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Parabolic Q-minima and minimal solutions to variational flow. (English) Zbl 0674.35042
The purpose of the paper is to provide a unified version of some regularity results on parabolic systems of the form \[ u'-\sum D_{x_ i}(D_{p_ i}F(x,u,Du))+D_ uF(x,u,Du)=0. \] To this aim, the notion of parabolic Q-minimum is introduced, where Q is a real number greater than or equal to 1. It is an adaptation to the parabolic case of the notion of (elliptic) Q-minimum introduced in M. Giaquinta and E. Giusti [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 79-104 (1984; Zbl 0541.49008)].
Results on \(L^ p\)-summability of the spatial derivatives of u, Hölder continuity of u (in the scalar case) and partial Hölder continuity of u (in the vector case) are given for parabolic Q-minima.
More special results are proved for the so-called minimal solutions, namely Q-minima with \(Q=1\). A characterization of the existence of such minimal solutions is given in terms of the convexity of the associated functional \(\int F(x,u,Du)dx\).
Reviewer: M.Degiovanni

MSC:
35K55 Nonlinear parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35A15 Variational methods applied to PDEs
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References:
[1] BREZIS, H., Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, Amsterdam: North-Holland, 1973
[2] CAMPANATO, S., Equazione paraboliche del secondo ordine e spaci LZ, ? (?, ?). Ann. Mat. Pura Appl.73 55-102 (1966) · Zbl 0144.14101
[3] DA PRATO, G., Spazi L(p, ?) (?, ?) e loro propriétà, Ann. Mat. Pura Appl.69, 383-392 (1965) · Zbl 0145.16207
[4] DI BENEDETTO, E.; TRUDINGER, N.S., Harnack inequalities for quasi-minima of variational integrals, Ann. Inst. H. Poincaré Analyse non linéaire1, 295-308 (1984)
[5] EKELAND, J.; TEMAM, R., Convex Analysis and Variational Problems, Amsterdam: North-Holland 1976. · Zbl 0322.90046
[6] GEHRING, F.W., The LP-integrability of the partial derivatives of a quasi conformal mapping, Acta Math.130, 265-277 (1973) · Zbl 0258.30021
[7] GIAQUINTA, M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematical Studies 105, Princeton, 1983 · Zbl 0516.49003
[8] GIAQUINTA, M., The Regularity Problem of Extremals of Variational Integrals, in: J.M. Ball (Ed), Systems of Nonlinear PDE, Reidel Publ. Comp., Dordrecht 1983 · Zbl 0528.49005
[9] GIAQUINTA, M.; GIUSTI, E., Quasi-minima, Ann. Inst. Henri Poincaré, Analyse non linéaire1. 79-107 (1984) · Zbl 0541.49008
[10] GIAQUINTA, M.; MODICA, G., Regularity results for some classes of higher order non linear elliptic systems, J. Reine Angew. Math.311/312, 145-169 (1979) · Zbl 0409.35015
[11] GIAQUINTA, M.; STRUWE, M., On the Partial Regularity of Weak Solutions of Nonlinear Parabolic Systems, Math. Z.179, 437-451 (1982) · Zbl 0477.35028
[12] GIUSTI, E., Some aspects of the Regularity Theory for Nonlinear Elliptic Systems, in: J.M. Ball (Ed.) Systems of Nonlinear PDE, Reidel, Dordrecht (1983) · Zbl 0524.35040
[13] LADYSHENSKAYA, O.A.; SOLONNIKOV, V.A.; URAL’CEVA, N, N., Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, AMS, Providence (1968)
[14] LIONS, J.L.; MAGENES, E., Non-Homogeneous Boundary Value Problems and Applications, Springer, Berlin, Heidelberg, New York (1972) · Zbl 0227.35001
[15] STRUWE, M., Some regularity results for quasilinear parabolic systems, to appear in Commentat. Math. Univ. Carol
[16] TREVES, F., Basic Linear Partial Differential Equations, Acad. Press, New York (1975)
[17] WIESER, W., Partial regularity of parabolic quasiminimizers, Inst. Mittag-Leffler, Report No. 8 (1985)
[18] HILDEBRANDT, S., Über Flächen konstanter Krümmung, Math. Z.112, 107-144 (1969) · Zbl 0183.39501
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