laplacian infinity
209516427
Biased infinity Laplacian Boundary Problem on finite graphs
We provide an algorithm, running in polynomial time in the number of vertices, computing the unique solution to the biased infinity Laplacian Boundary Problem on finite graphs. The algorithm is based on the general outline and approach taken in the corresponding algorithm for the unbiased case provided by Lazarus et al. The new ingredient is an adjusted (biased) notion of a slope of a function on a path in a graph. The algorithm can be used to determine efficiently numerical approximations to the viscosity solutions of biased infinity Laplacian PDEs.
Yuval Peres Zoran Sunic
132184484
Bernoulli's Problem for the Infinity Laplacian Near a Set with Positive Reach
We consider the exterior as well as the interior free boundary Bernoulli problem associated with the infinity Laplacian under a non autonomous boundary condition. Recall that the Bernoulli problem involves two domains: one is given, the other is unknown. Concerning the exterior problem we assume that the given domain has a positive reach, and prove an existence and uniqueness result together with an explicit representation of the solution. Concerning the interior problem, we obtain a similar result under the assumption that the complement of the given domain has a positive reach. In particular, for the interior problem we show that uniqueness holds in contrast to the usual problem associated to the Laplace operator.
Symmetry
Antonio Greco
149600308
The Infinity Laplacian
The limit equation of the p Laplace equation as (p\rightarrow \infty is a very fascinating one. In two variables it is the equation.
Peter Lindqvist
119175558
A Holder infinity Laplacian obtained as limit of Orlicz fractional Laplacians
This paper concerns with the study of the asymptotic behavior of the solutions to a family of fractional type problems on a bounded domain, satisfying homogeneous Dirichlet boundary conditions. The family of differential operators includes the fractional $p_n$ Laplacian when $p_n\to\infty$ as a particular case, tough it could be extended to a function of the H\"older quotient of order $s$ whose primitive is an Orlicz function satisfying appropriated growth conditions. The limit equation involves the H\"older infinity Laplacian.
Juli'an Fern'andez Bonder Mayte P'erez-Llanos Ariel Martin Salort
202144018
Harnack inequality and an asymptotic mean value property for the Finsler infinity Laplacian
Abstract In this paper, we prove a Harnack inequality for nonnegative viscosity supersolutions of nonhomogeneous equations associated with normalized Finsler infinity Laplace operators. Viscosity solutions to homogeneous equations are also characterized via an asymptotic mean value property, understood in a viscosity sense.
Advances in Calculus of Variations
Benyam Mebrate Ahmed Mohammed
202152011
Some estimates and properties of solutions to the biased infinity Laplacian equations
Liu Fang Yang Xiaoping Jiang Feida
119318684
Optimal regularity for a two phase free boundary problem ruled by the infinity Laplacian
In this paper we consider a non variational two phase free boundary problem ruled by the infinity Laplacian. Our main result states that normalized viscosity solutions in $B_1$ are universally Lipschitz continuous in $B_{1/2} which is the optimal regularity for the problem. We make a new use of the Ishii Lions' method, which works as a surrogate for the lack of a monotonicity formula and is bound to be applicable in related problems.
Damiao J Ara'ujo Eduardo V Teixeira Jose Miguel Urbano
125647936
Parabolic Biased Infinity Laplacian Equation Related to the Biased Tug of War
Abstract In this paper, we study the parabolic inhomogeneous b biased infinity Laplacian equation arising from the b biased tug of war u t D b u f x t {u_{t} \Delta_{\infty}\beta}u=f(x,t) where b is a fixed constant and D b \Delta_{\infty}\beta} is the b biased infinity Laplacian operator D b u D N u b D u \Delta_{\infty}\beta}u=\Delta_{\infty}{N}u+\beta\lvert Du\rvert related to the game theory named b biased tug of war. We first establish a comparison principle of viscosity solutions when the inhomogeneous term f does not change its sign. Based on the comparison principle, the uniqueness of viscosity solutions of the Cauchy Dirichlet boundary problem and some stability results are obtained. Then the existence of viscosity solutions of the corresponding Cauchy Dirichlet problem is established by a regularized approximation method when the inhomogeneous term is constant. We also obtain an interior gradient estimate of the viscosity solutions by Bernstein's method. This means that when f is Lipschitz continuous, a viscosity solution u is also Lipschitz in both the time variable t and the space variable x. Finally, when f 0 {f=0} we show some explicit solutions.
Fang Liu Feida Jiang
40630397
Multiple solutions for the fractional Laplacian problems with different asymptotic limits near infinity
Abstract In this paper we obtain the existence of two one sign nontrivial solutions for the fractional Laplacian equations with the nonlinearity having different asymptotic limits at infinity via the mountain pass theorem and the cut off techniques.
Appl. Math. Lett.
Yutong Chen Jiabao Su
125215404
An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term
In this paper, we prove the uniqueness and stability of viscosity solutions of the following initial boundary problem related to the random game named tug of war with a transport term \begin{document}\left\ \begin{array}{35}{l} {u}_{t} \Delta _{\infty {N}u \langle \xi ,Du\rangle f(x,t) \text{in} {Q}_{T} u g, \text{on} \partial }_{p}{Q}_{T} \end{array} \right. \end{document} where \begin{document} \Delta _{\infty {N}u \frac{1}\left| Du \right|{2}\sum\limits_{i,j 1}{n}{u}_{{x}_{i}{u}_{{x}_{j}{u}_{{x}_{i}{x}_{j}\end{document} denotes the normalized infinity Laplacian, \begin{document} x: Q_T\to R^n$\end{document} is a continuous vector field, \begin{document} f$\end{document} and \begin{document} g$\end{document} are continuous. When \begin{document} x$\end{document} is a fixed field and the inhomogeneous term \begin{document} f$\end{document} is constant, the existence is obtained by the approximate procedure. When \begin{document} f$\end{document} and \begin{document} x$\end{document} are Lipschitz continuous, we also establish the Lipschitz continuity of the viscosity solutions. Furthermore we establish the comparison principle of the generalized equation with the first order term with initial boundary condition \begin{document}{u_t}(x,t) D _ ^N u (x,t) H(x,t,Du(x,t) f(x,t) \end{document} where \begin{document} H(x,t,p):Q_Tx R^n\to R$\end{document} is continuous, \begin{document} H(x,t,0) 0$\end{document} and grows at most linearly at infinity with respect to the variable \begin{document} p$\end{document} And the existence result is also obtained when \begin{document} H(x,t,p) H(p)\end{document} and \begin{document} f$\end{document} is constant for the generalized equation.
Fang Liu