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Maximum Principle and Its Application for the Time-Fractional Diffusion Equations

Luchko, Yury (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversaryIn the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann-Liouville fractional derivative, the Caputo fractional...

Maximum principle for viscosity sub solutions and viscosity sub solutions of the Laplacian

Sundararaja Ramaswamy (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The aim of this paper is to characterize the u.s.c. (resp. l.s.c.) viscosity sub (resp. super) solutions of the Laplacian which do not take the value + ∞ (resp. - ∞ ) as precisely the sub (resp. super) harmonic functions.

Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

Dario Daniele Monticelli (2010)

Journal of the European Mathematical Society

We deal with maximum principles for a class of linear, degenerate elliptic differential operators of the second order. In particular the Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypothesis on the degeneracy set of the operator. We derive, as consequences of these principles, some generalized maximum principles and an a priori estimate on the solutions of the Dirichlet problem for the linear equation....

Monge solutions for discontinuous hamiltonians

Ariela Briani, Andrea Davini (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an Hamilton-Jacobi equation of the form H ( x , D u ) = 0 x ∈ Ω ⊂ ℝ N , ( 1 ) where H ( x , p ) is assumed Borel measurable and quasi-convex in p . The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation (1) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed.

Monge solutions for discontinuous Hamiltonians

Ariela Briani, Andrea Davini (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an Hamilton-Jacobi equation of the form

 H ( x , D u ) = 0 x ∈ Ω ⊂ ℝ N , ( 1 ) 
 where H(x,p) is assumed Borel measurable and quasi-convex in p. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation ([see full text]) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also...

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