The Porous Media Schrödinger Equation : Feynman-type Motivation, Well-Posedness and Control Interpretation
Résumé
This paper's aim is threefold. First, using Feynman's path approach to the derivation of the classical Schrödinger's equation in [7] and by introducing a slight path (or wave) dependency of the action, we derive a new class of equations of Schrödinger type where the driving operator is no longer the Laplace one but rather of complex porous media-type. Second, using suitable concepts of monotonicity in the complex setting and on appropriate functional spaces, we show the existence and uniqueness of the equation we have previously introduced. Finally, using Fitzpatrick's characterization of maximal monotone operators (cf. [8]), we propose a Brézis-Ekeland type characterization of the solution via a control problem.
Mots clés
40.12 Non-linear PDE
60.07 Quantum dynamics
equations of Schrödinger type Feynman's approach complex porous media equation Fitzpatrick's characterization of complex maximal monotone operators control problem MSC2020: Primary 35J10 81-10 76S99
equations of Schrödinger type
Feynman's approach
complex porous media equation
Fitzpatrick's characterization of complex maximal monotone operators
control problem
MSC2020: Primary 35J10 81-10 76S99
secondary 47H05 49J20 Classification: 40.09 Mathematical physics 40.12 Non-linear PDE 60.07 Quantum dynamics
control problem MSC2020: Primary 35J10
81-10
76S99
secondary 47H05 49J20 Classification: 40.09 Mathematical physics
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