August  2019, 39(8): 4487-4518. doi: 10.3934/dcds.2019184

A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

1. 

Dipartimento di Scienze di Base e Applicate per I'Ingegneria, Sapienza Università di Roma Roma, Italy, Via A. Scarpa 16, 00161 Roma, Italy

2. 

Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Mayagüez, Puerto Rico, 00681, USA

* Corresponding author: A. Vélez-Santiago

Received  August 2018 Published  May 2019

We investigate the solvability of the Ambrosetti-Prodi problem for the p-Laplace operator ∆p with Venttsel' boundary conditions on a twodimensional open bounded set with Koch-type boundary, and on an open bounded three-dimensional cylinder with Koch-type fractal boundary. Using a priori estimates, regularity theory and a sub-supersolution method, we obtain a necessary condition for the non-existence of solutions (in the weak sense), and the existence of at least one globally bounded weak solution. Moreover, under additional conditions, we apply the Leray-Schauder degree theory to obtain results about multiplicity of weak solutions.

Citation: Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4487-4518. doi: 10.3934/dcds.2019184
References:
[1]

R. Adams, Sobolev Spaces, New York-London, 1975. Google Scholar

[2]

S. AizicoviciN. S. Papageorgiou and V. Staicu, Sublinear and superlinear Ambrosetti–Prodi problems for the Dirichlet $p$-Laplacian, Nonlinear Analysis, 95 (2014), 263-280. doi: 10.1016/j.na.2013.08.026. Google Scholar

[3]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Annali Mat. Pura Appl., 93 (1972), 231-246. doi: 10.1007/BF02412022. Google Scholar

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D. Arcoya and D. Ruiz, The Ambrosetti–Prodi problem for the p-Laplacian operator, Comm. Partial Differential Equations, 31 (2006), 849-865. doi: 10.1080/03605300500394447. Google Scholar

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M. Arias and M. Cuesta, A one side superlinear Ambrosetti–Prodi problem for the Dirichlet $p$-laplacian, J. Math. Anal. Appl., 367 (2010), 499-507. doi: 10.1016/j.jmaa.2010.01.031. Google Scholar

[6]

M. Biegert, A priori estimate for the difference of solutions to quasi-linear elliptic equations, Manuscripta Math., 133 (2010), 273-306. doi: 10.1007/s00229-010-0367-z. Google Scholar

[7]

M. Biegert, On trace of Sobolev functions on the boundary of extension domains, Proc. Amer. Math. Soc., 137 (2009), 4169-4176. doi: 10.1090/S0002-9939-09-10045-X. Google Scholar

[8]

M. Biegert and P. Vernole, Strongly local nonlinear Dirichlet functionals and forms, Adv. Math. Sci. Appl., 15 (2005), 655-682. Google Scholar

[9]

V. I. Burenkov, Sobolev Spaces on Domains, TEUBNER-TEXTE zur Mathematik, Vol. 137, 1998. doi: 10.1007/978-3-663-11374-4. Google Scholar

[10]

R. Capitanelli, Homogeneous p-Lagrangians and self-similarity, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 215-235. Google Scholar

[11]

R. Capitanelli, Nonlinear energy forms on certain fractal curves, J. Nonlinear Convex Anal., 3 (2002), 67-80. Google Scholar

[12]

R. Capitanelli and M. R. Lancia, Nonlinear energy forms and Lipschitz spaces on the Koch curve, J. Convex Anal., 9 (2002), 245-257. Google Scholar

[13]

M. CefaloG. Dell'acqua and M. R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, AMC, 218 (2012), 5453-5473. doi: 10.1016/j.amc.2011.11.033. Google Scholar

[14]

M. Cefalo and M. R. Lancia, An optimal mesh generation algorithm for domains with Koch type boundaries, Math. Comput. Simulation, 106 (2014), 133-162. doi: 10.1016/j.matcom.2014.04.009. Google Scholar

[15]

M. CefaloM. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differential and Integral equations, 26 (2013), 1027-1054. Google Scholar

[16]

S. CreoM. R. LanciaA. Vélez-Santiago and P. Vernole, Approximation of a nonlinear fractal energy functional on varying Hilbert spaces, Comm. Pure Appl. Anal., 17 (2018), 647-669. doi: 10.3934/cpaa.2018035. Google Scholar

[17]

S. Creo and V. Regis Durante, Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 65-90. doi: 10.3934/dcdss.2019005. Google Scholar

[18]

D. Danielli, N. Garofalo and D.-H. Nhieu, Non-doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Carathéodory Spaces, Mem. Amer. Math. Soc., 182 (2006), x+119 pp. doi: 10.1090/memo/0857. Google Scholar

[19]

F. O.de Paiva and M. Montenegro, An Ambrosetti-Prodi-type result for a quasilinear Neumann problem, Proc. Edinburgh Math. Soc., 55 (2012), 771-780. doi: 10.1017/S0013091512000041. Google Scholar

[20]

F. O. de Paiva and A. E. Presoto, A Neumann problem of Ambrosetti-Prodi type, J. Fixed Point Theory Appl., 18 (2016), 189-200. doi: 10.1007/s11784-015-0277-5. Google Scholar

[21]

U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendungen., 23 (2004), 115-137. doi: 10.4171/ZAA/1190. Google Scholar

[22]

M. Fukushima, Y. Oshima and M. Takeda,, Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics, vol. 19, Berlin Eds. Bauer Kazdan, Zehnder, 1994. doi: 10.1515/9783110889741. Google Scholar

[23]

P. HajłaszP. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Functional Analysis, 254 (2008), 1217-1234. doi: 10.1016/j.jfa.2007.11.020. Google Scholar

[24]

A. Jonsson and H. Wallin, Function Spaces on Subsets of  $\mathbb{R\!}^{\, n} $., Math. Rep. Vol. 2 Part Ⅰ, Academic Publisher, Harwood, 1984. Google Scholar

[25]

E. Koizumi and K. Schmitt, Ambrosetti–Prodi-type problems for quasilinear elliptic problems, Diff. Integ. Equations, 18 (2005), 241-262. Google Scholar

[26]

O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, in: Mathematics in Science and Engineering, 46. Academic Press, New York-London, 1968. Google Scholar

[27]

M. R. LanciaA. Vélez-Santiago and P. Vernole, Quasi-linear Venttsel' problems with nonlocal boundary conditions on fractal domains, Nonlinear Anal. Real Worl Appl., 35 (2017), 265-291. doi: 10.1016/j.nonrwa.2016.11.002. Google Scholar

[28]

M. R. Lancia and P. Vernole, Semilinear Venttsel problems in fractal domains, Applied Mathematics, 5 (2014), 1820-1833. Google Scholar

[29]

M. R. Lancia and P. Vernole, Venttsel problems in fractal domains, J. Evolution Equations, 14 (2014), 681-712. doi: 10.1007/s00028-014-0233-7. Google Scholar

[30]

M. R. LanciaV. Regis Durante and P. Vernole, Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1493-1520. doi: 10.3934/dcdss.2016060. Google Scholar

[31]

M. R. Lancia and M. A. Vivaldi, Lipschitz spaces and Besov traces on self-similar fractals, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 23 (1999), 101-116. Google Scholar

[32]

V. Lappalainen and A. Lehtonen, Embedding of Orliz-Sobolev spaces in Hölder spaces, Annales Academiæ Scientiarum Fennicæ, 14 (1989), 41-46. doi: 10.5186/aasfm.1989.1417. Google Scholar

[33]

V. K. Le, On a sub-supersolutions method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Analysis, 71 (2009), 3305-3321. doi: 10.1016/j.na.2009.01.211. Google Scholar

[34]

V. K. Le and K. Schmitt, Some general concepts of sub- and supersolutions for nonlinear elliptic problems, Topol. Methods Nonlinear Anal., 28 (2006), 87-103. Google Scholar

[35]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969. Google Scholar

[36]

J. Maly and U. Mosco, Remarks on measure-valued Lagrangians on homogeneous spaces, Ricerche Mat., 48 (1999), 217-231. Google Scholar

[37]

J. Mawhin, Ambrosetti–Prodi-type results in nonlinear boundary value problems, Lecture Notes in Mathematics, 1285 (1987), 290-313. doi: 10.1007/BFb0080609. Google Scholar

[38]

V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-09922-3. Google Scholar

[39]

T. J. Miotto, Superlinear Ambrosetti–Prodi problem for the p-Laplacian operator, Nonlinear Differ. Equ. Appl. NoDEA, 17 (2010), 337-353. doi: 10.1007/s00030-010-0057-2. Google Scholar

[40]

T. J. Miotto, On the Ambrosetti-Prodi problem for a system involving p-Laplacian operator, Math Nachr., 289 (2016), 67-84. doi: 10.1002/mana.201300350. Google Scholar

[41]

U. Mosco, Lagrangian metrics on fractals, Proc. Symp. Appl. Math., 54, Amer. Math. Soc., R.Spigler and S. Venakides eds., (1998), 301–323.Google Scholar

[42]

M. R. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Annali Mat. Pura Appl., 80 (1968), 1-122. doi: 10.1007/BF02413623. Google Scholar

[43]

J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967. Google Scholar

[44]

D. D. Repovš, Ambrosetti-Prodi problem with degenerate potential and Neumann boundary conditions, Electronic J. Differential Equations, 2018 (2018), Paper No. 41, 10 pp. Google Scholar

[45]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997. Google Scholar

[46]

E. Sovrano, Ambrosetti-Prodi type result to a Neumann problem via a topological approach, Discrete Contin. Dyn. Syst. Ser. S, 11 (2009), 345-355. doi: 10.3934/dcdss.2018019. Google Scholar

[47]

A. Vélez-Santiago, A quasi-linear Neumann problem of Ambrosetti–Prodi type on extension domains, Nonlinear Analysis, 160 (2017), 191-210. doi: 10.1016/j.na.2017.05.012. Google Scholar

[48]

A. Vélez-Santiago, Ambrosetti–Prodi-type problems for quasi-linear elliptic equations with nonlocal boundary conditions, Calc. Var. PDEs, 54 (2015), 3439-3469. doi: 10.1007/s00526-015-0910-6. Google Scholar

[49]

A. Vélez-Santiago, Global regularity for a class of quasi-linear local and nonlocal elliptic equations on extension domains, J. Functional Analysis, 269 (2015), 1-46. doi: 10.1016/j.jfa.2015.04.016. Google Scholar

[50]

A. Vélez-Santiago and M. Warma, A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions, J. Math. Anal. Appl., 372 (2010), 120-139. doi: 10.1016/j.jmaa.2010.07.003. Google Scholar

[51]

H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125. doi: 10.1007/BF02567633. Google Scholar

[52]

M. Warma, The fractional relative capacity and the fractional Laplacian with Neuman and Robin boundary conditions on open sets, Potential Analysis, 42 (2015), 499-547. doi: 10.1007/s11118-014-9443-4. Google Scholar

[53]

W. P. Ziemer, Weakly Differentiable Functions, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3. Google Scholar

show all references

References:
[1]

R. Adams, Sobolev Spaces, New York-London, 1975. Google Scholar

[2]

S. AizicoviciN. S. Papageorgiou and V. Staicu, Sublinear and superlinear Ambrosetti–Prodi problems for the Dirichlet $p$-Laplacian, Nonlinear Analysis, 95 (2014), 263-280. doi: 10.1016/j.na.2013.08.026. Google Scholar

[3]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Annali Mat. Pura Appl., 93 (1972), 231-246. doi: 10.1007/BF02412022. Google Scholar

[4]

D. Arcoya and D. Ruiz, The Ambrosetti–Prodi problem for the p-Laplacian operator, Comm. Partial Differential Equations, 31 (2006), 849-865. doi: 10.1080/03605300500394447. Google Scholar

[5]

M. Arias and M. Cuesta, A one side superlinear Ambrosetti–Prodi problem for the Dirichlet $p$-laplacian, J. Math. Anal. Appl., 367 (2010), 499-507. doi: 10.1016/j.jmaa.2010.01.031. Google Scholar

[6]

M. Biegert, A priori estimate for the difference of solutions to quasi-linear elliptic equations, Manuscripta Math., 133 (2010), 273-306. doi: 10.1007/s00229-010-0367-z. Google Scholar

[7]

M. Biegert, On trace of Sobolev functions on the boundary of extension domains, Proc. Amer. Math. Soc., 137 (2009), 4169-4176. doi: 10.1090/S0002-9939-09-10045-X. Google Scholar

[8]

M. Biegert and P. Vernole, Strongly local nonlinear Dirichlet functionals and forms, Adv. Math. Sci. Appl., 15 (2005), 655-682. Google Scholar

[9]

V. I. Burenkov, Sobolev Spaces on Domains, TEUBNER-TEXTE zur Mathematik, Vol. 137, 1998. doi: 10.1007/978-3-663-11374-4. Google Scholar

[10]

R. Capitanelli, Homogeneous p-Lagrangians and self-similarity, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 215-235. Google Scholar

[11]

R. Capitanelli, Nonlinear energy forms on certain fractal curves, J. Nonlinear Convex Anal., 3 (2002), 67-80. Google Scholar

[12]

R. Capitanelli and M. R. Lancia, Nonlinear energy forms and Lipschitz spaces on the Koch curve, J. Convex Anal., 9 (2002), 245-257. Google Scholar

[13]

M. CefaloG. Dell'acqua and M. R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, AMC, 218 (2012), 5453-5473. doi: 10.1016/j.amc.2011.11.033. Google Scholar

[14]

M. Cefalo and M. R. Lancia, An optimal mesh generation algorithm for domains with Koch type boundaries, Math. Comput. Simulation, 106 (2014), 133-162. doi: 10.1016/j.matcom.2014.04.009. Google Scholar

[15]

M. CefaloM. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differential and Integral equations, 26 (2013), 1027-1054. Google Scholar

[16]

S. CreoM. R. LanciaA. Vélez-Santiago and P. Vernole, Approximation of a nonlinear fractal energy functional on varying Hilbert spaces, Comm. Pure Appl. Anal., 17 (2018), 647-669. doi: 10.3934/cpaa.2018035. Google Scholar

[17]

S. Creo and V. Regis Durante, Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 65-90. doi: 10.3934/dcdss.2019005. Google Scholar

[18]

D. Danielli, N. Garofalo and D.-H. Nhieu, Non-doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Carathéodory Spaces, Mem. Amer. Math. Soc., 182 (2006), x+119 pp. doi: 10.1090/memo/0857. Google Scholar

[19]

F. O.de Paiva and M. Montenegro, An Ambrosetti-Prodi-type result for a quasilinear Neumann problem, Proc. Edinburgh Math. Soc., 55 (2012), 771-780. doi: 10.1017/S0013091512000041. Google Scholar

[20]

F. O. de Paiva and A. E. Presoto, A Neumann problem of Ambrosetti-Prodi type, J. Fixed Point Theory Appl., 18 (2016), 189-200. doi: 10.1007/s11784-015-0277-5. Google Scholar

[21]

U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendungen., 23 (2004), 115-137. doi: 10.4171/ZAA/1190. Google Scholar

[22]

M. Fukushima, Y. Oshima and M. Takeda,, Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics, vol. 19, Berlin Eds. Bauer Kazdan, Zehnder, 1994. doi: 10.1515/9783110889741. Google Scholar

[23]

P. HajłaszP. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Functional Analysis, 254 (2008), 1217-1234. doi: 10.1016/j.jfa.2007.11.020. Google Scholar

[24]

A. Jonsson and H. Wallin, Function Spaces on Subsets of  $\mathbb{R\!}^{\, n} $., Math. Rep. Vol. 2 Part Ⅰ, Academic Publisher, Harwood, 1984. Google Scholar

[25]

E. Koizumi and K. Schmitt, Ambrosetti–Prodi-type problems for quasilinear elliptic problems, Diff. Integ. Equations, 18 (2005), 241-262. Google Scholar

[26]

O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, in: Mathematics in Science and Engineering, 46. Academic Press, New York-London, 1968. Google Scholar

[27]

M. R. LanciaA. Vélez-Santiago and P. Vernole, Quasi-linear Venttsel' problems with nonlocal boundary conditions on fractal domains, Nonlinear Anal. Real Worl Appl., 35 (2017), 265-291. doi: 10.1016/j.nonrwa.2016.11.002. Google Scholar

[28]

M. R. Lancia and P. Vernole, Semilinear Venttsel problems in fractal domains, Applied Mathematics, 5 (2014), 1820-1833. Google Scholar

[29]

M. R. Lancia and P. Vernole, Venttsel problems in fractal domains, J. Evolution Equations, 14 (2014), 681-712. doi: 10.1007/s00028-014-0233-7. Google Scholar

[30]

M. R. LanciaV. Regis Durante and P. Vernole, Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1493-1520. doi: 10.3934/dcdss.2016060. Google Scholar

[31]

M. R. Lancia and M. A. Vivaldi, Lipschitz spaces and Besov traces on self-similar fractals, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 23 (1999), 101-116. Google Scholar

[32]

V. Lappalainen and A. Lehtonen, Embedding of Orliz-Sobolev spaces in Hölder spaces, Annales Academiæ Scientiarum Fennicæ, 14 (1989), 41-46. doi: 10.5186/aasfm.1989.1417. Google Scholar

[33]

V. K. Le, On a sub-supersolutions method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Analysis, 71 (2009), 3305-3321. doi: 10.1016/j.na.2009.01.211. Google Scholar

[34]

V. K. Le and K. Schmitt, Some general concepts of sub- and supersolutions for nonlinear elliptic problems, Topol. Methods Nonlinear Anal., 28 (2006), 87-103. Google Scholar

[35]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969. Google Scholar

[36]

J. Maly and U. Mosco, Remarks on measure-valued Lagrangians on homogeneous spaces, Ricerche Mat., 48 (1999), 217-231. Google Scholar

[37]

J. Mawhin, Ambrosetti–Prodi-type results in nonlinear boundary value problems, Lecture Notes in Mathematics, 1285 (1987), 290-313. doi: 10.1007/BFb0080609. Google Scholar

[38]

V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-09922-3. Google Scholar

[39]

T. J. Miotto, Superlinear Ambrosetti–Prodi problem for the p-Laplacian operator, Nonlinear Differ. Equ. Appl. NoDEA, 17 (2010), 337-353. doi: 10.1007/s00030-010-0057-2. Google Scholar

[40]

T. J. Miotto, On the Ambrosetti-Prodi problem for a system involving p-Laplacian operator, Math Nachr., 289 (2016), 67-84. doi: 10.1002/mana.201300350. Google Scholar

[41]

U. Mosco, Lagrangian metrics on fractals, Proc. Symp. Appl. Math., 54, Amer. Math. Soc., R.Spigler and S. Venakides eds., (1998), 301–323.Google Scholar

[42]

M. R. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Annali Mat. Pura Appl., 80 (1968), 1-122. doi: 10.1007/BF02413623. Google Scholar

[43]

J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967. Google Scholar

[44]

D. D. Repovš, Ambrosetti-Prodi problem with degenerate potential and Neumann boundary conditions, Electronic J. Differential Equations, 2018 (2018), Paper No. 41, 10 pp. Google Scholar

[45]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997. Google Scholar

[46]

E. Sovrano, Ambrosetti-Prodi type result to a Neumann problem via a topological approach, Discrete Contin. Dyn. Syst. Ser. S, 11 (2009), 345-355. doi: 10.3934/dcdss.2018019. Google Scholar

[47]

A. Vélez-Santiago, A quasi-linear Neumann problem of Ambrosetti–Prodi type on extension domains, Nonlinear Analysis, 160 (2017), 191-210. doi: 10.1016/j.na.2017.05.012. Google Scholar

[48]

A. Vélez-Santiago, Ambrosetti–Prodi-type problems for quasi-linear elliptic equations with nonlocal boundary conditions, Calc. Var. PDEs, 54 (2015), 3439-3469. doi: 10.1007/s00526-015-0910-6. Google Scholar

[49]

A. Vélez-Santiago, Global regularity for a class of quasi-linear local and nonlocal elliptic equations on extension domains, J. Functional Analysis, 269 (2015), 1-46. doi: 10.1016/j.jfa.2015.04.016. Google Scholar

[50]

A. Vélez-Santiago and M. Warma, A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions, J. Math. Anal. Appl., 372 (2010), 120-139. doi: 10.1016/j.jmaa.2010.07.003. Google Scholar

[51]

H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125. doi: 10.1007/BF02567633. Google Scholar

[52]

M. Warma, The fractional relative capacity and the fractional Laplacian with Neuman and Robin boundary conditions on open sets, Potential Analysis, 42 (2015), 499-547. doi: 10.1007/s11118-014-9443-4. Google Scholar

[53]

W. P. Ziemer, Weakly Differentiable Functions, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3. Google Scholar

Figure 1.  Pre-fractal Koch snowflake
Figure 2.  Surface S3
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