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February  2019, 12(1): 57-64. doi: 10.3934/dcdss.2019004

On two-dimensional nonlocal Venttsel' problems in piecewise smooth domains

Simone Creo 1, , Maria Rosaria Lancia 1, , Alexander Nazarov 2, and  Paola Vernole 3,,

1. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università degli studi di Roma Sapienza, Via A. Scarpa 16, 00161 Roma, Italy
2. 

St. Petersburg Department of Steklov Mathematical Institute, and St. Petersburg State University, Fontanka 27, and Universitetskii pr. 28, 191023 St. Petersburg, Russia and 198504 St. Petersburg, Russia
3. 

Dipartimento di Matematica, Università degli Studi di Roma Sapienza, Piazzale Aldo Moro 2, 00185 Roma, Italy

* Corresponding author: Paola Vernole

Received  February 2017 Revised  August 2017 Published  July 2018

We establish the regularity results for solutions of nonlocal Venttsel' problems in polygonal and piecewise smooth two-dimensional domains.

Keywords: Venttsel' problems, nonlocal operators, piecewise smooth domains.
Mathematics Subject Classification: Primary: 35J25, 35B65; Secondary: 35R02, 35B45.
Citation: Simone Creo, Maria Rosaria Lancia, Alexander Nazarov, Paola Vernole. On two-dimensional nonlocal Venttsel' problems in piecewise smooth domains. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 57-64. doi: 10.3934/dcdss.2019004
References:
[1]

D. E. Apushkinskaya and A. I. Nazarov, A survey of results on nonlinear Venttsel' problems, Application of Mathematics, 45 (2000), 69-80.  doi: 10.1023/A:1022288717033.

[2]

D. E. Apushkinskaya and A. I. Nazarov, Linear two-phase Venttsel' problems, Ark. Mat., 39 (2001), 201-222.  doi: 10.1007/BF02384554.

[3]

W. Arendt, G. Metafune, D. Pallara and S. Romanelli, The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions, Semigroup Forum, 67 (2003), 247-261.  doi: 10.1007/s00233-002-0010-8.

[4]

F. Brezzi and G. Gilardi, Fundamentals of P. D. E. for Numerical Analysis, in: Finite Element Handbook (ed.: H. Kardestuncer and D. H. Norrie), McGraw-Hill Book Co., New York, 1987.

[5]

M. Cefalo, G. Dell'Acqua and M. R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, Applied Mathematics and Computation, 218 (2012), 5453-5473.  doi: 10.1016/j.amc.2011.11.033.

[6]

M. Cefalo, M. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differ. Integral Equ., 26 (2013), 1027-1054. 

[7]

G. Goldstein Ruiz, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. 

[8]

G. Hardy, J. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1952.

[9]

V. A. Kondrat'ev,, Boundary-value problems for elliptic equations in domains with conical or angular point, Trans. Moscow Math. Soc., 16 (1967), 209-292. 

[10]

M. R. Lancia, 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.

[11]

M. R. Lancia, A. Vélez-Santiago and P. Vernole, Quasi{linear Venttsel' problems with nonlocal boundary conditions, Nonlinear Anal. Real World Appl., 35 (2017), 265-291.  doi: 10.1016/j.nonrwa.2016.11.002.

[12]

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

[13]

V. G. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer-Verlag, 2011. doi: 10. 1007/978-3-642-15564-2.

[14]

A. I. Nazarov, On the nonstationary two-phase Venttsel problem in the transversal case, Problems in Mathematical Analysis, J. Math. Sci. (N. Y.), 122 (2004), 3251-3264.  doi: 10.1023/B:JOTH.0000031019.56619.4d.

[15]

S. A. Nazarov, B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Gruyter, Berlin-New York, 1994. doi: 10. 1515/9783110848915. 525.

[16]

J. Necas, Les Mèthodes Directes en Thèorie des Èquationes Elliptiques, Masson, Paris, 1967.

[17]

A. Vélez-Santiago, Quasi-linear variable exponent boundary value problems with Wentzell-Robin and Wentzell boundary conditions, J. Functional Analysis, 266 (2014), 560-615.  doi: 10.1016/j.jfa.2013.10.017.

[18]

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.

[19]

A. D. Venttsel', On boundary conditions for multidimensional diffusion processes, Teor. Veroyatnost. i Primenen, 4 (1959), 172-185; English translation: Theor. Probability Appl., 4 (1959), 164-177.

[20]

M. Warma, An ultracontractivity property for semigroups generated by the p-Laplacian with nonlinear Wentzell-Robin boundary conditions, Adv. Differential Equations, 14 (2009), 771-800. 

[21]

M. Warma, The p-Laplace operator with the nonlocal Robin boundary conditions on arbitrary open sets, Ann. Mat. Pura Appl., 193 (2014), 203-235.  doi: 10.1007/s10231-012-0273-y.

show all references

References:
[1]

D. E. Apushkinskaya and A. I. Nazarov, A survey of results on nonlinear Venttsel' problems, Application of Mathematics, 45 (2000), 69-80.  doi: 10.1023/A:1022288717033.

[2]

D. E. Apushkinskaya and A. I. Nazarov, Linear two-phase Venttsel' problems, Ark. Mat., 39 (2001), 201-222.  doi: 10.1007/BF02384554.

[3]

W. Arendt, G. Metafune, D. Pallara and S. Romanelli, The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions, Semigroup Forum, 67 (2003), 247-261.  doi: 10.1007/s00233-002-0010-8.

[4]

F. Brezzi and G. Gilardi, Fundamentals of P. D. E. for Numerical Analysis, in: Finite Element Handbook (ed.: H. Kardestuncer and D. H. Norrie), McGraw-Hill Book Co., New York, 1987.

[5]

M. Cefalo, G. Dell'Acqua and M. R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, Applied Mathematics and Computation, 218 (2012), 5453-5473.  doi: 10.1016/j.amc.2011.11.033.

[6]

M. Cefalo, M. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differ. Integral Equ., 26 (2013), 1027-1054. 

[7]

G. Goldstein Ruiz, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. 

[8]

G. Hardy, J. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1952.

[9]

V. A. Kondrat'ev,, Boundary-value problems for elliptic equations in domains with conical or angular point, Trans. Moscow Math. Soc., 16 (1967), 209-292. 

[10]

M. R. Lancia, 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.

[11]

M. R. Lancia, A. Vélez-Santiago and P. Vernole, Quasi{linear Venttsel' problems with nonlocal boundary conditions, Nonlinear Anal. Real World Appl., 35 (2017), 265-291.  doi: 10.1016/j.nonrwa.2016.11.002.

[12]

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

[13]

V. G. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer-Verlag, 2011. doi: 10. 1007/978-3-642-15564-2.

[14]

A. I. Nazarov, On the nonstationary two-phase Venttsel problem in the transversal case, Problems in Mathematical Analysis, J. Math. Sci. (N. Y.), 122 (2004), 3251-3264.  doi: 10.1023/B:JOTH.0000031019.56619.4d.

[15]

S. A. Nazarov, B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Gruyter, Berlin-New York, 1994. doi: 10. 1515/9783110848915. 525.

[16]

J. Necas, Les Mèthodes Directes en Thèorie des Èquationes Elliptiques, Masson, Paris, 1967.

[17]

A. Vélez-Santiago, Quasi-linear variable exponent boundary value problems with Wentzell-Robin and Wentzell boundary conditions, J. Functional Analysis, 266 (2014), 560-615.  doi: 10.1016/j.jfa.2013.10.017.

[18]

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.

[19]

A. D. Venttsel', On boundary conditions for multidimensional diffusion processes, Teor. Veroyatnost. i Primenen, 4 (1959), 172-185; English translation: Theor. Probability Appl., 4 (1959), 164-177.

[20]

M. Warma, An ultracontractivity property for semigroups generated by the p-Laplacian with nonlinear Wentzell-Robin boundary conditions, Adv. Differential Equations, 14 (2009), 771-800. 

[21]

M. Warma, The p-Laplace operator with the nonlocal Robin boundary conditions on arbitrary open sets, Ann. Mat. Pura Appl., 193 (2014), 203-235.  doi: 10.1007/s10231-012-0273-y.

Figure 1.  A possible example of domain $\Omega$. In this case $N=9$ and $\alpha=\alpha_7$
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