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\(L^1\) existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions. (English) Zbl 1123.35016

The authors study quasilinear elliptic equations with nonlinear boundary conditions of the form \[ -\text{div}\;\mathbf{a} (x, Du) + \gamma(u) \ni \phi,\;\; \text{in}\; \Omega, \]
\[ \mathbf{a} (x, Du) + \beta(u) \ni \psi,\;\; \text{on}\; \partial \Omega, \] where \(\Omega\) is an open bounded domain of \(\mathbb{R}^N\), with smooth boundary \(\partial \Omega\), \(\phi \in L^1 (\Omega)\) and \(\psi \in L^1 (\partial \Omega)\). The nonlinearities \(\gamma\), \(\beta\) are maximal monotone graphs in \(\mathbb{R}^2\), such that \(0 \in \gamma(0)\) and \(0 \in \beta(0)\). More precisely, they prove existence and uniqueness results of weak and entropy solutions in the case of general nonlinear operators \(\mathbf{a} (x, Du)\) with nonhomogeneous boundary conditions and general nonlinearities \(\beta\) and \(\gamma\). The existence results are obtained with the help of appropriate approximating problems whose solutions satisfy certain estimates and monotone properties, which allow to pass to the limit.

MSC:

35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
76D27 Other free boundary flows; Hele-Shaw flows
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References:

[1] K. Ammar, F. Andreu, J. Toledo, Quasi-linear elliptic problems in \(L^1\) with non homogeneous boundary conditions, Rend. Mat. Univ. Roma, in press · Zbl 1153.35026
[2] F. Andreu, N. Igbida, J.M. Mazón, J. Toledo, A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions, in preparation · Zbl 1116.35073
[3] Andreu, F.; Mazón, J.M.; Segura de León, S.; Toledo, J., Quasi-linear elliptic and parabolic equations in \(L^1\) with nonlinear boundary conditions, Adv. math. sci. appl., 7, 1, 183-213, (1997) · Zbl 0882.35048
[4] Ph. Bénilan, Equations d’évolution dans un espace de Banach quelconque et applications, Thesis, Univ. Orsay, 1972
[5] Bénilan, Ph.; Boccardo, L.; Gallouët, Th.; Gariepy, R.; Pierre, M.; Vázquez, J.L., An \(L^1\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. scuola norm. sup. Pisa cl. sci. (4), 22, 2, 241-273, (1995) · Zbl 0866.35037
[6] Benilan, Ph.; Brezis, H.; Crandall, M.G., A semilinear equation in \(L^1(R^N)\), Ann. scuola norm. sup. Pisa cl. sci. (4), 2, 4, 523-555, (1975) · Zbl 0314.35077
[7] Bénilan, Ph.; Crandall, M.G., Completely accretive operators, (), 41-75 · Zbl 0895.47036
[8] Ph. Bénilan, M.G. Crandall, A. Pazy, Evolution Equations governed by accretive operators, in press
[9] Bénilan, Ph.; Crandall, M.G.; Sacks, P., Some \(L^1\) existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions, Appl. math. optim., 17, 3, 203-224, (1988) · Zbl 0652.35043
[10] Boccardo, L.; Gallouët, Th., Nonlinear elliptic equations with right-hand side measures, Comm. partial differential equations, 17, 641-655, (1992) · Zbl 0812.35043
[11] Brezis, H., Problémes unilatéraux, J. math. pures appl., 51, 1-168, (1972) · Zbl 0237.35001
[12] Brezis, H., Opérateur maximaux monotones et semi-groupes de contractions dans LES espaces de Hilbert, (1984), Oxford Univ. Press Oxford
[13] Crandall, M.G., An introduction to evolution governed by accretive operators, (), 131-165, Dekker, New York, 1991
[14] Crank, J., Free and moving boundary problems, (1977), North-Holland Amsterdam · Zbl 0367.65050
[15] DiBenedetto, E.; Friedman, A., The ill-posed hele – shaw model and the Stefan problem for supercooler water, Trans. amer. math. soc., 282, 183-204, (1984) · Zbl 0621.35102
[16] Duvaux, G.; Lions, J.L., Inequalities in mechanics and physiscs, (1976), Springer-Verlag
[17] Elliot, C.M.; Janosky, V., A variational inequality approach to the hele – shaw flow with a moving boundary, Proc. roy. soc. edinburg sect. A, 88, 93-107, (1981) · Zbl 0455.76043
[18] Igbida, N.; Kirane, M., A degenerate diffusion problem with dynamical boundary conditions, Math. ann., 323, 2, 377-396, (2002) · Zbl 1001.35072
[19] N. Igbida, The Hele-Shaw problem with dynamical boundary conditions, Preprint · Zbl 1127.35023
[20] N. Igbida, Nonlinear heat equation with fast/logarithmic diffusion, Preprint · Zbl 1180.35300
[21] Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities and their applications, Pure appl. math., vol. 88, (1980), Academic Press Inc. New York · Zbl 0457.35001
[22] Lieberman, G.M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear anal., 12, 1203-1219, (1988) · Zbl 0675.35042
[23] Lions, J.L., Quelques méthodes de résolution de problémes aux limites non linéaires, (1968), Dunod-Gauthier-Vilars Paris · Zbl 0189.40603
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