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| | === Description === | | === Description === |
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| − | == Chris Grant's Proposed Core Topics for Math 647/648 ==
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| − | <div style="-moz-column-count:2; column-count:2;">
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| − | # Linear elliptic operators of order ''n''
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| − | #* Classification
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| − | #* Strong and weak solutions
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| − | #* Gårding's inequality
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| − | #* Existence of weak solutions for the Dirichlet and Neumann problems
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| − | #* Agmon-Douglis-Nirenberg regularity
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| − | #* Green's formula
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| − | # Fundamental solutions for general linear differential operators
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| − | # Green's functions for general linear BVPs
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| − | # Dirichlet's Principle for Laplace’s equation in '''R'''<sup>''n''</sup>
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| − | # Poisson's Equation
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| − | #* Newtonian Potential
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| − | #* Local existence for the Dirichlet Problem with locally Hölder boundary data
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| − | #* Interior Hölder estimates
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| − | #* Kellogg's Theorem
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| − | # Second-order linear elliptic operators
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| − | #* Weak Maximum Principle
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| − | #* Perron's Method
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| − | #* Uniqueness for the Dirichlet Problem
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| − | #* Hopf's bondary-point lemma
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| − | #* Hopf's Strong Maximum Principle
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| − | #* Alexandroff Maximum Principle
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| − | #* Gidas-Ni-Nirenberg
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| − | #* Uniqueness for the Neumann Problem
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| − | #* Harnack inequality
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| − | #* Finite difference methods
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| − | #* Interior regularity
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| − | #* Schauder estimates
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| − | #* Moser iteration
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| − | #* De Giorgi's theorem
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| − | #* Boundary/Global regularity
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| − | # Second-order quasilinear equations in divergence form
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| − | #* Existence of weak solutions for the Dirichlet problem via the Browder-Minty theorem
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| − | #* Local-in-time existence for reaction-diffusion IBVPs and systems using the contraction mapping principle
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| − | # Abstract evolution equations
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| − | #* General theory
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| − | #* Existence and reqularity for parabolic IVPs
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| − | #* Existence for hyperbolic IVPs
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| − | # Viscosity solutions
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| − | </div>
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| | == Desired Learning Outcomes == | | == Desired Learning Outcomes == |
