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doi: 10.3934/dcdss.2022047
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Transmission problems for the fractional $ p $-Laplacian across fractal interfaces

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 10, 00161 Roma, Italy

* Corresponding author: Maria Rosaria Lancia

Received  October 2021 Revised  January 2022 Early access March 2022

We consider a parabolic transmission problem, involving nonlinear fractional operators of different order, across a fractal interface $ \Sigma $. The transmission condition is of Robin type and it involves the jump of the $ p $-fractional normal derivatives on the irregular interface. After proving existence and uniqueness results for the weak solution of the problem at hand, via a semigroup approach, we investigate the regularity of the nonlinear fractional semigroup.

Citation: Simone Creo, Maria Rosaria Lancia, Paola Vernole. Transmission problems for the fractional $ p $-Laplacian across fractal interfaces. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022047
References:
[1]

S. Abe and S. Thurner, Anomalous diffusion in view of Einstein's 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407. 

[2]

D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03282-4.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, (translated from the Romanian), Noordhoff International Publishing, Leiden, 1976.

[4]

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.

[5]

H. Brézis, Propriétés régularisantes de certains semi-groupes non linéaires, Israel J. Math., 9 (1971), 513-534.  doi: 10.1007/BF02771467.

[6]

F. Brezzi and G. Gilardi, Fundamentals of PDEs for numerical analysis, in Finite Element Handbook (eds. H. Kardestuncer and D. H. Norrie), McGraw-Hill Book Co., New York, 1987.

[7]

F. Cipriani and G. Grillo, Uniform bounds for solutions to quasilinear parabolic equations, J. Differential Equations, 177 (2001), 209-234.  doi: 10.1006/jdeq.2000.3985.

[8]

S. Creo and M. R. Lancia, Fractional $(s, p)$-Robin-Venttsel' problems on extension domains, NoDEA Nonlinear Differential Equations Appl., 28 (2021), paper no. 31, 33 pp. doi: 10.1007/s00030-021-00692-w.

[9]

S. CreoM. R. Lancia and P. Vernole, Convergence of fractional diffusion processes in extension domains, J. Evol. Equ., 20 (2020), 109-139.  doi: 10.1007/s00028-019-00517-5.

[10]

S. CreoM. R. Lancia and P. Vernole, M-convergence of $p$-fractional energies in irregular domains, J. Convex Anal., 28 (2021), 509-534. 

[11]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology Vol. 2: Functional and Variational Methods, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-61566-5.

[12]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[13]

A. A. DubkovB. Spagnolo and V. V. Uchaikin, Lèvy flight superdiffusion: An introduction, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 2649-2672.  doi: 10.1142/S0218127408021877.

[14] K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1986. 
[15]

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

[16]

C. G. Gal and M. Warma, Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions, Evol. Equat. Control Theory, 5 (2016), 61-103.  doi: 10.3934/eect.2016.5.61.

[17]

C. G. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.

[18]

C. G. Gal and M. Warma, Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations, 42 (2017), 579-625.  doi: 10.1080/03605302.2017.1295060.

[19]

R. GorenfloF. Mainardi and A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion, Chaos Solitons Fractals, 34 (2007), 87-103.  doi: 10.1016/j.chaos.2007.01.052.

[20]

Q.-Y. Guan, Integration by parts formula for regional fractional Laplacian, Commun. Math. Phys., 266 (2006), 289-329.  doi: 10.1007/s00220-006-0054-9.

[21]

Q.-Y. Guan and Z.-M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.  doi: 10.1142/S021949370500150X.

[22]

P. H. Hung and E. Sanchez-Palencia, Phénomènes des transmission à travers des couches minces de conductivité élevée, J. Math. Anal. Appl., 47 (1974), 284-309.  doi: 10.1016/0022-247X(74)90023-7.

[23]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.  doi: 10.1002/cpa.20253.

[24]

P. W. Jones, Quasiconformal mapping and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88.  doi: 10.1007/BF02392869.

[25]

A. Jonsson, Besov spaces on closed subsets of $\mathbb{R}^n$, Trans. Amer. Math. Soc., 341 (1994), 355-370.  doi: 10.2307/2154626.

[26]

A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb{R}^n$, Part 1, Math. Reports, Vol. 2, Harwood Acad. Publ., London, 1984.

[27]

A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia Math., 112 (1995), 285-300.  doi: 10.4064/sm-112-3-285-300.

[28]

M. R. Lancia, A transmission problem with a fractal interface, Z. Anal. Anwendungen, 21 (2002), 113-133.  doi: 10.4171/ZAA/1067.

[29]

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.

[30]

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

[31]

M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. Math. Anal., 42 (2010), 1539-1567.  doi: 10.1137/090761173.

[32]

M. R. Lancia and P. Vernole, Semilinear fractal problems: Approximation and regularity results, Nonlinear Anal., 80 (2013), 216-232.  doi: 10.1016/j.na.2012.08.020.

[33]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093.

[34]

A. MelletS. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.

[35]

W. R. Schneider, Grey noise, in Stochastic Processes, Physics and Geometry (eds. S. Albeverio, G. Casati, U. Cattaneo, D. Merlini and R. Moresi), World Scientific, Teaneck, NJ, USA, 1990,676–681.

[36]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs 49, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/049.

[37]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33-44. 

[38]

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

[39]

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.

[40]

M. Warma, The fractional Neumann and Robin type boundary conditions for the regional fractional $p$-Laplacian, NoDEA Nonlinear Differential Equations Appl., 23 (2016), paper no. 1, 46 pp. doi: 10.1007/s00030-016-0354-5.

[41]

M. Warma, On a fractional $(s, p)$-Dirichlet-to-Neumann operator on bounded Lipschitz domains, J. Elliptic Parabol. Equ., 4 (2018), 223-269.  doi: 10.1007/s41808-018-0017-2.

[42]

J. L. Zolesio, Multiplication dans les espaces de Besov, Proc. Roy. Soc. Edinburgh Sect. A, 78 (1977/78), 113-117.  doi: 10.1017/S0308210500009872.

show all references

References:
[1]

S. Abe and S. Thurner, Anomalous diffusion in view of Einstein's 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407. 

[2]

D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03282-4.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, (translated from the Romanian), Noordhoff International Publishing, Leiden, 1976.

[4]

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.

[5]

H. Brézis, Propriétés régularisantes de certains semi-groupes non linéaires, Israel J. Math., 9 (1971), 513-534.  doi: 10.1007/BF02771467.

[6]

F. Brezzi and G. Gilardi, Fundamentals of PDEs for numerical analysis, in Finite Element Handbook (eds. H. Kardestuncer and D. H. Norrie), McGraw-Hill Book Co., New York, 1987.

[7]

F. Cipriani and G. Grillo, Uniform bounds for solutions to quasilinear parabolic equations, J. Differential Equations, 177 (2001), 209-234.  doi: 10.1006/jdeq.2000.3985.

[8]

S. Creo and M. R. Lancia, Fractional $(s, p)$-Robin-Venttsel' problems on extension domains, NoDEA Nonlinear Differential Equations Appl., 28 (2021), paper no. 31, 33 pp. doi: 10.1007/s00030-021-00692-w.

[9]

S. CreoM. R. Lancia and P. Vernole, Convergence of fractional diffusion processes in extension domains, J. Evol. Equ., 20 (2020), 109-139.  doi: 10.1007/s00028-019-00517-5.

[10]

S. CreoM. R. Lancia and P. Vernole, M-convergence of $p$-fractional energies in irregular domains, J. Convex Anal., 28 (2021), 509-534. 

[11]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology Vol. 2: Functional and Variational Methods, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-61566-5.

[12]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[13]

A. A. DubkovB. Spagnolo and V. V. Uchaikin, Lèvy flight superdiffusion: An introduction, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 2649-2672.  doi: 10.1142/S0218127408021877.

[14] K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1986. 
[15]

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

[16]

C. G. Gal and M. Warma, Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions, Evol. Equat. Control Theory, 5 (2016), 61-103.  doi: 10.3934/eect.2016.5.61.

[17]

C. G. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.

[18]

C. G. Gal and M. Warma, Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations, 42 (2017), 579-625.  doi: 10.1080/03605302.2017.1295060.

[19]

R. GorenfloF. Mainardi and A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion, Chaos Solitons Fractals, 34 (2007), 87-103.  doi: 10.1016/j.chaos.2007.01.052.

[20]

Q.-Y. Guan, Integration by parts formula for regional fractional Laplacian, Commun. Math. Phys., 266 (2006), 289-329.  doi: 10.1007/s00220-006-0054-9.

[21]

Q.-Y. Guan and Z.-M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.  doi: 10.1142/S021949370500150X.

[22]

P. H. Hung and E. Sanchez-Palencia, Phénomènes des transmission à travers des couches minces de conductivité élevée, J. Math. Anal. Appl., 47 (1974), 284-309.  doi: 10.1016/0022-247X(74)90023-7.

[23]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.  doi: 10.1002/cpa.20253.

[24]

P. W. Jones, Quasiconformal mapping and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88.  doi: 10.1007/BF02392869.

[25]

A. Jonsson, Besov spaces on closed subsets of $\mathbb{R}^n$, Trans. Amer. Math. Soc., 341 (1994), 355-370.  doi: 10.2307/2154626.

[26]

A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb{R}^n$, Part 1, Math. Reports, Vol. 2, Harwood Acad. Publ., London, 1984.

[27]

A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia Math., 112 (1995), 285-300.  doi: 10.4064/sm-112-3-285-300.

[28]

M. R. Lancia, A transmission problem with a fractal interface, Z. Anal. Anwendungen, 21 (2002), 113-133.  doi: 10.4171/ZAA/1067.

[29]

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.

[30]

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

[31]

M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. Math. Anal., 42 (2010), 1539-1567.  doi: 10.1137/090761173.

[32]

M. R. Lancia and P. Vernole, Semilinear fractal problems: Approximation and regularity results, Nonlinear Anal., 80 (2013), 216-232.  doi: 10.1016/j.na.2012.08.020.

[33]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093.

[34]

A. MelletS. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.

[35]

W. R. Schneider, Grey noise, in Stochastic Processes, Physics and Geometry (eds. S. Albeverio, G. Casati, U. Cattaneo, D. Merlini and R. Moresi), World Scientific, Teaneck, NJ, USA, 1990,676–681.

[36]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs 49, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/049.

[37]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33-44. 

[38]

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

[39]

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.

[40]

M. Warma, The fractional Neumann and Robin type boundary conditions for the regional fractional $p$-Laplacian, NoDEA Nonlinear Differential Equations Appl., 23 (2016), paper no. 1, 46 pp. doi: 10.1007/s00030-016-0354-5.

[41]

M. Warma, On a fractional $(s, p)$-Dirichlet-to-Neumann operator on bounded Lipschitz domains, J. Elliptic Parabol. Equ., 4 (2018), 223-269.  doi: 10.1007/s41808-018-0017-2.

[42]

J. L. Zolesio, Multiplication dans les espaces de Besov, Proc. Roy. Soc. Edinburgh Sect. A, 78 (1977/78), 113-117.  doi: 10.1017/S0308210500009872.

Figure 1.  The domain $ \Omega $
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