A strongly degenerate quasilinear equation : the elliptic case

Andreu, Fuensanta; Caselles, Vicent; Mazón, José

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 555-587.

We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation $u-\mathrm{div}𝐚(u,Du)=v$, where $v\in L^1$, $𝐚(z,\xi )={\nabla }_{\xi }f(z,\xi )$, and $f$ is a convex function of $\xi$ with linear growth as $\parallel \xi \parallel \to \infty$, satisfying other additional assumptions. In particular, this class includes the case where $f(z,\xi )=\varphi \left(z\right)\psi \left(\xi \right)$, $\varphi >0$, $\psi$ being a convex function with linear growth as $\parallel \xi \parallel \to \infty$. In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic problem with initial data in $L^1$.

Classification : 35J60, 47H06, 47H20

[1] L. Ambrosio - N. Fusco - D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs, The Clarendom Press, Oxford University Press, 2000. | MR | Zbl

[2] F. Andreu - V. Caselles - J.M. Mazón, A parabolic quasilinear problem for linear growth functionals, Rev. Mat. Iberoamericana 18 (2002), 135-185. | MR | Zbl

[3] F. Andreu - V. Caselles - J.M. Mazón, Existence and uniqueness of solution for a parabolic quasilinear problem for linear growth functionals with $L^1$ data, Math. Ann. 322 (2002), 139-206. | MR | Zbl

[4] F. Andreu - V. Caselles - J.M. Mazón, The Cauchy Problem for Linear Growth Functionals with, J. Evol. Equ. 3 (2003), 39-65. | MR | Zbl

[5] F. Andreu - V. Caselles - J.M. Mazón, “Parabolic Quasilinear Equations Minimizing Linear Growth Functionals”, Progress in Mathematics, vol. 223, Birkhäuser-Verlag, Basel, 2004. | MR | Zbl

[6] F. Andreu - V. Caselles - J.M. Mazón, A Strongly Degenerate Quasilinear Equation: the Parabolic Case, preprint, 2003.

[7] G. Anzellotti, Pairings Between Measures and Bounded Functions and Compensated Compactness, Ann. Mat. Pura Appl. (4) 135 (1983), 293-318. | MR | Zbl

[8] G.I. Barenblatt - M. Bertsch - R. Dal Passo - V.M. Prostokishin - M. Ughi, A mathematical Model of Turbulent Heat and Mass Transfer in Stable Stratified Shear Flow, J. Fluid Mech. 253 (1993), 341-358. | MR | Zbl

[9] Ph. Bénilan - L. Boccardo - T. Gallouet - R. Gariepy - M. Pierre - J.L Vazquez, An $L^1$-Theory of Existence and Uniqueness of Solutions of Nonlinear Elliptic Equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), 241-273. | Numdam | MR | Zbl

[10] Ph. Bénilan - M.G. Crandall, Completely Accretive Operators, In: “Semigroups Theory and Evolution Equations”, Ph. Clement et al. (eds.), Marcel Dekker, 1991, pp. 41-76. | MR | Zbl

[11] Ph. Bénilan - M.G. Crandall - A. Pazy, “Evolution Equations Governed by Accretive Operators”, Book in preparation.

[12] M. Bertsch - R. Dal Passo, Hyperbolic Phenomena in a Strongly Degenerate Parabolic Equation, Arch. Ration. Mech. Anal. 117 (1992), 349-387. | MR | Zbl

[13] M. Bertsch - R. Dal Passo, A Parabolic Equation with Mean-Curvature Type Operator, In: “Progres Nonlinear Differential Equation Appl.”, F. Browder (ed.), 7 Birkhäuser, 1992, 89-97. | MR | Zbl

[14] Ph. Blanc, On the regularity of the solutions of some degenerate parabolic equations, Comm. Partial Differential Equations 18 (1993), 821-846. | MR | Zbl

[15] Ph. Blanc, Sur une classe d'equations paraboliques degeneréesa une dimension d'espace possedant des solutions discontinues, Ph.D. Thesis, number 798, Ecole Polytechnique Federale de Lausanne, 1989. | Zbl

[16] D. Blanchard - F. Murat, Renormalized solutions on nonlinear parabolic problems with $L^1$ data: existence and uniqueness, Proc. Royal Soc. Edinburgh Sect. A 127 (1997), 1137-1152. | MR | Zbl

[17] J. Carrillo - P. Wittbold, Uniqueness of Renormalized Solutions of Degenerate Elliptic-Parabolic problems, J. Differential Equations 156 (1999), 93-121. | MR | Zbl

[18] M. G. Crandall, Nonlinear Semigroups and Evolution Governed by Accretive Operators, In: “Proceeding of Symposium in Pure Mat.”, Part I, F. Browder (ed.) A.M.S., Providence 1986, 305-338. | MR | Zbl

[19] M. G. Crandall - T. M. Liggett, Generation of Semigroups of Nonlinear Transformations on General Banach Spaces, Amer. J. Math. 93 (1971), 265-298. | MR | Zbl

[20] G. Dal Maso, Integral representation on $BV\left(\Omega \right)$ of $\Gamma$-limits of variational integrals, Manuscripta Math. 30 (1980), 387-416. | MR | Zbl

[21] R. Dal Passo, Uniqueness of the entropy solution of a strongly degenerate parabolic equation, Comm. Partial Differential Equations 18 (1993), 265-279. | MR | Zbl

[22] E. De Giorgi - L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur., Rend. Lincei (9) Mat. Appl., (8) 82 (1988), 199-210. | MR | Zbl

[23] J. J. Duderstadt - G. A. Moses, “Inertial Confinement Fusion”, John Wiley & Sons, 1982.

[24] L. C. Evans - R. F. Gariepy, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Math., CRC Press, 1992. | MR | Zbl

[25] M. A. Krasnosel'Skii - Ya. B. Rutickii, “Convex Functions and Orliz Spaces”, P. Noordholff, Groningen, 1961. | Zbl

[26] R. Kohn - R. Temam, Dual space of stress and strains with application to Hencky plasticity, Appl. Math. Optim. 10 (1983), 1-35. | MR | Zbl

[27] S. N. Kruzhkov, First order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970), 217-243. | Zbl

[28] J. Leray - J. L. Lions, Quelques resultats de Visik sur les problemes elliptiques semi-lineaires par les metode de Minty et Browder, Bull. Soc. Math. France 93 (1965), 97-107. | Numdam | MR | Zbl

[29] P. Rosenau, Free Energy Functionals at the High Gradient Limit, Phys. Rev. A 41 (1990), 2227-2230.

[30] W.P. Ziemer, “Weakly Differentiable Functions”, GTM 120, Springer Verlag, 1989. | MR | Zbl