Felsinger, Matthieu: Parabolic equations associated with symmetric nonlocal operators. 2013
Inhalt
- Contents
- Introduction
- 1 Integration theory & Lebesgue spaces
- 1.1 Measurable functions
- 1.2 The Bochner-Lebesgue integral
- 1.3 Spaces of integrable functions
- 1.4 Steklov averages
- 2 Distributions & Sobolev spaces
- 2.1 The spaces D(Omega) and generalized derivatives
- 2.2 The spaces S(Rd) and the Fourier transform
- 2.3 Sobolev spaces of integer order
- 2.4 The constant A
- 2.5 Sobolev spaces of fractional order
- 2.6 Characterization of (fractional) Sobolev spaces by Fourier transform
- 2.7 The fractional Laplacian
- 3 Existence and uniqueness of solutions to local and nonlocal parabolic differential equations
- 3.1 Generalized derivatives of abstract functions
- 3.2 Evolution triplets and the space W(0,T)
- 3.3 Hilbert space methods for parabolic equations
- 3.4 The bilinear forms associated to Ls
- 3.5 Weak formulation of the initial boundary value problem
- 3.6 Well-posedness result
- 4 Set-up & Main results
- 4.1 Assumptions on k
- 4.2 Local weak solutions
- 4.3 Main results: Weak Harnack inequality and Hölder regularity for fractional order parabolic equations
- 5 Auxiliary Results
- 5.1 Standard cylindrical domains and scaling property
- 5.2 Alternative formulation in terms of Steklov averages
- 5.3 Some algebraic inequalities
- 5.4 Sobolev and weighted Poincaré inequalities
- 5.5 Abstract Moser iteration
- 5.6 A lemma by Bombieri and Giusti
- 6 Proof of the main results for fractional order parabolic equations
- 6.1 Basic step of Moser's iteration
- 6.2 An estimate for the infimum of supersolutions
- 6.3 An estimate for the L1-norm of a supersolution
- 6.4 An inequality for log u
- 6.5 Proof of the weak Harnack inequality
- 6.6 Proof of Hölder regularity
- 7 Proof of the main results for second order parabolic equations
- 7.1 Basic step of Moser's iteration
- 7.2 Estimates for inf u of a supersolution and sup u of a solution
- 7.3 An estimate for the L1-norm of a supersolution
- 7.4 An inequality for log u
- 7.5 Strong Harnack inequality for solutions
- 7.6 Hölder regularity for weak solutions
- Notation
- Bibliography