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Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measures. (English) Zbl 0843.35127

The author studies an elliptic Dirichlet problem

$-{\partial }_{{x}_{i}}\left({a}_{ij}{\partial }_{{x}_{j}}u\right)=f\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{1.em}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}u=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{1.em}{0ex}}\partial {\Omega }$

with ${a}_{ij}\in {L}^{\infty }\left({\Omega }\right)$ and $f\in {\left(C\left(\overline{{\Omega }}\right)\right)}^{\text{'}}$ the space of Radon measures and ${\Omega }$ an open bounded set of ${ℝ}^{N}$ with $N\ge 2$. For $f\in {H}^{1}\left({\Omega }\right)$ there was a unique variational solution in ${H}_{0}^{1}\left({\Omega }\right)$. For $f\notin {H}^{1}\left({\Omega }\right)$ one did not find solutions in ${H}_{0}^{1}\left({\Omega }\right)$ and some weaker formulations had been introduced. Solutions were given by Stampacchia or by Boccardo and Gallouët by an approximation of $f$. The author shows equivalences to other weaker formulations to see that the one of Boccardo and Gallouët is weaker than the one used by Stampacchia and to see that the solutions of Boccardo and Gallouët are solutions of Stampacchia. Because the Stampacchia-formulation ensures the uniqueness of $u$, the approximative solution of Boccardo and Gallouët is unique.

Uniqueness for the Boccardo-Gallouët-formulation is not ensured and for $N>2$ and ${\Omega }$ the unit ball of ${ℝ}^{N}$ the author gives an adaption of the counterexample of Serrin for getting a nontrivial solution $u$ for $f=0$ such that $u\in {W}^{1,q}\left({\Omega }\right)$ for all $q and $u\notin {H}^{1}\left({\Omega }\right)$ but with trace in ${H}^{1/2}\left(\partial {\Omega }\right)$ which gives two solutions in this space.

##### MSC:
 35R05 PDEs with discontinuous coefficients or data 35J25 Second order elliptic equations, boundary value problems 35A35 Theoretical approximation to solutions of PDE