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Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains
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On two-dimensional nonlocal Venttsel' problems in piecewise smooth domains
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 |
We establish the regularity results for solutions of nonlocal Venttsel' problems in polygonal and piecewise smooth two-dimensional domains.
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. |
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