Related papers: Composite media, almost touching disks and the max…
In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a…
We establish a spatial gradient maximum principle for classical solutions to the initial and Neumann boundary value problem of some quasilinear parabolic equations on smooth convex domains.
We prove gradient estimates for solutions of the oblique derivative problem for a large class of elliptic and parabolic quasilinear PDEs. In particular, we expand on previous work of the author using a maximum principle argument. In…
The maximum principle for hyperbolic inversive distance circle packings on polyhedral surfaces is established,which unifies and generalizes existing maximum principles for various types of circle packings in the literature.As an application…
In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the…
We prove the validity of maximum principles for a class of fully nonlinear operators on unbounded subdomains $\Omega \subset \mathbb R^n$ of cylindrical type. The main structural assumption is the uniform ellipticity of the operator along…
In this note, we establish a boundary maximum principle for a class of stationary pairs of varifolds satisfying a fixed contact angle condition in any compact Riemannian manifold with smooth boundary.
This is a generalization of our prior work on the compact fixed point theory for the elliptic Rosseland-type equations. We obtain the maximum principle without the technical Steklov techniques. Inspired by the Rosseland equation in the…
A mesh condition is developed for linear finite element approximations of anisotropic diffusion-convection-reaction problems to satisfy a discrete maximum principle. Loosely speaking, the condition requires that the mesh be simplicial and…
We study the perfect conductivity problem when two perfectly conducting inclusions are closely located to each other in an anisotropic background medium. We establish optimal upper and lower gradient bounds for the solution in any dimension…
It has been a widely belief that for a planar convex domain with two coordinate axes of symmetry, the location of maximal norm of gradient of torsion function is either linked to contact points of largest inscribed circle or connected to…
We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields.…
When a convex perfectly conducting inclusion is closely spaced to the boundary of the matrix domain, a bigger convex domain containing the inclusion, the electric field can be arbitrary large. We establish both the pointwise upper bound and…
In this paper, we are concerned with the gradient estimate of the electric field due to two nearly touching dielectric inclusions, which is a central topic in the theory of composite materials. We derive accurate quantitative…
We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form equations, and the local boundedness turns out to be sharp in more than two dimensions, answering the `Moser…
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…
We present an asymptotic study of the Dirichlet to Neumann map of high contrast composite media with perfectly conducting inclusions that are close to touching. The result is an explicit characterization of the map in the asymptotic limit…
We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic…
We study the second order elliptic equations of non-divergence form in a planar domain with complicated geometry. In this case the domain winds around a fixed circle infinitely many times and converges to it when the rotating angle goes to…
We formulate a finite-particle method of mass transport that accounts for general mixed boundary conditions. The particle method couples a geometrically-exact treatment of advection; Wasserstein gradient-flow dynamics; and a…