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.


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