# zbMATH — the first resource for mathematics

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)

##### MSC:
 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