Title of Project:<\/b>
\nViscosity Solutions and Nonlinear Partial Differential Equations
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Code: <\/b>FD2122B
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Supervisor:\u00a0<\/b>Dr. F. Dragoni
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Project description:<\/b>
\nNonlinear partial differential equations (PDEs) describe many phenomena
\nin sciences and economics, e.g. the value of an optimal control problem,
\nminimal surfaces, porous media, conservation laws, etc. Solutions to these
\nequations are often not sufficiently regular in order to write all the derivatives
\nappearing in the equation (e.g. they can be just continuous and not
\ndifferentiable). So the solutions have to be understood in a suitable generalized
\nsense. We consider the so-called viscosity solutions which consist
\nin using sufficiently smooth test functions to approximate from above and
\nbelow a unique continuous solution for a very large class of first-order and
\nsecond-order PDEs.
\nA project would consist of an introduction to the general theory of viscosity
\nsolutions and applications to a specific nonlinear PDE (e.g. Hamilton-Jacobi
\neq.s, infinite-Laplace eq., p-Laplace eq., etc.). One possibility is to investigate
\nthe relation between viscosity solutions and control theory or games
\ntheory. Another possibility is to focus on degenerate elliptic\/parabolic equations
\n(e.g. the evolution by mean curvature flow) or the relations with convexity.
\nIf required, it will be possible to deal with more geometric equations
\n(space-dependent) and metric formulas. One possibility is to investigate the
\nrelation between viscosity solutions and control theory or games theory.
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Project offered a double module, single module, or both:<\/b>
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Prerequisite 3rdd year modules:<\/b>
\nThe following classes may be helpful:
\nMA3303 Theoretical and Computational PDEs
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Number of students who can be supervised on this project:<\/b>
\n1<\/p>\n","protected":false},"excerpt":{"rendered":"
Title of Project: Viscosity Solutions and Nonlinear Partial Differential Equations Code: FD2122B Supervisor:\u00a0Dr. F. Dragoni Project description: Nonlinear partial differential equations (PDEs) describe many phenomena in sciences and economics, e.g. the value of an optimal control problem, minimal surfaces, porous media, conservation laws, etc. Solutions to these equations are often not sufficiently regular in order … Continue reading Viscosity solutions and homogenisation – Supervisor: Dr Federica Dragoni<\/span>