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September  2016, 15(5): 1643-1659. doi: 10.3934/cpaa.2016018

On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China, China, China

Received  November 2015 Revised  May 2016 Published  July 2016

Under some reasonable conditions, some trace embedding properties of Musielak-Sobolev spaces in a bounded domain are given, including the trace on the inner lower dimensional hyperplane and the trace on the boundary. Furthermore, a compact trace embedding on the boundary is given.
Citation: Duchao Liu, Beibei Wang, Peihao Zhao. On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1643-1659. doi: 10.3934/cpaa.2016018
References:
[1]

R. Adams, Sobolev Spaces, Academic Press, New York, 1975.  Google Scholar

[2]

A. Benkirane and M. S. Vally, An existence result for nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces, Bulletin of the Belgian Mathematical Society, 20 (2013), 1-187.  Google Scholar

[3]

S. Byun, J. Ok and L. Wang, $W^{1,p(\cdot)}$-Regularity for elliptic eqautions with measurable coefficients in nonsmooth domains, Communications in Mathematical Physics, 329 (2014), 937-958. doi: 10.1007/s00220-014-1962-8.  Google Scholar

[4]

F. Cammaroto and L. Vilasi, Multiple solutions for a Kirchhoff-type problem involving the $p(x)$-Laplacian operator, Nonlinear Analysis, 74 (2011), 1841-1852. doi: 10.1016/j.na.2010.10.057.  Google Scholar

[5]

T. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Func. Anal., 8 (1971), 52-75.  Google Scholar

[6]

X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., 339 (2008), 1395-1412. doi: 10.1016/j.jmaa.2007.08.003.  Google Scholar

[7]

X. Fan and C. Guan, Uniform convexity of Musielak-Orlicz-Sobolev spaces and applications, Nonlinear Analysis, 73 (2010), 163-175. doi: 10.1016/j.na.2010.03.010.  Google Scholar

[8]

X. Fan, Differential equations of divergence form in Musielak-Sobolev spaces and sub-supersolution method, J. Math. Anal. Appl., 386 (2012), 593-604. doi: 10.1016/j.jmaa.2011.08.022.  Google Scholar

[9]

X. Fan, An imbedding theorem for Musielak-Sobolev spaces, Nonlinear Analysis, 75 (2012), 1959-1971. doi: 10.1016/j.na.2011.09.045.  Google Scholar

[10]

X. Fan and D. Zhao, On the generalized Orlicz-Sobolev space $W^{k,p(x)}(\Omega)$, J. Gansu Educ. College, 12 (1998), 1-6. Google Scholar

[11]

M. G. Huidobro, V. K. Le, R. Manásevich and K. Schmitt, On principle eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting, Nonlinear differ. equ. appl., 6 (1999), 207-225. doi: 10.1007/s000300050073.  Google Scholar

[12]

D. Liu and P. Zhao, Solutions for a quasilinear elliptic equation in Musielak-Sobolev spaces, Nonlinear Analysis: RWA, 26 (2015), 315-329. doi: 10.1016/j.nonrwa.2015.06.002.  Google Scholar

[13]

J. Musielak, Orlicz spaces and modular spaces, in Lecture Notes in Math., volume 1034}, Springer-Verlag, Berlin, 1983.  Google Scholar

show all references

References:
[1]

R. Adams, Sobolev Spaces, Academic Press, New York, 1975.  Google Scholar

[2]

A. Benkirane and M. S. Vally, An existence result for nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces, Bulletin of the Belgian Mathematical Society, 20 (2013), 1-187.  Google Scholar

[3]

S. Byun, J. Ok and L. Wang, $W^{1,p(\cdot)}$-Regularity for elliptic eqautions with measurable coefficients in nonsmooth domains, Communications in Mathematical Physics, 329 (2014), 937-958. doi: 10.1007/s00220-014-1962-8.  Google Scholar

[4]

F. Cammaroto and L. Vilasi, Multiple solutions for a Kirchhoff-type problem involving the $p(x)$-Laplacian operator, Nonlinear Analysis, 74 (2011), 1841-1852. doi: 10.1016/j.na.2010.10.057.  Google Scholar

[5]

T. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Func. Anal., 8 (1971), 52-75.  Google Scholar

[6]

X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., 339 (2008), 1395-1412. doi: 10.1016/j.jmaa.2007.08.003.  Google Scholar

[7]

X. Fan and C. Guan, Uniform convexity of Musielak-Orlicz-Sobolev spaces and applications, Nonlinear Analysis, 73 (2010), 163-175. doi: 10.1016/j.na.2010.03.010.  Google Scholar

[8]

X. Fan, Differential equations of divergence form in Musielak-Sobolev spaces and sub-supersolution method, J. Math. Anal. Appl., 386 (2012), 593-604. doi: 10.1016/j.jmaa.2011.08.022.  Google Scholar

[9]

X. Fan, An imbedding theorem for Musielak-Sobolev spaces, Nonlinear Analysis, 75 (2012), 1959-1971. doi: 10.1016/j.na.2011.09.045.  Google Scholar

[10]

X. Fan and D. Zhao, On the generalized Orlicz-Sobolev space $W^{k,p(x)}(\Omega)$, J. Gansu Educ. College, 12 (1998), 1-6. Google Scholar

[11]

M. G. Huidobro, V. K. Le, R. Manásevich and K. Schmitt, On principle eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting, Nonlinear differ. equ. appl., 6 (1999), 207-225. doi: 10.1007/s000300050073.  Google Scholar

[12]

D. Liu and P. Zhao, Solutions for a quasilinear elliptic equation in Musielak-Sobolev spaces, Nonlinear Analysis: RWA, 26 (2015), 315-329. doi: 10.1016/j.nonrwa.2015.06.002.  Google Scholar

[13]

J. Musielak, Orlicz spaces and modular spaces, in Lecture Notes in Math., volume 1034}, Springer-Verlag, Berlin, 1983.  Google Scholar

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