Abstract
The goal of this paper is to show the existence of a bounded variation solution, which is based on the Anzellotti pairing to an evolution problem associated with the minimal surface equations. A key ingredient in the proof is to approximate the parabolic minimal surface problem by a quasilinear parabolic problem involving a parameter \(p>1\), and then by establishing some energy estimates independent of p, we take the limit as \(p\rightarrow 1^{+}\) to obtain the desired result.
Similar content being viewed by others
Availability of data and materials
Data sharing not applicable to this paper as no datasets were generated or analyzed during the current study.
References
Azzollini, A.: On a prescribed mean curvature equation in Lorentz–Minkowski space. J. Math. Pures Appl. (9) 106, 1122–1140 (2016)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems, In: Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, ISBN: 0-19-850245-1, (2000), xviii+434 pp
Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 135(1), 293–318 (1983)
Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. MPS-SIAM, Philadelphia (2006)
Andreu, F., Caselles, V., Mazón, J.M.: Parabolic Quasilinear Equations Minimizing Linear Growth Functional. Progress in Mathematics, vol. 223. Birkhäuser, Basel (2004)
Bereanu, C., Jebelean, P., Mawhin, J.: The Dirichlet problem with mean curvature operator in Minkowski space–a variational approach. Adv. Nonlinear Stud. 14, 315–326 (2014)
Corsato, C., Obersnel, F., Omari, P., Rivetti, S.: On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Discrete Contin. Dyn. Syst. 2013, 159–169 (2013)
Coffman, C.V., Ziemer, W.K.: A prescribed mean curvature problem on domains without radial symmetry. SIAM J. Math. Anal. 22, 982–990 (1991)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)
Engler, H., Kawohl, B., Luckhaus, S.: Gradient estimates for solutions of parabolic equations and systems. J. Math. Anal. Appl. 147, 309–329 (1990)
Ecker, K.: Estimates for Evolutionary surfaces of prescribed mean curvature. Math. Z. 180, 179–192 (1982)
Finn, R.: On the equations of capillarity. J. Math. Fluid Mech. 3, 139–151 (2001)
Finn, R.: Capillarity problems for compressible fluids. Mem. Differ. Equ. Math. Phys. 33, 47–55 (2004)
Finn, R., Luli, G.: On the capillary problem for compressible fluids. J. Math. Fluid Mech. 9, 87–103 (2007)
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser, Boston/Basel/Stuttgart (1984)
Gerhardt, C.: Evolutionary surfaces of prescribed mean curvature. J. Differ. Equ. 36, 139–172 (1980)
Giga, Y.: Interior derivative blow-up for quasilinear parabolic equations. Discrete Contin. Dyn. Syst. 1, 449–461 (1995)
Kinnunen, J., Scheven, C.: On the definition of solution to the total variation flow, arXiv:2106.05711 [math.AP]
Kawohl, B., Kutev, N.: Global behaviour of solutions to a parabolic mean curvature equation. Differ. Integral Equ. 8, 1923–1946 (1995)
Kawohl, B., Kutev, N.: Global behaviour of solutions to a parabolic mean curvature equation. Differ. Integral Equ. 8, 1923–1946 (1995)
Lopéz, R.: An existence theorem of constant mean curvature graphs in Euclidean space. Glasg. Math. J. 44, 455–461 (2002)
Lichnewsky, A., Temam, R.: Pseudosolutions of the time-dependent minimal surface problem. J. Differ. Equ. 30, 340–364 (1978)
Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux limites non linéaires. Dounod, Paris (1969)
Lieberman, G.M.: Interior Gradient bounds for non-uniformly parabolic equations. Indiana Univ. Math. J. 32, 579–601 (1983)
Marcellini, P., Miller, K.: Asymptotic growth for the parabolic equation of prescribed mean curvature. J. Differ. Equ. 51, 326–358 (1984)
Marcellini, P., Miller, K.: Elliptic versus parabolic regularization for the equation of prescribed mean curvature. J. Differ. Equ. 137(1), 1–53 (1997)
Nakao, M., Ohara, Y.: Gradient estimates for a quasilinear parabolic equation of the mean curvature type. J. Math. Soc. Jpn. 48, 455–466 (1996)
Nakao, M., Chen, C., Ohara, Y.: Global existence and gradient estimates for a quasilinear parabolic equation of the mean curvature type with a strong perturbation. Differ. Integral Equ. 14(1), 59–74 (2001)
Pimenta, M.T.O., Montenegro, M.: Existence of a BV solution for a mean curvature equation. J. Differ. Equ. 299, 51–64 (2021)
Robinson, J.C.: Infinite-Dimensional Dynamical Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge University Press, Cambridge (2001)
Showalter, E.R.: Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations. (Mathematical Surveys and Monographs), vol. 49. American Mathematical Society, Providence (1997)
Simon, J.: Compact sets in the space \(L^{p}(0, T, B)\). Ann. Math. Pura. Appl. (4) 146, 65–96 (1987)
Scheven, C., Schmidt, T.: On the dual formulation of obstacle problems for the total variation and the area functional. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 35, 1175–1207 (2018)
Scheven, C., Schmidt, T.: BV supersolutions to equations of 1-Laplace and minimal surface type. J. Differ. Equ. 261(3), 1904–1932 (2016)
Serrin, J.: The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. R. Soc. Lond. Ser. A 264, 413–496 (1969)
Trudinger, N.S.: Gradient estimates and mean curvature. Math. Z. 131, 165–175 (1972)
Temam, R.: Applications de l’analyse convexe au calcul des variations. In: Gossez, J.P., et al. (eds.) Les Opérateurs Non-Linéaires et le Calcul de Variation. Lecture Notes in Mathematics, vol. 543. Springer, Berlin (1976)
Ziemer, W.P.: Weakly Differentiable Functions, GTM 120. Springer, Berlin (1989)
Zheng, S.: Nonlinear Evolution Equations, Chapman & Hall/CRC Monographs and surveys in Pure and Applied Mathematics, vol. 133. Chapman & Hall/CRC, Boca Raton (2004)
Funding
Funding information is not applicable/No funding was received.
Author information
Authors and Affiliations
Contributions
CA and TB wrote the main manuscript text. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
There is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Alves, C.O., Boudjeriou, T. Existence of solutions for a class of heat equations involving the mean curvature operator. Anal.Math.Phys. 13, 13 (2023). https://doi.org/10.1007/s13324-022-00774-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-022-00774-7