A two-phase free boundary problem involving exponential operator
DOI:
https://doi.org/10.12775/TMNA.2025.027Keywords
Free boundaries problems, degenerate operator, exponential functional, finite perimeterAbstract
In this paper we are interested in the study of a two-phase problem equipped with the $\Phi$-Laplacian operator $$ \Delta_\Phi u \coloneqq \mbox{div} \bigg(\varphi(|\nabla u|) \dfrac{\nabla u}{|\nabla u|}\bigg), $$% where $\Phi(s)=e^{s^2}-1$ and $\varphi=\Phi'$. We obtain the existence, boundedness, and Log-Lipschitz regularity of the minimizers of the energy functional associated to the two-phase problem. Furthermore, we also prove that the free boundaries of these minimizers have locally finite perimeter and Hausdorff dimension at most $(N-1)$.References
R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
C.O. Alves, A.R.F de Holanda and J.A. Santos, Existence of positive solutions for a class of semipositone quasilinear problems through Orlicz–Sobolev space, Proc. Amer. Math. Soc. 147 (2019), 285–299.
M.D. Amaral and E.T. Teixeira, Free transmission problems, Comm. Math. Phys. 337 (2015), no. 3, 1465–1489.
D.M. Anderson, G.B. McFadden and A.A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech. 30 (1998), 139–165.
S.N. Antontsev, J.I. Dı́az and S. Shmarev, Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics, Progress in Nonlinear Differential Equations and their Applications, vol. 48, Birkhäuser Boston, Inc., Boston, MA, 2002, xii+329 pp.
J.C. Arciero, P. Causin and F. Malgaroli, Mathematical methods for modeling the microcirculation, AIMS Biophysics 4 (2017), no. 3, 362–399.
J. Azzam and J. Bedrossian, Bounded mean oscillation and the uniqueness of active scalar equations, Trans. Amer. Math. Soc. 367 (2015), 3095-3118.
J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Springer Science & Business Media, 1990.
J. Bear and A. Verruijt, Modeling Groundwater Flow and Pollution, Springer Science & Business Media, 1987.
M. Bocea and M. Mihăilescu, On the existence and uniqueness of exponentially harmonic maps and some related problems, Israel J. Math. 230 (2019), no. 2, 795–812.
J.E.M. Braga, On the Lipschitz regularity and asymptotic behaviour of the free boundary for classes of minima of inhomogeneous two-phase Alt–Caffarelli functionals in Orlicz spaces, Ann. Mat. Pura Appl. (4) 197 (2018).
J.E.M. Braga and D.R. Moreira, Uniform Lipschitz regularity for classes of minimizers in two phase free boundary problems in Orlicz spaces with small density on the negative phase, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 4, 823–850.
N. Cantizano, A. Salort and J. Spedaletti, Continuity of solutions for the ∆φ Laplacian operator, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 4, 1355–1382.
R.S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003, xvi+411 pp.
D.M. Duc and J. Eells, Regularity of exponetional harmonic functions, Int. J. Math. 2 (1991), 395–408.
N. Fukagai, M. Ito and K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz–Sobolev nonlinearity on RN , Funkcial. Ekvac. 49 (2006), 235–267.
A. Hastings, Population Biology: Concepts and Models, Springer New York, NY, 1997, xvi+220 pp.
O.A. Ladyzhenskaya and N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering, vol. 46, Academic Press, New York, 1968.
R. Leitão, O.S. de Queiroz and E.V. Teixeira, Regularity for degenerate two-phase free boundary problems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 32 (2015), no. 4, 741–762.
G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.
G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), no. 2–3, 311–361.
F. Maggi, Sets of finite perimeter and geometric variational problems, Cambridge Studies in Advanced Mathematics, vol. 135, An Introduction to Geometric Measure Theory, Cambridge University Press, Cambridge, 2012, xx+454 pp.
S. Martı́nez and N. Wolanski, A minimum problem with free boundary in Orlicz spaces, Adv. Math. 218 (2008), no. 6, 1914–1971.
H. Naito, On a local Hölder continuity for a minimizer of the exponential energy functional, Nagoya Math. J. 129 (1993), 97–113.
M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker, Inc., New York, 1991.
J.A. Santos and S.H.M. Soares, A limiting free boundary problem for a degenerate operator in Orlicz–Sobolev spaces, Rev. Mat. Iberoam. 36 (2020), no. 6, 1687–1720.
J.A. Santos and S.H.M. Soares, Optimal design problems for a degenerate operator in Orlicz–Sobolev spaces, Calc. Var. Partial Differential Equations 59 (2020), no. 6, paper no. 183, 23.
M. Sharan and A.S. Popel, A two-phase model for flow of blood in narrow tubes with increased effective viscosity near the wall, Biorheology 38 (2001), no. 5–6, 415-28.
H. Shrivastava, A non-isotropic free transmission problem governed by quasi-linear operators, Ann. Mat. Pura Appl. (4) 200 (2021), no. 6, 2455–2471.
J. Simon, Régularité de la solution d’une équation non linéaire dans RN , Journées d’Analyse Non Linéaire (Proc. Conf., Besançon, 1977), Lecture Notes in Math., vol. 665, Springer, Berlin, 1978, pp. 205–227.
C. Zhan and S. Zhou, On a class of non-uniformly elliptic equations, NoDEA Nonlinear Differential Equations Appl. 19 (2012), no. 3, 345–363.
J. Zheng and L.S. Tavares, A free boundary problem with subcritical exponents in Orlicz spaces, Ann. Mat. Pura Appl. (4) 201 (2022), no. 2, 695–731.
J. Zheng, Z. Zhang and P. Zhao, A minimum problem with two-phase free boundary in Orlicz spaces, Monatsh. Math. 172 (2013), no. 3–4, 441–475.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
