Regularity for minimizers for functionals of double phase with variable exponents

Abstract The functionals of double phase type H(u):=∫|Du|p+a(x)|Du|qdx,     (q>p>1,  a(x)≥0) $$\begin{array}{} \displaystyle {\cal H} (u):= \int \left(|Du|^{p} + a(x)|Du|^{q} \right) dx, ~~ ~~~(q \gt p \gt 1,~~a(x)\geq 0) \end{array}$$ are introduced in the epoch-making paper by Colombo-Mingione [1] for constants p and q, and investigated by them and Baroni. They obtained sharp regularity results for minimizers of such functionals. In this paper we treat the case that the exponents are functions of x and partly generalize their regularity results.

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

[2]  M. Eleuteri.  Holder continuity results for a class of functionals with non standard growth , 2002 .

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

[4]  Paolo Marcellini.  Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions , 1989 .

[5]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[6]  M. Giaquinta,et al.  Regularity results for some classes of higher order non linear elliptic systems. , 1979 .

[7]  Thomas Schmidt.  Regularity theorems for degenerate quasiconvex energies with (p, q)-growth , 2008 .

[8]  E. Giusti.  Direct methods in the calculus of variations , 2003 .

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

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

[11]  G. Mingione.  Short Tales from Nonlinear Calderón-Zygmund Theory , 2017 .

[12]  Maria Alessandra Ragusa,et al.  Partial regularity of $p(x)$-harmonic maps , 2011, 1108.2947.

[13]  Jan Malý,et al.  Fine Regularity of Solutions of Elliptic Partial Differential Equations , 1997 .

[14]  M. Giaquinta,et al.  An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs , 2005 .

[15]  Thomas Schmidt.  Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (p, q)-growth , 2009 .

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

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

[18]  M. Fuchs,et al.  C1,α-solutions to non-autonomous anisotropic variational problems , 2005 .

[19]  Pablo Pedregal,et al.  Some Variational Problems , 1997 .

[20]  G. Mingione,et al.  Manifold Constrained Non-uniformly Elliptic Problems , 2019, The Journal of Geometric Analysis.

[21]  iuseppe,et al.  Regularity for double phase variational problems , 2014 .

[22]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[23]  C. Tintarev,et al.  Compactness properties and ground states for the affine Laplacian , 2017, 1708.02413.

[24]  Erika Nio,et al.  Partial regularity of minimizers of p(x)-growth functionals with p(x)>1 , 2017 .

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

[26]  Dušan D. Repovš,et al.  Double-phase problems with reaction of arbitrary growth , 2018, Zeitschrift für angewandte Mathematik und Physik.

[27]  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.

[28]  Giuseppe Mingione,et al.  Regularity of minima: An invitation to the dark side of the calculus of variations , 2006 .

[29]  Paolo Marcellini,et al.  Lipschitz estimates for systems with ellipticity conditions at infinity , 2016 .

[30]  P. Hästö,et al.  Hölder regularity of quasiminimizers under generalized growth conditions , 2017 .

[31]  Kunihiro Usuba.  Partial regularity of minimizers of p(x)‐growth functionals with 1 < p(x) < 2 , 2015 .

[32]  Lars Diening,et al.  Fractional estimates for non-differentiable elliptic systems with general growth , 2008 .

[33]  Paolo Marcellini,et al.  Lipschitz continuity for energy integrals with variable exponents , 2016 .

[34]  M. Ružička,et al.  Mathematical modeling of electrorheological materials , 2001 .

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

[36]  D. Adams.  FINE REGULARITY OF SOLUTIONS OF ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS (Mathematical Surveys and Monographs 51) By J AN M ALY and W ILLIAM P. Z IEMER : 291 pp., US$75.00, ISBN 0 8218 0335 2 (American Mathematical Society, 1997). , 1999 .

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

[38]  D. Mucci,et al.  Integral representation and Γ-convergence of variational integrals with p(x)-growth , 2002 .

[39]  M. Ragusa,et al.  On interior regularity of minimizers of p(x)-energy functionals , 2013 .

[40]  A. Tachikawa,et al.  Regularity results up to the boundary for minimizers of p(x)-energy with $$p(x)>1$$p(x)>1 , 2017 .

[41]  Giuseppe Mingione,et al.  Regularity Results for Stationary Electro-Rheological Fluids , 2002 .

[42]  A. Tachikawa.  On the singular set of minimizers of $$p(x)$$p(x)-energies , 2013 .

[43]  H. Choe.  Interior behaviour of minimizers for certain functionals with nonstandard growth , 1992 .

[44]  P. Hästö,et al.  Local higher integrability of the gradient of a quasiminimizer under generalized Orlicz growth conditions , 2017, Nonlinear Analysis.

[45]  E. Acerbi,et al.  Regularity results for a class of quasiconvex functionals with nonstandard growth , 2001 .

[46]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

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

[48]  Paolo Marcellini,et al.  Regularity for scalar integrals without structure conditions , 2018 .

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

[50]  Matija Cencelj,et al.  Double phase problems with variable growth , 2018, Nonlinear Analysis.

[51]  B. Dacorogna.  Introduction to the calculus of variations , 2004 .

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

[53]  Gebräuchliche Fertigarzneimittel,et al.  V , 1893, Therapielexikon Neurologie.

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

[55]  Thomas Schmidt.  Regularity of minimizers of W1,p-quasiconvex variational integrals with (p,q)-growth , 2008 .

[56]  Vicentiu D. Rădulescu.  Isotropic and anisotropic double-phase problems: old and new , 2019, Opuscula Mathematica.

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

[58]  M. Ragusa,et al.  Boundary regularity of minimizers of p(x)-energy functionals , 2016 .

[59]  C. De Filippis,et al.  Regularity for multi-phase variational problems , 2018, Journal of Differential Equations.

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

[61]  Christoph Scheven,et al.  Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients , 2012 .

[62]  Giuseppe Mingione,et al.  Regularity Results for a Class of Functionals with Non-Standard Growth , 2001 .

[63]  Giuseppe Mingione,et al.  Hölder continuity of the gradient of p(x)-harmonic mappings , 1999 .

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

[65]  D. Breit.  New regularity theorems for non-autonomous variational integrals with (p, q)-growth , 2011, Calculus of Variations and Partial Differential Equations.