The limit case in a nonlocal $p$-Laplacian equation with dynamical boundary conditions
DOI:
https://doi.org/10.33044/revuma.4631Abstract
In this paper we deal with the limit as $p\to \infty$ for the nonlocal analogous to the $p$-Laplacian evolution with dynamic boundary conditions. Our main result demonstrates this limit in both the elliptic and parabolic cases. We are interested in smooth and singular kernels and show the existence and uniqueness of a limit solution. We obtain that the limit solution of the elliptic problem turns out to be also a viscosity solution of a corresponding problem. We prove that the natural energy functionals associated with this problem converge, in the sense of Mosco, to a limit functional and therefore we obtain convergence of solutions to the evolution problems in the parabolic case. For the limit problem, we provide examples of explicit solutions for some particular data.
Downloads
References
F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, The limit as $p→∞$ in a nonlocal $p$-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles, Calc. Var. Partial Differential Equations 35 no. 3 (2009), 279–316. DOI MR Zbl
F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, A nonlocal $p$-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions, SIAM J. Math. Anal. 40 no. 5 (2009), 1815–1851. DOI MR Zbl
F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, The Neumann problem for nonlocal nonlinear diffusion equations, J. Evol. Equ. 8 no. 1 (2008), 189–215. DOI MR Zbl
F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi, and J. J. Toledo-Melero, Nonlocal diffusion problems, Math. Surv. Monogr. 165, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. DOI MR Zbl
G. Aronsson, L. C. Evans, and Y. Wu, Fast/slow diffusion and growing sandpiles, J. Differential Equations 131 no. 2 (1996), 304–335. DOI MR Zbl
H. Attouch, Familles d'opérateurs maximaux monotones et mesurabilité, Ann. Mat. Pura Appl. (4) 120 (1979), 35–111. DOI MR Zbl
P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: stationary solutions in higher space dimensions, J. Statist. Phys. 95 no. 5-6 (1999), 1119–1139. DOI MR Zbl
P. W. Bates, P. C. Fife, X. Ren, and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal. 138 no. 2 (1997), 105–136. DOI MR Zbl
P. M. Berná and J. D. Rossi, Nonlocal diffusion equations with dynamical boundary conditions, Nonlinear Anal. 195 (2020), article no. 111751, 25 pp. DOI MR Zbl
T. Bhattacharya, E. DiBenedetto, and J. Manfredi, Limits as $p→∞$ of $Δ_pu_p=f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, special issue (1991), 15–68. MR
H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble) 18 no. 1 (1968), 115–175. MR Zbl Available at http://www.numdam.org/item?id=AIF_1968__18_1_115_0.
H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies 5, Notas de Matemática 50, North-Holland, Amsterdam-London; American Elsevier, New York, 1973. MR Zbl
H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Funct. Anal. 9 (1972), 63–74. DOI MR Zbl
C. Carrillo and P. Fife, Spatial effects in discrete generation population models, J. Math. Biol. 50 no. 2 (2005), 161–188. DOI MR Zbl
A. Chambolle, E. Lindgren, and R. Monneau, A Hölder infinity Laplacian, ESAIM Control Optim. Calc. Var. 18 no. 3 (2012), 799–835. DOI MR Zbl
X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations 2 no. 1 (1997), 125–160. DOI MR Zbl
C. Cortazar, M. Elgueta, J. D. Rossi, and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations 234 no. 2 (2007), 360–390. DOI MR Zbl
C. Cortazar, M. Elgueta, J. D. Rossi, and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal. 187 no. 1 (2008), 137–156. DOI MR Zbl
L. C. Evans, M. Feldman, and R. F. Gariepy, Fast/slow diffusion and collapsing sandpiles, J. Differential Equations 137 no. 1 (1997), 166–209. DOI MR Zbl
P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in nonlinear analysis, Springer, Berlin, 2003, pp. 153–191. MR Zbl
P. Fife and X. Wang, A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions, Adv. Differential Equations 3 no. 1 (1998), 85–110. DOI MR Zbl
E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 no. 1-2 (2014), 795–826. DOI MR Zbl
J. M. Mazón, M. Solera-Diana, and J. J. Toledo-Melero, Variational and diffusion problems in random walk spaces, Prog. Nonlinear Differ. Equ. Appl. 103, Birkhäuser, Cham, 2023. DOI MR Zbl
U. Mosco, Approximation of the solutions of some variational inequalities, Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 373–394; erratum: 21 (1967), 765. MR Zbl
U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (1969), 510–585. DOI MR Zbl
E. Öztürk and J. D. Rossi, Limit for the $p$-Laplacian equation with dynamical boundary conditions, Electron. J. Differential Equations no. 1, Special issue (2021), 135–147. DOI MR
M. Solera and J. Toledo, Nonlocal doubly nonlinear diffusion problems with nonlinear boundary conditions, J. Evol. Equ. 23 no. 2 (2023), article no. 24, 83 pp. DOI MR Zbl
X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations 183 no. 2 (2002), 434–461. DOI MR Zbl
L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, J. Differential Equations 197 no. 1 (2004), 162–196. DOI MR Zbl
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Eylem Öztürk
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.
