Maximum principles, gradient estimates, and weak solutions for second-order partial differential equations
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- by William Bertiger PDF
- Trans. Amer. Math. Soc. 238 (1978), 213-227 Request permission
Abstract:
Weak solutions to second order elliptic equations and the first derivatives of these solutions are shown to satisfy ${L^p}$ bounds. Classical second order equations with nonnegative characteristic form are also considered. It is proved that auxiliary functions of the gradient of a solution must satisfy a maximum principle. This result is extended to higher order derivatives and systems.References
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- Carlo Miranda, Sul teorema del massimo modulo per una classe di sistemi ellittici di equazioni del secondo ordine e per le equazioni a coefficienti complessi, Ist. Lombardo Accad. Sci. Lett. Rend. A 104 (1970), 736–745 (Italian). MR 296497
- Guido Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 189–258 (French). MR 192177, DOI 10.5802/aif.204
- A. M. Il′in, A. S. Kalašnikov, and O. A. Oleĭnik, Second-order linear equations of parabolic type, Uspehi Mat. Nauk 17 (1962), no. 3 (105), 3–146 (Russian). MR 0138888 O. A. Oleĭnik and E. V. Radkevič, Equations of second order with nonnegative characteristic form, Itogi Nauki. Mat. Anal. 1969, VINITI, Moscow, 1971; English transl., Plenum Press, New York, 1973.
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- M. H. Protter and H. F. Weinberger, A maximum principle and gradient bounds for linear elliptic equations, Indiana Univ. Math. J. 23 (1973/74), 239–249. MR 324204, DOI 10.1512/iumj.1973.23.23020
- Jean-Michel Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), no. fasc. 1, 277–304 xii (French, with English summary). MR 262881, DOI 10.5802/aif.319
- A. D. Aleksandrov, Investigations on the maximum principle. I, Izv. Vysš. Učebn. Zaved. Matematika 1958 (1958), no. 5 (6), 126–157 (Russian). MR 0133569
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 238 (1978), 213-227
- MSC: Primary 35B45; Secondary 35D99, 35J15
- DOI: https://doi.org/10.1090/S0002-9947-1978-0482916-6
- MathSciNet review: 482916