Search Advanced

Mathematics in Engineering

2021, Volume 3, Issue 1: 1-39. doi: 10.3934/mine.2021009
Previous Article Next Article
Research article Special Issues

Lipschitz continuity of minimizers in a problem with nonstandard growth

  • IMAS - CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina
  • Received: 07 February 2020 Accepted: 14 October 2020 Published: 21 October 2020

  • In this paper we obtain the Lipschitz continuity of nonnegative local minimizers of the functional $J(v)=\int_\Omega\big(F(x, v, \nabla v)+\lambda(x)\chi_{\{v>0\}}\big)\, dx$, under nonstandard growth conditions of the energy function $F(x, s, \eta)$ and $0<\lambda_{\min}\le \lambda(x)\le \lambda_{\max}<\infty$. This is the optimal regularity for the problem. Our results generalize the ones we obtained in the case of the inhomogeneous $p(x)$-Laplacian in our previous work. Nonnegative local minimizers $u$ satisfy in their positivity set a general nonlinear degenerate/singular equation ${\rm div}A(x, u, \nabla u)=B(x, u, \nabla u)$ of nonstandard growth type. As a by-product of our study, we obtain several results for this equation that are of independent interest.

    Citation: Claudia Lederman, Noemi Wolanski. Lipschitz continuity of minimizers in a problem with nonstandard growth[J]. Mathematics in Engineering, 2021, 3(1): 1-39. doi: 10.3934/mine.2021009

    Related Papers:

  • In this paper we obtain the Lipschitz continuity of nonnegative local minimizers of the functional $J(v)=\int_\Omega\big(F(x, v, \nabla v)+\lambda(x)\chi_{\{v>0\}}\big)\, dx$, under nonstandard growth conditions of the energy function $F(x, s, \eta)$ and $0<\lambda_{\min}\le \lambda(x)\le \lambda_{\max}<\infty$. This is the optimal regularity for the problem. Our results generalize the ones we obtained in the case of the inhomogeneous $p(x)$-Laplacian in our previous work. Nonnegative local minimizers $u$ satisfy in their positivity set a general nonlinear degenerate/singular equation ${\rm div}A(x, u, \nabla u)=B(x, u, \nabla u)$ of nonstandard growth type. As a by-product of our study, we obtain several results for this equation that are of independent interest.


    加载中


    [1] Aboulaich R, Meskine D, Souissi A (2008) New diffusion models in image processing. Comput Math Appl 56: 874-882.
    [2] Alt HW, Caffarelli LA (1981) Existence and regularity for a minimum problem with free boundary. J Reine Angew Math 325: 105-144.
    [3] Alt HW, Caffarelli LA, Friedman A (1984) A free boundary problem for quasilinear elliptic equations. Ann Scuola Norm Sci 11: 1-44.
    [4] Antontsev SN, Rodrigues JF (2006) On stationary thermo-rheological viscous flows. Ann Univ Ferrara 52: 19-36.
    [5] Antontsev SN, Shmarev SI (2005) A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions. Nonlinear Anal 60: 515-545.
    [6] Bonder JF, Martínez S, Wolanski N (2010) A free boundary problem for the p(x)-Laplacian. Nonlinear Anal 72: 1078-1103.
    [7] Chen Y, Levine S, Rao M (2006) Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math 66: 1383-1406.
    [8] Danielli D, Petrosyan A (2005) A minimum problem with free boundary for a degenerate quasilinear operator. Calc Var 23: 97-124.
    [9] Diening L, Harjulehto P, Hasto P, et al. (2011) Lebesgue and Sobolev Spaces with Variable Exponents, Berlin: Springer.
    [10] Fan X (2007) Global C1, α regularity for variable exponent elliptic equations in divergence form. J Differ Equations 235: 397-417.
    [11] Fan X, Zhao D (1999) A class of De Giorgi type and Hölder continuity. Nonlinear Anal 36: 295-318.
    [12] Gustafsson B, Shahgholian H (1996) Existence and geometric properties of solutions of a free boundary problem in potential theory. J Reine Angew Math 473: 137-179.
    [13] Harjulehto P, Hästö P (2019) Orlicz Spaces and Generalized Orlicz Spaces, Berlin: Springer.
    [14] Kováčik O, Rákosník J (1991) On spaces Lp(x) and Wk, p(x). Czechoslovak Math J 41: 592-618.
    [15] Lederman C (1996) A free boundary problem with a volume penalization. Ann Scuola Norm Sci 23: 249-300.
    [16] Lederman C, Wolanski N (2017) Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the p(x)-Laplacian. Interface Free Bound 19: 201-241.
    [17] Lederman C, Wolanski N (2019) Inhomogeneous minimization problems for the p(x)-Laplacian. J Math Anal Appl 475: 423-463.
    [18] Maly J, Ziemer WP (1997) Fine Regularity of Solutions of Elliptic Partial Differential Equations, Providence, RI: American Mathematical Society.
    [19] Martínez S, Wolanski N (2008) A minimum problem with free boundary in Orlicz spaces. Adv Math 218: 1914-1971.
    [20] Radulescu VD, Repovs DD (2015) Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Boca Raton, FL: Chapman & Hall / CRC Press.
    [21] Ruzicka M (2000) Electrorheological Fluids: Modeling and Mathematical Theory, Berlin: Springer-Verlag.
    [22] Serrin J (1964) Local behavior of solutions of quasi-linear equations. Acta Math 111: 247-302.
    [23] Wolanski N (2015) Local bounds, Harnack inequality and Hölder continuity for divergence type elliptic equations with non-standard growth. Rev Un Mat Argentina 56: 73-105.
    [24] Zhikov VV (1987) Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv 29: 33-66.
    [25] Zhikov VV (2008) Solvability of the three-dimensional thermistor problem. Proc Steklov Inst Math 261: 98-111.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematics in Engineering
1.4 2.2

Metrics

Article views(3759) PDF downloads(568) Cited by(1)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog