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On nonhomogeneous quasilinear elliptic equations. (English) Zbl 0911.35048
Annales Academiæ Scientiarum Fennicæ. Mathematica. Dissertationes. 117. Helsinki: Suomalainen Tiedeakatemia. 46 p. (1998).
The aim of the paper is to investigate the Dirichlet problem \[ Tu=-\text{div} A(x,\nabla u)=f\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega, \] where \(\Omega\subset\mathbb{R}^n\) is a bounded domain, \(n\geq 2\), and \(Tu\) is similar in some sense to the \(p\)-Lacplacian \(-\text{div}(|\nabla u|^{p-2}\nabla u)\), \(1<p<\infty\) (we omit here the list of technical assumptions on \(A)\). First, the author proves the uniqueness of the solution in appropriate Sobolev spaces, and establishes the existence of solutions in these spaces for \(1<p\leq n\). Then the author studies the variations of the domain \(\Omega\) and obtains the continuity of the map \(\Omega\to u\) for some families of open sets \(\Omega\) which satisfy a capacity density condition. These results confirm a conjecture made recently by V. Šverák [J. Math. Pures Appl., IX. Ser. 72, 537-551 (1993; Zbl 0849.49021)] in relation to optimal shape design. Finally, the author considers a special class of the above equations where \(f\in\text{weak}-L^{n/p}(\Omega)\), and shows that in this case the solution \(u\) is locally of bounded mean oscillation.
Reviewer: O.Titow (Berlin)

35J60 Nonlinear elliptic equations
35J70 Degenerate elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs