Abstract
The expression “maximum principle” is a generic term that covers a set of results of two kinds. One kind concerns the points of maximum or minimum of solutions of certain boundary value problems, elliptic in our case. The other kind is about monotone dependence of the solutions with respect to the data. The two aspects are naturally related to one another. There are furthermore two general contexts, the so-called “strong” context in which classical solutions are considered, and the “weak” context in which variational solutions are considered. This terminology is not universal however.
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Notes
- 1.
Caution: p is not a mapping, the distance may be attained at several points of M.
- 2.
When T is moreover \(L^1_{\mathrm {loc}}\), it is then nonnegative almost everywhere.
- 3.
This is very surprising: we only know that a very specific linear combination of second order derivatives is square integrable, and it turns out that all individual second order derivatives are square integrable.
- 4.
By hypothesis, \(u\in L^2({\mathbb R}^d)\) so that we already know that \(\hat u\) is an L 2-function and that writing \(\hat u(\xi )\) is licit.
- 5.
This is a sort of discrete integration by parts.
- 6.
Under the assumption that u and Ω also have some minimal regularity in this context.
- 7.
We can also add first order terms.
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)
Ball, J.M., Currie, J.C., Olver, P.J.: Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41, 135–174 (1981)
Bourbaki, N.: Espaces vectoriels topologiques. Hermann, Paris (1967)
Brezis, H.: Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble) 18, 115–175 (1968)
Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, vol. 5. Notas de Matemática, vol. 50. North-Holland Publishing Co., Amsterdam; American Elsevier Publishing Co., Inc., New York (1973)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)
Browder, F.: Problèmes non linéaires. Université de Montréal (1966)
Buttazzo, G., Giaquinta, M., Hildebrandt, S.: One-Dimensional Variational Problems, An Introduction. Oxford Lecture Series in Mathematics and Its Applications, vol. 15. Clarendon Press, Oxford (1998)
Caffarelli, L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 4, 383–402 (1988)
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Le Dret, H. (2018). The Maximum Principle, Elliptic Regularity, and Applications. In: Nonlinear Elliptic Partial Differential Equations. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-78390-1_5
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