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Some \(L^ 1\) existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions. (English) Zbl 0652.35043

The authors study semilinear elliptic boundary value problems of the general type \[ f\in -\Delta u+\beta (u)\quad on\quad \Omega;\quad 0\in \partial u/\partial \nu +\gamma (u)\quad on\quad \partial \Omega \] where \(\beta\) and \(\gamma\) are maximal monotone graphs with \(D(\beta)\cap \gamma^{-1}(0)\neq \emptyset\) and \(f\in L_ 1(\Omega)\). A triple \((u,v,w)\in W_{1,1}(\Omega)\times L_ 1(\Omega)\times L_ 1(\partial \Omega)\) is called a solution, if v(x)\(\in \beta (u(x))\) a.e. on \(\Omega\), w(x)\(\in \gamma (u(x))\) a.e. on \(\partial \Omega\) and \[ \int_{\Omega}v\phi +\int_{\Omega}\nabla u\nabla \phi +\int_{\partial \Omega}v\phi =\int_{\Omega}f\phi \quad for\quad all\quad \gamma \in W_{1,\infty}(\Omega). \] They show that existence of a solution implies \(\int_{\Omega}f\in range(B)\) with \(B=| \partial \Omega | \gamma +| \Omega | \beta,\) while conversely \(\int_{\Omega}\in int(range(B))\) implies existence - the borderline cases \(\int_{\Omega}f=\sup range(B)\) (resp. inf) are characterized, too. The basis is an approximation theorem for such solutions which has its own interest. From the vast literature concerning these problems let us just recall H. Brezis [J. Math. Pures Appl., IX. Ser. 51, 1-168 (1972; Zbl 0237.35001)].
Reviewer: M.Wiegner

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
47H05 Monotone operators and generalizations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)

Citations:

Zbl 0237.35001
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