Meinert, Melissa: Partial differential equations on fractals. Existence, Uniqueness and Approximation results. 2020
Content
- Introduction
- I Tools and preliminaries
- Dirichlet forms
- Appendix to Part II
- Resistance forms
- Appendix to Part III
- Vector analysis for resistance forms
- Universal derivation
- Energy measures and discrete approximations
- Energy measures and discrete approximations in the local case
- Energy measures and discrete approximations in the general case
- Derivations and generators associated with different energies
- First order derivatives and measurable bundles
- Examples of resistance spaces
- II Existence and uniqueness results
- Linear equations of elliptic and parabolic type on resistance spaces
- The viscous Burgers equation
- Different formulations of the formal problem
- Heat and Burgers equation on metric graphs
- Heat and Burgers equations on resistance spaces
- Existence of solutions to the continuity equation
- Calculus of variations on fractals
- III Approximation results
- Generalized strong resolvent convergence for linear PDEs on compact resistance spaces
- KS-generalized Mosco convergence for non-symmetric Dirichlet forms
- Convergence of solutions on a single space
- Convergence of solutions on varying spaces
- Setup and basic assumptions
- Some consequences of the assumptions
- Boundedness and compatibility of vector fields
- Accumulation points
- Spectral convergence
- Approximations
- Generalized norm resolvent convergence and metric graph approximation for Cole-Hopf solutions to the Burgers equation
- Generalized norm resolvent convergence
- Metric graph approximation of solutions to the heat equation
- Metric graph approximation of Cole-Hopf solutions to the Burgers equation
- Discrete graph approximation for continuity equations on finitely ramified spaces
- Convergence in the sense of Kuwae and Shioya
- Choice of vector fields
- Uniform bounds
- Accumulation point along a subsequence to the solution of the continuity equation
- KS-generalized strong resolvent convergence and P-generalized norm resolvent convergence
- Proof of Theorem 11.1
- Auxiliary results from functional analysis
- Bibliography