Double phase problems with variable growth

Abstract We consider a class of double phase variational integrals driven by nonhomogeneous potentials. We study the associated Euler equation and we highlight the existence of two different Rayleigh quotients. One of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues. The notion of eigenvalue is understood in the sense of pairs of nonlinear operators, as introduced by Fucik, Necas, Soucek, and Soucek. The analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the p ( x ) -Laplace operator, the generalized mean curvature operator, or the capillarity differential operator with variable exponent. The results contained in this paper complement the pioneering contributions of Marcellini, Mingione et al. in the field of variational integrals with unbalanced growth.

[1]  Jessica Schulze,et al.  Spectral Analysis Of Nonlinear Operators , 2016 .

[2]  Julian Musielak,et al.  Orlicz Spaces and Modular Spaces , 1983 .

[3]  Jacques Simon,et al.  Regularite de la solution d’une equation non lineaire dans ℝN , 1978 .

[4]  N. Fusco,et al.  Some remarks on the regularity of minima of anisotropic integrals , 1993 .

[5]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[6]  Giuseppe Mingione,et al.  Gradient estimates for the p (x)-Laplacean system , 2005 .

[7]  M. Ruzicka,et al.  Electrorheological Fluids: Modeling and Mathematical Theory , 2000 .

[8]  Paolo Marcellini,et al.  Everywhere regularity for a class of elliptic systems without growth conditions , 1996 .

[9]  G. Mingione,et al.  Calderón–Zygmund estimates and non-uniformly elliptic operators , 2016 .

[10]  Paolo Marcellini.  On the definition and the lower semicontinuity of certain quasiconvex integrals , 1986 .

[11]  G. Mingione,et al.  Non-autonomous functionals, borderline cases and related function classes , 2016 .

[12]  C. Filippis.  Higher Integrability for Constrained Minimizers of Integral Functionals with (p,q)-Growth in low dimension , 2017, 1712.03252.

[13]  G. Mingione,et al.  Regularity for general functionals with double phase , 2017, 1708.09147.

[14]  J. Ball,et al.  Discontinuous equilibrium solutions and cavitation in nonlinear elasticity , 1982, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[15]  Zhikov.  On Lavrentiev's Phenomenon. , 1995 .

[16]  I. Kim,et al.  Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents , 2015 .

[17]  G. Mingione,et al.  Harnack inequalities for double phase functionals , 2015 .

[18]  Dun Zhao,et al.  On the Spaces L and W , 2001 .

[19]  P. Hästö,et al.  Lebesgue and Sobolev Spaces with Variable Exponents , 2011 .

[20]  Giuseppe Mingione,et al.  Sharp regularity for functionals with (p,q) growth , 2004 .

[21]  D. E. Edmunds,et al.  On Lp(x)norms , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[22]  Vicentiu D. Rădulescu,et al.  Double phase anisotropic variational problems and combined effects of reaction and absorption terms , 2018, Journal de Mathématiques Pures et Appliquées.

[23]  Dušan D. Repovš,et al.  A weighted anisotropic variant of the Caffarelli–Kohn–Nirenberg inequality and applications , 2018, 1803.05687.

[24]  Jiří Rákosník,et al.  Density of smooth functions in Wk, p(x) (Ω) , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[25]  M. Squassina,et al.  Eigenvalues for double phase variational integrals , 2015, 1507.01959.

[26]  G. Mingione,et al.  Regularity for Double Phase Variational Problems , 2015 .

[27]  Vicentiu D. Rădulescu,et al.  Non-autonomous Eigenvalue Problems with Variable (p1,p2)-Growth , 2017 .

[28]  Dušan D. Repovš,et al.  Multiple solutions of double phase variational problems with variable exponent , 2018, Advances in Calculus of Variations.

[29]  V. Zhikov,et al.  AVERAGING OF FUNCTIONALS OF THE CALCULUS OF VARIATIONS AND ELASTICITY THEORY , 1987 .

[30]  Vicentiu D. Rădulescu,et al.  Existence and Non-Existence Results for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions , 2007, 0706.2403.

[31]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[32]  Paolo Marcellini.  Regularity and existence of solutions of elliptic equations with p,q-growth conditions , 1991 .

[33]  G. Mingione,et al.  Bounded Minimisers of Double Phase Variational Integrals , 2015 .

[34]  Xianling Fan,et al.  On the Spaces Lp(x)(Ω) and Wm, p(x)(Ω) , 2001 .

[35]  D. Edmunds,et al.  Sobolev Embeddings with Variable Exponent, II , 2002 .

[36]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[37]  Jiří Rákosník,et al.  Sobolev embeddings with variable exponent , 2000 .

[38]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[39]  Vicentiu D. Radulescu,et al.  Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis , 2015 .

[40]  Vicentiu D. Rădulescu.  Nonlinear elliptic equations with variable exponent: Old and new , 2015 .

[41]  M. Mihăilescu,et al.  Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential , 2010 .

[42]  V. V. Zhikov.  Density of smooth functions in weighted Sobolev spaces , 2013 .

[43]  T. Halsey.  Electrorheological Fluids , 1992, Science.

[44]  Yunmei Chen,et al.  Variable Exponent, Linear Growth Functionals in Image Restoration , 2006, SIAM J. Appl. Math..

[45]  W. Orlicz,et al.  Über konjugierte Exponentenfolgen , 1931 .