On removable singularities for solutions of Neumann problem for elliptic equations involving variable exponent. (English) Zbl 1517.35009
Summary: We study the removability of a singular set in the boundary of Neumann problem for elliptic equations with variable exponent. We consider the case where the singular set is compact, and give sufficient conditions for removability of this singularity for equations in the variable exponent Sobolev space \(W^{1,p(\cdot)}(\Omega)\).
MSC:
| 35A21 | Singularity in context of PDEs |
| 35D30 | Weak solutions to PDEs |
| 35J57 | Boundary value problems for second-order elliptic systems |
| 35J60 | Nonlinear elliptic equations |
References:
| [1] | Lars Diening, Petteri Harjulehto, Peter Hästö, and Michael Ruzicka: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, 2011 · Zbl 1222.46002 |
| [2] | Fabes, EB; Stroock, DW, The \(L^p\)-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J., 51, 4, 997-1016 (1984) · Zbl 0567.35003 · doi:10.1215/S0012-7094-84-05145-7 |
| [3] | Fan, Xianling, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., 339, 2, 1395-1412 (2008) · Zbl 1136.46025 · doi:10.1016/j.jmaa.2007.08.003 |
| [4] | Fan, Xianling; Zhao, Dun, On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\), J. Math. Anal. Appl., 263, 2, 424-446 (2001) · Zbl 1028.46041 · doi:10.1006/jmaa.2000.7617 |
| [5] | Yongqiang, Fu; Shan, Yingying, On the removability of isolated singular points for elliptic equations involving variable exponent, Adv. Nonlinear Anal., 5, 2, 121-132 (2016) · Zbl 1338.35014 · doi:10.1515/anona-2015-0055 |
| [6] | Kováčik, Ondrej; Rákosník, Jiří, On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\), Czechoslov. Math. J., 41, 4, 592-618 (1991) · Zbl 0784.46029 · doi:10.21136/CMJ.1991.102493 |
| [7] | Mamedov, Farman I.; Harman, Aziz, On the removability of isolated singular points for degenerating nonlinear elliptic equations, Nonlinear Anal. Theory Methods Appl., 71, 12, 6290-6298 (2009) · Zbl 1180.35254 · doi:10.1016/j.na.2009.06.034 |
| [8] | Samko, Stefan G., Convolution type operators in \(L^{p(x)}\), Integral Transforms Spec Funct, 7, 1-2, 123-144 (1998) · Zbl 0934.46032 · doi:10.1080/10652469808819191 |
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