Abstract
If f is a mapping from a set E into itself, any element x of E such that f(x) = x is called a fixed point of f. Many problems, including nonlinear partial differential equations problems, may be recast as problems of finding a fixed point of a certain mapping in a certain space. We will see several examples of this a little later on. It is therefore interesting to have fixed point theorems at our disposal that are as general as possible.
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Notes
- 1.
To some extent. In the same vein, there is also a stirred coffee cup example.
- 2.
Keeping in mind that this is pretty subjective.
- 3.
In fact, a polynomial, hence C ∞ mapping.
- 4.
Indeed, if x ∈ S d−1, u(x) is not a tangent vector by construction.
- 5.
In the sense that the infimum is not + ∞.
- 6.
This is more striking for an inviscid coffee that does not adhere to the side of the cup…
- 7.
In fact, this sum is even bounded below on K by some constant δ > 0.
- 8.
Or equivalently, the smallest closed convex set containing A.
- 9.
Actually, it is compact.
- 10.
Actually, here f(v) ∈ L ∞( Ω).
- 11.
This is where the hypothesis f bounded is used.
References
Acerbi, E., Fusco, N.: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 62, 371–387 (1984)
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Commun. Pure Appl. Math. 12, 623–727 (1959)
Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math. 17, 35–92 (1964)
Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)
Ball, J.M.: A version of the fundamental theorem for Young measures. In: PDEs and Continuum Models of Phase Transitions (Nice, 1988). Lecture Notes in Physics, vol. 344, pp. 207–215. Springer, Berlin (1989)
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Le Dret, H. (2018). Fixed Point Theorems and Applications. In: Nonlinear Elliptic Partial Differential Equations. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-78390-1_2
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DOI: https://doi.org/10.1007/978-3-319-78390-1_2
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