Article
Full entry |
PDF
(1.0 MB)
Feedback
PDF
(1.0 MB)
Feedback
Keywords:
bounded sequences in Lebesgue spaces; oscillations; Young measures; DiPerna and Majda measures; rays; extreme points; extreme rays; concentrations
bounded sequences in Lebesgue spaces; oscillations; Young measures; DiPerna and Majda measures; rays; extreme points; extreme rays; concentrations
Summary:
DiPerna and Majda generalized Young measures so that it is possible to describe "in the limit" oscillation as well as concentration effects of bounded sequences in $L^p$-spaces. Here the complete description of all such measures is stated, showing that the "energy" put at "infinity" by concentration effects can be described in the limit basically by an arbitrary positive Radon measure. Moreover, it is shown that concentration effects are intimately related to rays (in a suitable locally convex geometry) in the set of all DiPerna-Majda measures. Finally, a complete characterization of extreme points and extreme rays is established.
References:
[1] J. J. Alibert G. Bouchitté: Non uniform integrability and generalized Young measures. J. Convex. Anal. 4 (1997), 1-19. MR 1459885
[2] J. P. Aubin H. Frankowska: Set-valued Analysis. Birkhäuser, 1990. MR 1048347
[3] E. J. Balder: New existence results for optimal controls in the absence of convexity: the importance of extremality. SIAM J. Control Anal. 32 (1994), 890-916. DOI 10.1137/S0363012990193099 | MR 1269997 | Zbl 0813.49004
[4] J. M. Ball: PDEs and Continuum Models of Phase Transition. (M. Rascle, D. Serre, M. Slemrod., eds.). Lecture Notes in Physics 344, Springer, Berlin, 1989, pp. 207-215. MR 1036070
[5] R. J. DiPerna A. J. Majda: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Physics 108 (1987), 667-689. DOI 10.1007/BF01214424 | MR 0877643
[6] R. J. DiPerna A. J. Majda: Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. 40 (1987), 301-345. DOI 10.1002/cpa.3160400304 | MR 0882068
[7] R. J. DiPerna A. J. Majda: Reduced Hausdorff dimension and concentration cancellation for 2-D incompressible flow. J. Amer. Math. Soc. 1 (1988), 59-95. MR 0924702
[8] N. Dunford J. T. Schwartz: Linear Operators. Part I. Interscience, New York, 1967.
[9] C. Greengard E. Thomann: On DiPerna-Majda concentration sets for two-dimensional incompresssible flow. Comm. Pure Appl. Math. 41 (1988), 295-303. DOI 10.1002/cpa.3160410303 | MR 0929281
[10] P. R. Halmos: Measure Theory. D. van Nostrand, 1950. MR 0033869 | Zbl 0040.16802
[11] G. Köthe: Topological Vector Spaces I. 2nd ed. Springer, Berlin, 1983. MR 0551623
[12] M. Kružík T. Roubíček: Explicit characterization of $L^p$-Young measures. J. Math. Anal. Appl. 198 (1996), 830-843. DOI 10.1006/jmaa.1996.0115 | MR 1377827
[13] T. Roubíček: Relaxation in Optimization Theory and Variational Calculus. W. de Gruyter, Berlin, 1997. MR 1458067
[14] T. Roubíček: Nonconcentrating generalized Young functionals. Comment. Math. Univ. Carolin. 38 (1997), 91-99. MR 1455472
[15] M. E. Schonbek: Convergence of solutions to nonlinear dispersive equations. Comm. Partial Differential Equations 7 (1982), 959-1000. DOI 10.1080/03605308208820242 | MR 0668586 | Zbl 0496.35058
[16] L. Tartar: Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics (R. J. Knops, ed.). Heriott-Watt Symposium IV, Pitman Res. Notes in Math. 39, San Francisco, 1979. MR 0584398 | Zbl 0437.35004
[17] M. Valadier: Young measures. Methods of Nonconvex Analysis (A. Cellina, ed.). Lecture Notes in Math. 1446, Springer, Berlin, 1990, pp. 152-188. MR 1079763 | Zbl 0742.49010
[18] J. Warga: Optimal Control of Differential and Functional Equations. Academic Press, New York, 1972. MR 0372708 | Zbl 0253.49001
[19] L. C. Young: Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, Classe III 30 (1937), 212-234. Zbl 0019.21901
Search
Partner of
