15 September, at 18:00 MSK
The Lavrentiev phenomenon is the difference between the minimums/minimizers of integral functionals when minimized over "wide" and "narrow" Sobolev spaces. In particular, the "narrow" space can be understood as the closure of smooth function in the "wide" Sobolev space (the natural energy space of functions, where the integral functional is finite), and in this case the Lavrentiev phenomenon happens when smooth functions are not dense in the "wide" Sobolev space. For variable exponent Sobolev spaces (integral functional with integrand including the gradient of a function, raised to the variable power p(x)) examples for the Lavrentiev phenomenon were constructed by V.V. Zhikov, he also obtained the famous log-condition guaranteeing the density of smooth functions (and as a consequence the absence of the Laverntiev phenomenon). The crucial property of Zhikov's examples was that the exponent had to pass the dimensional threshold (the example was constructed around a saddle point). For a long time it remained unknown whether this (i.e. for the variable exponent to take values both above and below the space dimension) is essential for the Lavrentiev phenomenon, at least for variable exponent Sobolev spaces. I will present new fractal examples for the Lavrentiev phenomenon constructed jointly with Anna Balci and Lars Diening (Bielefeld), where the exponent can take values in any given open interval. I will also discuss similar questions for other models (weighted Sobolev spaces, double-phase functionals, Sobolev-Orlicz spaces with non-power functions).
Dr. of Sc. M.D. Surnachev (Keldysh Institute of Applied Mathematics of RAS, Moscow).
Title of the talk: ON LAVRENTIEV PHENOMENON.