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Existence of renormalized solutions to quasilinear elliptic problems with general measure data. (English) Zbl 1413.35428

Summary: We consider a Dirichlet problem in divergence form with variable growth. We obtain existence of a renormalized solution for general measure data whose model is \[ \begin{aligned} - \operatorname{div} a(x,\nabla u)= & \mu\quad\text{in } \Omega \\ u= & 0 \quad\text{on }\partial \Omega, \end{aligned} \] for any, possibly general, Radon measure \(\mu \) with bounded total variation on \(\Omega \). The proofs rely crucially on a priori estimates.

MSC:

35R06 PDEs with measure
35A35 Theoretical approximation in context of PDEs
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
35Q35 PDEs in connection with fluid mechanics
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