Peter I. Kogut, Olha P. Kupenko, Guenter Leugering, Yue Wang


In this paper we discuss some issues related to Poincar´e’s inequality for a
special class of weighted Sobolev spaces. A common feature of these spaces is that they can be naturally associated with differential operators with variable diffusion coefficients that are not uniformly elliptic. We give a classification of these spaces in the 1-D case bases on a measure of degeneracy of the corresponding weight coefficient and study their key properties.

Ключові слова

Degenerate equation; weighted Sobolev spaces; Poincar´e’s inequality

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F. Alabau-Boussouira, P. Cannarsa, G. Leugering, Control and stabilization of degenerate wave equations, SIAM J. Control Optim., 55 (3) (2017), 2052–2087.

F. Alabau-Boussouira, P. Cannarsa, G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161–204.

P. Cannarsa, P. Martinez, J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47(2008), 1–19.

V. Chiad´o Piat,F. Serra Cassano, Some remarks about the density of smooth functions in weighted Sobolev spaces, J. Convex Analysis, 1 (2) (1994), 135–142.

M. Colombo, A. Figalli, Regularity results for very degenerate elliptic equations, J. Math. Pures Appl., 101(2014), 94–117.

P. Drabek, A. Kufner, F. Nicolosi, On the solvability of degenerate quasilinear elliptic equations of higher order, J. Differential Equations, 109(1994), 325–347.

M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52(2014), 2037–2054.

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev., 51(2009), 3–33.

P.I. Kogut, G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Approximation and Asymptotic Analysis, Series: Systems and Control, Birkh¨auser Verlag, Boston, 2011.

P. I. Kogut, G. Leugering, Optimal L1-control in coefficients for Dirichlet elliptic problems: W-optimal solutions, Journal of Optimization Theory and Applications, 150(2)(2011), 205–232.

P. I. Kogut, G. Leugering, Optimal L1-control in coefficients for Dirichlet elliptic problems: H-optimal solutions, Zeitschrift f¨ur Analysis und ihre Anwendungen, 31(1)(2012), 31–53.

P. I. Kogut, G. Leugering, Matrix-valued L1-optimal controls in the coefficients of linear elliptic problems, Zeitschrift f¨ur Analysis und ihre Anwendungen, 32(4)(2013), 433–456.

G. Marinoschi, R. M. Mininni, S. Romanelli, An optimal control problem in coefficients for a strongly degenerate parabolic equation with interior degeneracy, J. Optim. Theory Appl, 173(2017), 56–77.

F. Santambrogio, V. Vespri, Continuity in two dimensions for a very degenerate elliptic equation, Nonlinear Anal., 73(2010), 3832–3841.

V. V. Zhikov, On Lavrentiev phenomenon, Russian J. Math. Phys. 3 (2) (1994), 249–269.

V. V. Zhikov, Weighted Sobolev spaces, : Mathematics, 189 (8) (1998), 27–58.

DOI: http://dx.doi.org/10.15421/141905


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Адреса редколегії: 49050, Україна, Дніпровський національний університет імені Олеся Гончара, вул. Козакова 18, корп. 14, механіко-математичний факультет, д-р фіз.-мат. наук, проф. Когут П.І. 

email: p.kogut@i.ua



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