Infinity Laplace Equation Research Articles

Asymptotic estimates of large solutions to the infinity Laplacian equations

Asymptotic estimates of large solutions to the infinity Laplacian equations

Convexity Property of Finsler Infinity Harmonic Functions

In this paper, we investigate the convexity property of viscosity solutions to a homogeneous normalized Finsler infinity Laplacian equation. Weak and strong forms for convexity property have been addressed.

Boundary Behavior of Large Solutions to the Infinity Laplace Equations on the Half-Line

Boundary Behavior of Large Solutions to the Infinity Laplace Equations on the Half-Line

Viscosity solutions to the infinity Laplacian equation with lower terms

We establish the existence and uniqueness of viscosity solutions tothe Dirichlet problem $$\displaylines{ \Delta_\infty^h u=f(x,u), \quad \hbox{in } \Omega,\cr u=q, \quad\hbox{on }\partial\Omega,}$$ where \(q\in C(\partial\Omega)\), \(h>1\), \(\Delta_\infty^h u=|Du|^{h-3}\Delta_\infty u\). The operator \(\Delta_\infty u=\langle D^2uDu,Du \rangle\) is the infinity Laplacian which is strongly degenerate, quasilinear and it is associated with the absolutely minimizing Lipschitz extension. When the nonhomogeneous term \(f(x,t)\) is non-decreasing in \(t\), we prove the existence of the viscosity solution via Perron's method. We also establish a uniqueness result based on the perturbation analysis of the viscosity solutions. If the function \(f(x,t)\) is nonpositive (nonnegative) and non-increasing in \(t\), we also give the existence of viscosity solutions by an iteration technique under the condition that the domain has small diameter. Furthermore, we investigate the existence and uniqueness of viscosity solutions to the boundary-value problem with singularity $$\displaylines{ \Delta_\infty^h u=-b(x)g(u), \quad \hbox{in } \Omega, \cr u>0, \quad \hbox{in } \Omega, \cr u=0, \quad \hbox{on }\partial\Omega, }$$ when the domain satisfies some regular condition. We analyze asymptotic estimates for the viscosity solution near the boundary.

Lipschitz Regularity of Viscosity Solutions to the Infinity Laplace Equation

Lipschitz Regularity of Viscosity Solutions to the Infinity Laplace Equation

Uniform convergence rates for Lipschitz learning on graphs

Abstract Lipschitz learning is a graph-based semisupervised learning method where one extends labels from a labeled to an unlabeled data set by solving the infinity Laplace equation on a weighted graph. In this work we prove uniform convergence rates for solutions of the graph infinity Laplace equation as the number of vertices grows to infinity. Their continuum limits are absolutely minimizing Lipschitz extensions (AMLEs) with respect to the geodesic metric of the domain where the graph vertices are sampled from. We work under very general assumptions on the graph weights, the set of labeled vertices and the continuum domain. Our main contribution is that we obtain quantitative convergence rates even for very sparsely connected graphs, as they typically appear in applications like semisupervised learning. In particular, our framework allows for graph bandwidths down to the connectivity radius. For proving this we first show a quantitative convergence statement for graph distance functions to geodesic distance functions in the continuum. Using the ‘comparison with distance functions’ principle, we can pass these convergence statements to infinity harmonic functions and AMLEs.

Infinity Laplacian equations with singular absorptions

In this work, we study regularity properties for nonvariational singular elliptic equations ruled by the infinity Laplacian. We obtain optimal $C^{1,\alpha}$ regularity along the free boundary. We also show existence of solutions, nondegeneracy properties and fine geometric estimates for the free boundary.

Continuum Limit of Lipschitz Learning on Graphs

Tackling semi-supervised learning problems with graph-based methods has become a trend in recent years since graphs can represent all kinds of data and provide a suitable framework for studying continuum limits, for example, of differential operators. A popular strategy here is p-Laplacian learning, which poses a smoothness condition on the sought inference function on the set of unlabeled data. For p<infty continuum limits of this approach were studied using tools from varGamma -convergence. For the case p=infty , which is referred to as Lipschitz learning, continuum limits of the related infinity Laplacian equation were studied using the concept of viscosity solutions. In this work, we prove continuum limits of Lipschitz learning using varGamma -convergence. In particular, we define a sequence of functionals which approximate the largest local Lipschitz constant of a graph function and prove varGamma -convergence in the L^{infty }-topology to the supremum norm of the gradient as the graph becomes denser. Furthermore, we show compactness of the functionals which implies convergence of minimizers. In our analysis we allow a varying set of labeled data which converges to a general closed set in the Hausdorff distance. We apply our results to nonlinear ground states, i.e., minimizers with constrained L^p-norm, and, as a by-product, prove convergence of graph distance functions to geodesic distance functions.

Viscosity solutions to the infinity Laplacian equation with strong absorptions

&lt;p style='text-indent:20px;'&gt;In this paper, we study the Dirichlet problem of the inhomogeneous infinity Laplace equation with strong absorptions:&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation*} \begin{cases} \ \ \Delta_{\infty}u(x)-\lambda(x)(u(x))^{\gamma}_+ = f(x), \ &amp;amp;{\rm{in}}\ \Omega, \\ \ \ u(x) = \phi(x), \ &amp;amp;{\rm{on}}\ \partial\Omega, \end{cases} \end{equation*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;where &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \gamma\in(0, 3). $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; We first prove the existence and uniqueness of the viscosity solution to the Dirichlet problem. Then, under additional structure conditions on &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ f $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, we establish the &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ C^{\frac{4}{3-\gamma}} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; regularity of the viscosity solution across the free boundary and the non-degeneracy. Based on the comparison principle, we also obtain the stability result, that is the viscosity solution of the Dirichlet problem converges uniformly to the solution to the corresponding Dirichlet problem of the homogeneous equation when the inhomogeneous term converges uniformly to &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ 0. $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;&lt;/p&gt;

Regularity of Viscosity Solutions of the Biased Infinity Laplacian Equation

Regularity of Viscosity Solutions of the Biased Infinity Laplacian Equation

Comparison principle to the infinity Laplacian equation with lower term

Comparison principle to the infinity Laplacian equation with lower term

The Boundary Regularity for Normalized Infinity Laplace Equations

Infinity Laplace equations, which derive from minimal Lipschitz extensions and absolutely minimal variational problems, have been widely applied in zero-sum tug-of-war game, optimal transport, shape deformation and so on. However, due to the quasi-linearity, extreme degeneration (non-degeneration only in the gradient direction) and non-divergence of the equations, it is difficult to define its classical or weak solutions. After introducing the idea of viscosity solutions, the theoretical research of infinite Laplace equations begin to develop. We study the boundary Hölder regularity of solutions for inhomogeneous normalized infinite Laplace equations on bounded domains. Main ideas are as follows: Firstly, we get bounded estimate of solutions through constructing barrier functions about super(sub)-solutions. Secondly, we use iterative method to approach the solutions of equations. Finally, we obtain regularity estimates near the boundary by calculating error between barrier functions and solutions of equations. This paper proves that the visvosity solutions of inhomogeneous normalized infinite Laplace equations are Hölder continious on Lipschitz boundary provided that the region boundary is Lipschitz continuous, the right inhomogeneous term is positive (negative) continuous and the boundary values are Hölder continuous. On the basis of our conclusion, the global Hölder regularity theory of the normalized infinite Laplace equations can be obtained combining with the internal regularity estimates. In addition, this method can be extended to the boundary estimates of the infinite fractional Laplace equations.

无穷Laplace方程的超定边值问题

无穷Laplace方程的超定边值问题

On Anisotropic Optical Flow Inpainting Algorithms

This work describes two anisotropic optical flow inpainting algorithms. The first one recovers the missing flow values using the Absolutely Minimizing Lipschitz Extension partial differential equation (also called infinity Laplacian equation) and the second one uses the Laplace partial differential equation, both defined on a Riemmanian manifold. The Riemannian manifold is defined by endowing the plane domain with an appropriate metric depending on the reference video frame. A detailed analysis of both approaches is provided and their results are compared on three different applications: flow densification, occlusion inpainting and large hole inpainting.

Reaction–diffusion equations for the infinity Laplacian

We derive sharp regularity for viscosity solutions of an inhomogeneous infinity Laplace equation across the free boundary, when the right hand side does not change sign and satisfies a certain growth condition. We prove geometric regularity estimates for solutions and conclude that once the source term is comparable to a homogeneous function, then the free boundary is a porous set and hence, has zero Lebesgue measure. Additionally, we derive a Liouville type theorem. When near the origin the right hand side grows not faster than third degree homogeneous function, we show that if a non-negative viscosity solution vanishes at a point, then it has to vanish everywhere.

Pointwise boundary differentiability for the infinity Laplace equations

In this paper, we study the boundary differentiability for infinity harmonic functions at some general corner points. We prove the slope estimate and provide a necessary and sufficient condition of the boundary differentiability for infinity harmonic function at less thanπcorner points or greater thanπcorner points (as defined in the Introduction). The boundary differentiable conclusions of the homogeneous infinity Laplace equations also apply to the inhomogeneous case.

Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation

In this paper, we investigate positive viscosity solutions of a third degree homogeneous parabolic equation \begin{document}$ u^{2}u_{t} = \Delta_{\infty}u $\end{document} . We prove a comparison principle, existence and uniqueness of continuous positive viscosity solutions.

Infinity Laplace equation with a gradient term and a nonlinear inhomogeneous term

Infinity Laplace equation with a gradient term and a nonlinear inhomogeneous term

Some sharp Sobolev regularity for inhomogeneous infinity Laplace equation in plane

Suppose Ω⋐R2 and f∈BVloc(Ω)∩C0(Ω) with |f|>0 in Ω. Let u∈C0(Ω) be a viscosity solution to the inhomogeneous ∞-Laplace equation−Δ∞u:=−12∑i=12(|Du|2)iui=−∑i,j=12uiujuij=finΩ. The following are proved in this paper:(i)For α>3/2, we have |Du|α∈Wloc1,2(Ω), which is (asymptotic) sharp when α→3/2. Indeed, the function w(x1,x2)=−x14/3 is a viscosity solution to −Δ∞w=4334 in R2. For any p>2, we have |Dw|α∉Wloc1,p(R2) whenever α∈(3/2,3−3/p).(ii)For α∈(0,3/2] and p∈[1,3/(3−α)), we have |Du|α∈Wloc1,p(Ω), which is sharp when p→3/(3−α). Indeed, |Dw|α∉Wloc1,3/(3−α)(R2).(iii)For ϵ>0, we have |Du|−3+ϵ∈Lloc1(Ω), which is sharp when ϵ→0. Indeed, |Dw|−3∉Lloc1(R2).(iv)For α>0, we have −(|Du|α)iui=2α|Du|α−2f almost everywhere in Ω. Some quantative bounds are also given.

Some estimates and properties of solutions to the biased infinity Laplacian equations

Some estimates and properties of solutions to the biased infinity Laplacian equations