The course was devoted to the study of regularity for solutions to quasi-linear parabolic equations of second order with growth of order 2. The prerequisites were the knowledge of the basic results in the theory of partial differential equations.
Topics
- Technical Preliminaries
- An Introduction to the Harnack Inequality
- Quasi-Linear Equations and Parabolic DeGiorgi Classes (PDG classes)
- Local Boundedness of Functions in the PDG Classes
-
Hölder Continuity of Functions in the PDG Classes
- Estimating the Values of u by the Measure of the Set Where u is Either Near μ+ (supremum of u) or Near μ− (infimum of u)
- Reducing the Measure of the Set Where u is Either Near μ+ or Near μ−
- Propagating in Time the Measure-Theoretical Information
- Proof of the Hölder Continuity
- Boundary Parabolic DeGiorgi Classes: Dirichlet Data
- Boundary Parabolic DeGiorgi Classes: Neumann Data
- The Harnack Inequality
- The Harnack Inequality Implies the Hölder Continuity
- A Consequence of the Harnack Inequality
- A More Straightforward Proof of the Hölder Continuity
-
Higher Integrability of the Gradients
- Higher Integrability in the Interior
- Higher Integrability at the Lateral Boundary
- Proof of the Giaquinta–Modica Lemma
References
- [1] E. DiBenedetto, Harnack Estimates in Certain Function Classes, Atti Sem. Mat. Fis. Univ. Modena, XXXVII, (1989), 173–182.
- [2] E. DiBenedetto and U. Gianazza, Partial Differential Equations. Third Edition, Cornerstones, Birkhäuser Cham, 2023.
- [3] L.C. Evans, Entropy and Partial Differential Equations, Lecture Notes, Department of Mathematics, UC Berkeley, (2001).
- [4] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, (AM-105). Princeton University Press, 1983.
- [5] M. Giaquinta and G. Modica, Regularity Results for Some Classes of Higher-Order Non-Linear Elliptic Systems, J. reine angew. Math., 311, (1979), 145–169.
- [6] M. Giaquinta and M. Struwe, On the Partial Regularity of Weak Solutions of Nonlinear Parabolic Systems, Math. Zeit., 179, (1982), 437–451.
- [7] O.A. Ladyzhenskaya and N.N. Ural'tzeva, Linear and Quasilinear Elliptic Equations, Academic Press, London–New York, 1968.
- [8] G.M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1998.
- [9] J. Nash, Continuity of Solutions of Parabolic and Elliptic Equations, Amer. J. Math., 80(4), (1958), 931–954.
- [10] L. Saloff-Coste, Aspects of Sobolev-Type Inequalities, London Mathematical Society Lecture Note Series #289, 2002.