## A $$p$$-Laplacian problem in $$L^ 1$$ with nonlinear boundary conditions.(English)Zbl 0858.35147

Let $$\Omega$$ be a bounded domain in $$\mathbb{R}^n$$ and $$(\Gamma_D,\Gamma_N)$$, resp. $$(\Gamma_E,\Gamma_I)$$ two different partitions of the boundary $$\partial\Omega$$. Let $$\alpha$$ be a maximal monotone graph on $$\mathbb{R}$$. The author studies the following problem: For $$H\in L^1$$, $$\overline{u},\overline{\varphi}\in W^{1,p}(\Omega)$$, find functions $$h,u,\varphi$$ such that \begin{aligned} -\text{div } a(x,u,\nabla u)&= \sigma(u)|\nabla \varphi|^2+H-h \quad\text{in }\Omega,\\ \text{div}(\sigma(u)\nabla\varphi)&= 0\quad\text{in }\Omega, \qquad h\in\alpha(u) \text{ a.e. in }\Omega,\\ u=\overline{u}\quad&\text{ on }\Gamma_D, \qquad a(x,u,\nabla u)\cdot n+f(x,u)=0 \quad\text{on }\Gamma_N\\ \varphi=\overline{\varphi}\quad &\text{ on }\Gamma_E, \qquad \sigma(u)\partial_n \varphi+g(x,u)=0 \quad\text{on }\Gamma_I.\end{aligned} The operator $$\text{div }a$$ is of $$p$$-Lapace type (with $$1<p<2)$$ and $$a$$ satisfies Leray-Lions type conditions. The author introduces the notion of a capacity solution which implies in particular that $$u$$ is not in a classical Sobolev space, but is a $$p$$-quasicontinuous function. The main result of the paper states the existence of a capacity solution of the above problem. The proof uses approximation by regularized problems and a priori estimates.
In the last section an initial value problem for the corresponding parabolic problem (with $$H=h=0$$) is studied and existence of a classical weak solution is proved.

### MSC:

 35R70 PDEs with multivalued right-hand sides 35J70 Degenerate elliptic equations 35K55 Nonlinear parabolic equations

### Keywords:

capacity solution; $$p$$-Laplacian; regularization
Full Text:

### References:

 [1] Atthey, D.R. 1974. ”A finite difference scheme for melting problems”. Vol. 13, 353–366. J. Inst. Maths. Applics. [2] Benilan, P., Crandall, M.G. and Sacks, P. 1988. ”Some L1existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions”. Vol. 17, 203–224. Appl. Math. Optim. · Zbl 0652.35043 [3] Boccardo L., J. Differential Equations · Zbl 0893.35131 [4] Brezis, H. and Strauss, W.A. 1973. ”Semilinear second-order elliptic equations in L1”. Vol. 25, 565–590. Japan: J.Math.Soc. [5] DiBenedetto, E. and Herrero, M.A. 1990. ”Non-negative solutions of the evolution p-Laplacian equation, initial traces and Cauchy problem when 1 < p > 2”. Vol. 111, 225–290. Arch. Rat. Mech. and Anal. · Zbl 0726.35066 [6] DiPerna, R.J. and Lions, P.L. 1989. ”Global existence for the Fokker-Planck-Boltzmann equations”. Vol. 11, 729–758. Comm.Pure Appl.Math. · Zbl 0698.35128 [7] DiPerna, R.J. and Majda, A. 1988. ”Reduced Hausdorff dimension and concentration-cancellation for two-dimensional incompressible flow”. Vol. 1, 59–95. J. American Math. Soc. · Zbl 0707.76026 [8] Evans, L.C. and Gariepy, R.F. 1992. ”Gariepy, Measure Theory and Fine Properties of Functions”. London: CRC Press. · Zbl 0804.28001 [9] Evans L.C., Measure Theory and Fine Properties of Functions (1992) · Zbl 0804.28001 [10] Evans, L.C. 1990. ”Weak Convergence Methods for Nonlinear Partial Differential Equations”. Island: AMS, Rhode. [11] Federer, H. and Ziemer, W.P. 1972. ”The Lebesgue set of a function whose distributional derivatives are pth-power summable”. Vol. 22, 139–158. Indiana Univ. Math. J. · Zbl 0238.28015 [12] Landes, R. 1989. ”Solvability of perturbed elliptic equations with critical growth exponent for the gradient”. Vol. 139, 63–77. J. Math. Anal. and Appl. · Zbl 0691.35038 [13] Morrey, C.B. 1966. ”Multiple Integrals in the Calculus of Variations”. New York: Springer-Verlag. · Zbl 0142.38701 [14] Oden, J.T. 1986. ”Qualitative Methods in Nonlinear Mechanics”. New Jersey: Prentice-Hall, Inc., Englewood Cliffs. · Zbl 0578.70001 [15] Shi P., J. Differential Equations 3 [16] Tayer, A.B. 1986. ”Mathematical Models in Applied Mechanics”. Oxford: Clarendon Press. [17] Temam, R. 1984. ”Navier-Stokes Equations”. North Holland · Zbl 0568.35002 [18] Xu, X. 1992. ”A degenerate Stefan-like problem with Joule’s heating”. Vol. 23, 1417–1438. SIAM J. Math. Anal. · Zbl 0768.35081 [19] Xu, X. 1993. ”A strongly degenerate system involving an equation of parabolic type and an equation of elliptic type”. Vol. 18, 199–213. Commun. in Partial Differential Equations. · Zbl 0814.35062 [20] Xu, X. 1990. ”Existence and convergence theorems for doubly nonlinear partial differential equations of elliptic-parabolic type”. Vol. 150, 205–223. J. Math. Anal. Appl. · Zbl 0772.35038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.