satisfy a discrete maximum principle. The condition is…
We study the mathematical theory of second order systems with two species, arising in the dynamics of interacting particles subject to linear damping, to nonlocal forces and to external ones, and resulting into a nonlocal version of the…
We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary.…
We employ a generalization of Einstein's random walk paradigm for diffusion to derive a class of multidimensional degenerate nonlinear parabolic equations in non-divergence form. Specifically, in these equations, the diffusion coefficient…
We justify rigorously the non-equilibrium-diffusion limit of the compressible Euler model coupled with a radiative transfer equation arising in radiation hydrodynamics. For general initial data, we establish the uniform existence of the…
We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second…
The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces…
A self-consistent theory for the classical description of the interaction of light and matter at the nano-scale is presented, which takes into account spatial dispersion. Up to now, the Maxwell equations in nanostructured materials with…
We study some maximum principle preserving and energy stable schemes for the Allen-Cahn-Ohta-Kawasaki model with fixed volume constraint. With the inclusion of a nonlinear term in the Ohta-Kawasaki free energy functional, we show that the…
We provide a framework for high-order discretizations of nonlinear scalar convection-diffusion equations that satisfy a discrete maximum principle. The resulting schemes can have arbitrarily high order accuracy in time and space, and can be…
In this survey we formulate our results on different forms of maximum principles for linear elliptic equations and systems. We start with necessary and sufficient conditions for validity of the classical maximum modulus principle for…
The initial boundary value problem for a Cahn-Hilliard system subject to a dynamic boundary condition of Allen-Cahn type is treated. The vanishing of the surface diffusion on the dynamic boundary condition is the point of emphasis. By the…
This work is a series of two articles. The main goal is to rigorously derive the degenerate parabolic-elliptic Keller-Segel system in the sub-critical regime from a moderately interacting stochastic particle system. In the first article…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…
We prove sharp regularity estimates for solutions of highly degenerate fully nonlinear elliptic equations. These are free boundary models in which a nonlinear diffusion process drives the system only in the region where the gradient…
In this paper, we demonstrate the existence of positive solutions for certain weakly coupled elliptic systems of sublinear growth under homogeneous Dirichlet boundary conditions. Our findings generalize existing results related to sublinear…