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Renormalization solutions of elliptic equations with general measure data. (English) Zbl 0958.35045
Authors’ abstract: We study the nonlinear monotone elliptic problem \[ \begin{cases} -\text{div}\bigl(a(x,\nabla u)\bigr)=\mu\quad & \text{ on }\Omega,\\ u=0\quad & \text{ on }\partial\Omega,\end{cases} \] when \(\Omega\subset \mathbb{R}^N\), \(\mu\) is a Radon measure with bounded total variation on \(\Omega\), \(1<p\leq N\), and \(u\mapsto -\text{div}(a(x,\nabla u))\) is a monotone operator acting on \(W_0^{1,p}(\Omega)\). We introduce a new definition of the solution (the renormalized solution) in four equivalent ways. We prove the existence of a renormalized solution by an approximation procedure, where the key point is a stability result (the strong convergence in \(W_0^{1,p}(\Omega)\) of the truncates). We also prove partial uniqueness results.
Reviewer: A.Doktor (Praha)

MSC:
35J60 Nonlinear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
35A15 Variational methods applied to PDEs
35J25 Boundary value problems for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
31C45 Other generalizations (nonlinear potential theory, etc.)
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