H-measures and variants: A short course by Nenad Antonic

H-measures and variants

Nenad Antonic

Weak convergences are the foundations of most successful methods in the investigation of partial differential equations today.  The approach known as Tartar's programme [15] proved particularly convenient for partial differential equations modelling physical laws; the basic assumption being that weak convergences of sequences of solutions model the transition from microscale to macroscale. We are particularly interested in mathematical objects being, in a generalised sense, accumulation points of weakly convergent sequences, such as defect measures or H-measures. H-measures are a microlocal analysis tool which has successfully been applied in a variety of applied analysis problems: in small amplitude homogenisation [16], in the theory of microstructure (problems with two or more potential wells) [10], in the study of propagation of oscillations and microlocal energy [17, 1], in controllability of partial differential equations [8], etc. The original H-measures are suited for the treatment of hyperbolic problems. Recently, different variants of the concept have been introduced, first allowing extension of the results to parabolic type problems [3, 5, 14], and later to problems of more general type. 
Another successful application of H-measures is in the velocity averaging theory, as they enable generalisation of results obtained for constant coefficients (the homogeneous setting) to variable ones (the heterogeneous setting). First results of this kind are found in [9] for hyperbolic problems, while recent development of variant H-measures enabled generalisations to problems of various types [12], including other linear and even fractional differential equations. The notion of H-measures is given in the L2 framework, and the above results are constrained to solutions bounded in Lp, p>=2. By developping and applying appropriate tools, namely H-distributions [6], applications outside the L2 framework are also possible. Quite often, the nature of the problem requires a particular variant to be developed, prompting further research in constructing new objects.

[1] N. Antonic: H-measures applied to symmetric systems, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 6, 1133-1155
[2] N. Antonic, M. Lazar: A parabolic variant of H-measures, Ann. Univ. Ferrara Sez. VII Sci. Marc. 54 (2008) 2, 183-201
[3] N. Antonic, M. Lazar: H-measures and  variants applied to parabolic equations, J.Math. Anal. Appl. 343 (2008) 1, 207-225
[4] N. Antonic, M. Lazar: Parabolic variant of H-measures in homogenization of a model problem based on Navier-Stokes equation, Nonlinear Anal. Real World Appl. 11 (2010) 6, 4500-4512
[5] N. Antonic, M. Lazar: Parabolic H-measures, J. Funct. Anal. 265 (2013) 7, 1190-1239
[6] N. Antonic, D. Mitrovic: H-distributions - an extension of H-measures to an Lp-Lq setting, Abstr. Appl. Anal. 2011 (2011), article ID 901084, 12 pages
[7] N. Antonic, M. Vrdoliak: Parabolic H-convergence and small amplitude homogenization, Appl. Anal. 88 (2009) 10-11, 1493-1508
[8] N. Burq, P. Gérard: Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Math. Acad. Sci. Paris, Série I, 325 (1997) 749-752
[9] P. Gérard: Microlocal defect measures, Comm. Partial Differential Equations 16 (1991) 1761-1794
[10] R. V. Kohn: The relaxation of a double-well energy, Contin. Mech. Thermodyn. 3 (1991) 193-236
[11] M. Lazar, D. Mitrovic: The velocity averaging for a heterogeneous heat type equation, Math. Commun. 16 (2011), 271-282
[12] M. Lazar, D. Mitrovic: Velocity averaging - a general framework, Dyn.Partial Differ. Equ. 9 (2012), 239-260
[13] M. Lazar, D. Mitrovic: On an extension of a bilinear functional on Lp(Rd) x E to a Bochner space with an application to velocity averaging, C. R. Math. Acad. Sci. Paris 351 (2013) 261-264
[14] E. Yu. Panov: Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property, J. Math. Sci. (N. Y.) 159 (2009), 180-228
[15] L. Tartar: Compensated compactness and applications to partial differential equations, Non-linear Analysis and Mechanics: Heriott-Watt symposium, Vol IV, Res. Notes in Math. 39, Pitman, Boston, 1979, 136-212
[16] L. Tartar: H-measures and small amplitude homogenization, Random Media and Composites, R.V. Kohn & G.W. Milton eds, 89-99, SIAM, Philadelphia, 1989
[17] L. Tartar: H-measures, a new approach for studying homogenisation, oscillation and concentration effects in PDEs, Proc. Roy. Soc. Edinburgh Sect. A 115:3-4 (1990) 193-230
[18] L. Tartar: Oscillations and concentration effects in partial differential equations: why waves may behave like particles. XVII CEDYA: Congress on Differential Equations and Applications/VII CMA: Congress of Applied Mathematics (Spanish) (Salamanca, 2001), 179-219, Dep. Mat. Apl., Univ. Salamanca, Salamanca, 2001.
[19] L. Tartar: The general theory of homogenization, Springer, 2009