• Corpus ID: 119660624

Nonlinear elliptic Partial Differential Equations and p-harmonic functions on graphs

@article{Manfredi2012NonlinearEP,
  title={Nonlinear elliptic Partial Differential Equations and p-harmonic functions on graphs},
  author={Juan J. Manfredi and Adam M. Oberman and Alex P. Svirodov},
  journal={arXiv: Analysis of PDEs},
  year={2012}
}
In this article we study the well-posedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit of the theory of viscosity solutions for PDEs. The equations include the graph Laplacian, the $p$-Laplacian, the Infinity Laplacian, the Mean Curvature equation, and the Eikonal operator on the graph… 

Figures from this paper

NONLINEAR MEAN-VALUE FORMULAS ON FRACTAL SETS
In this paper we study the solutions to nonlinear mean-value formulas on fractal sets. We focus on the mean-value problem [Formula: see text] in the Sierpiński gasket with prescribed values [Formula:
Convex and quasiconvex functions in metric graphs
TLDR
This work describes the largest convex function as the unique largest viscosity subsolution to a simple differential equation, u ′′ = 0 on the edges, plus nonlinear transmission conditions at the vertices, that is, below the prescribed datum.
A Finite Difference Method for the Variational p-Laplacian
TLDR
To the best of the knowledge, this is the first monotone finite difference discretization of the variational p -Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.
Adaptive Finite Difference Methods for Nonlinear Elliptic and Parabolic Partial Differential Equations with Free Boundaries
TLDR
This article combines monotone finite difference methods with an adaptive grid refinement technique to produce a PDE discretization and solver which is applied to a broad class of equations, in curved or unbounded domains which include free boundaries.
Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs
In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues
Γ-CONVERGENCE OF GRAPH GINZBURG–LANDAU FUNCTIONALS
We study Γ-convergence of graph-based Ginzburg–Landau functionals, both the limit for zero diffusive interface parameter ε → 0 and the limit for infinite nodes in the graph m→∞. For general graphs we
Monotone and consistent discretization of the Monge-Ampère operator
TLDR
A novel discretization of the Monge-Ampere operator is introduced, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications, and achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid.
Gamma-convergence of graph Ginzburg-Landau functionals
We study Gamma-convergence of graph based Ginzburg-Landau functionals, both the limit for zero diffusive interface parameter epsilon->0 and the limit for infinite nodes in the graph m -> infinity.
Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth
TLDR
It is shown that the continuum limit of the p-eikonal equation on a random geometric graph recovers a geodesic density weighted distance in the continuum, and the p→∞ limit recovers shortest path distances.
...
...

References

SHOWING 1-10 OF 44 REFERENCES
Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian
Certain fully nonlinear elliptic Partial Differential Equations can be written as functions of the eigenvalues of the Hessian. These include: the Monge-Ampere equation, Pucci’s Maximal and Minimal
User’s guide to viscosity solutions of second order partial differential equations
The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence
A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions
This article considers the problem of building absolutely minimizing Lipschitz extensions to a given function. These extensions can be characterized as being the solution of a degenerate elliptic
COMPUTING THE CONVEX ENVELOPE USING A NONLINEAR PARTIAL DIFFERENTIAL EQUATION
A fully nonlinear partial differential equation for the convex envelope was recently introduced by the author. In this paper, the equation is discretized using a finite difference method. The
Convergence of approximation schemes for fully nonlinear second order equations
The convergence of a wide class of approximation schemes to the viscosity solution of fully nonlinear second-order elliptic or parabolic, possibly degenerate, partial differential equations is
An infinity Laplace equation with gradient term and mixed boundary conditions
We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equation ―Δ ∞ u ― β|Du| = f, subject to Dirichlet or mixed Dirichlet-Neumann boundary
Finite difference methods for the Infinity Laplace and p-Laplace equations
TLDR
A semi-implicit solver is built, which solves the Laplace equation as each step, and is fast in the sense that the number of iterations is independent of the problem size.
Geometric partial differential equations and image analysis
1. Basic mathematical background 2. Geometric curve and surface evolution 3. Geodesic curves and minimal surfaces 4. Geometric diffusion of scalar images 5. Geometric diffusion of vector valued
An Asymptotic Mean Value Characterization for a Class of Nonlinear Parabolic Equations Related to Tug-of-War Games
TLDR
It is shown that the value functions for tug-of-war games with noise approximate a solution to the PDE above when the parameter that controls the size of the possible steps goes to zero.
...
...