Local boundedness of variational solutions to evolutionary problems with non-standard growth

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• 作者：Thomas Singer
• 关键词：Parabolic minimizers ; Regularity ; Non ; standard growth
• 刊名：NoDEA : Nonlinear Differential Equations and Applications
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：23
• 期：2
• 全文大小：628 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-9004

文摘

We prove the local boundedness of variational solutions and parabolic minimizers to evolutionary problems, where the integrand f is convex and satisfies a non-standard p, q-growth condition with$$1 < p \leq q \leq p \tfrac{n+2}{n}.$$A function \({u\colon \Omega_T := \Omega \times (0,T) \to \mathbb{R}}\) is called parabolic minimizer if it satisfies the minimality condition$$\int_{\Omega_T} u \cdot \partial_t \varphi +f(x, Du) {\rm d} z \leq \int_{\Omega_T} f(x, Du + D \varphi) {\rm d}z$$for every \({\varphi \in C^\infty_0(\Omega_T)}\). Moreover, we will show local boundedness for parabolic minimizers, if f satisfies an anisotropic growth condition. Mathematics Subject Classification 35A15 35K86 49J40 49N60 Keywords Parabolic minimizers Regularity Non-standard growth References1.Bögelein V., Duzaar F., Marcellini P.: Existence of evolutionary variational solutions via the calculus of variations. J. Differ. Equ. 256(12), 3912–3942 (2014)MathSciNetCrossRefMATH2.Bögelein, V., Duzaar, F., Marcellini, P.: A time dependent variational approach to image restoration. SIAM J. Imaging Sci. 8(2), 968–1006 (2015)3.Cupini G., Marcellini P., Mascolo E.: Regularity under sharp anisotropic general growth conditions. Discrete Contin. Dyn. Syst. Ser. B 11(1), 66–86 (2009)MathSciNetMATH4.Cupini G., Marcellini P., Mascolo E.: Local boundedness of solutions to quasilinear elliptic systems. Manuscripta Math. 137(3–4), 287–315 (2012)MathSciNetCrossRefMATH5.Cupini G., Marcellini P., Mascolo E.: Local boundedness of minimizers with limit growth conditions. J. Optim. Theory Appl. 166(1), 122 (2015)MathSciNetCrossRefMATH6.DiBenedetto, E.: Degenerate Parabolic Equations. Springer, New York (1993)7.Föglein, A.: Regularity results for minimizers of integrals with (2,q)-growth in the Heisenberg group. PhD thesis, Friedrich-Alexander-Universität, Erlangen-Nürnberg (2009)8.Fusco, N., Sbordone, C.: Local boundedness of minimizers in a limit case. Manuscripta Math. 69, 19–25 (1990)9.Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Mathematica 148, 31–46 (1982)10.Kinnunen, J., Lindqvist, P.: Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation. Ann. Mat. Pura Appl. (4) 185(3), 411–435 (2006)11.Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and quasilinear elliptic equations. Academic Press, New York (1968)12.Lichnewsky A., Temam R.: Pseudosolutions of the time-dependent minimal surface problem. J. Differ. Equ. 30(3), 340–364 (1978)MathSciNetCrossRefMATH13.Marcellini, P.: Un exemple de solution discontinue dun probleme variationnel dans le cas scalaire. Istituto Matematico “U. Dini”, n. 11, Universita di Firenze (1987)14.Marcellini P.: Regularity of minimizers of integrals of the calculus of variations with non standart growth conditions. Arch. Mat. Mech. Anal. 105, 267–284 (1989)MathSciNetMATH15.Mingqi Y., Xiting L.: Boundedness of solutions of parabolic equations with anisotropic growth conditions. Can. J. Math. 49(4), 798–809 (1997)MathSciNetCrossRefMATH16.Moscariello G., Nania L.: Hölder continuity of minimizers of functional with non standart growth conditions. Ricerce Mat. 40(2), 259–273 (1991)MathSciNetMATH17.Wieser W.: Parabolic Q-minima and minimal solutions to variational flow. Manuscripta Math. 59(1), 63–107 (1987)MathSciNetCrossRefMATHCopyright information© The Author(s) 2016 Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://​creativecommons.​org/​licenses/​by/​4.​0/​), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.