منابع مشابه
Inhomogeneous infinity Laplace equation ✩
We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation ∂xi u∂xj u∂ 2 xixj u = f in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the visc...
متن کاملInfinity Laplace Equation with Non-trivial Right-hand Side
We analyze the set of continuous viscosity solutions of the infinity Laplace equation −∆∞w(x) = f(x), with generally sign-changing right-hand side in a bounded domain. The existence of a least and a greatest continuous viscosity solutions, up to the boundary, is proved through a Perron’s construction by means of a strict comparison principle. These extremal solutions are proved to be absolutely...
متن کاملA stochastic differential game for the inhomogeneous -Laplace equation
Given a bounded C domain G ⊂ R and functions g ∈ C(∂G,R) and h ∈ C(Ḡ,R \ {0}), let u denote the unique viscosity solution to the equation −2∆∞u = h in G with boundary data g. We provide a representation for u as the value of a two-player zero-sum stochastic differential game. AMS 2000 subject classifications: 91A15, 91A23, 35J70, 49L20
متن کاملTug-of-war and Infinity Laplace Equation with Vanishing Neumann Boundary Condition
We study a version of the stochastic “tug-of-war” game, played on graphs and smooth domains, with the empty set of terminal states. We prove that, when the running payoff function is shifted by an appropriate constant, the values of the game after n steps converge in the continuous case and the case of finite graphs with loops. Using this we prove the existence of solutions to the infinity Lapl...
متن کاملComparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation∗
Within the framework of shifted-Laplace preconditioners [Erlangga, Vuik, Oosterlee, Appl. Numer. Math., 50(2004), pp.409–425] for the Helmholtz equation, different methods for the approximation of the inverse of a complex-valued Helmholtz operator are discussed. The performance of the preconditioner for Helmholtz problems at high wavenumbers in heterogeneous media is evaluated. Comparison with ...
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ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 2008
ISSN: 0001-8708
DOI: 10.1016/j.aim.2007.11.020