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Quasilinear elliptic systems in divergence form with weak monotonicity. (English) Zbl 0920.35060

Summary: We consider the Dirichlet problem for the quasilinear elliptic system \[ -\text{div }\sigma(x, u(x), Du(x))= f\quad\text{in }\Omega,\quad u(x)= 0\quad\text{on }\partial\Omega \] for a function \(u: \Omega\to \mathbb{R}^m\), where \(\Omega\) is a bounded open domain in \(\mathbb{R}^n\). For arbitrary right-hand side \(f\in W^{-1,p'}(\Omega)\) we prove existence of a weak solution under classical regularity, growth and coercivity conditions, but with only very mild monotonicity assumptions.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
47J05 Equations involving nonlinear operators (general)
35J55 Systems of elliptic equations, boundary value problems (MSC2000)

Keywords:

Young measures
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