For a bounded domain Ω ⊂ R 3 with Lipschitz boundary Γ and some relatively open Lipschitz subset Γ t = ∅ of Γ, we prove the existence of some c > 0, such thatholds for all tensor fields in H(Curl; Ω), i.e., for all square-integrable tensor fields T : Ω → R 3×3 with square-integrable generalized rotation Curl T : Ω → R 3×3 , having vanishing restricted tangential trace on Γ t . If Γ t = ∅, (0.1) still holds at least for simply connected Ω and for all tensor fields T ∈ H(Curl; Ω) which are L 2 (Ω)-perpendicular to so(3), i.e., to all skew-symmetric constant tensors. Here, both operations, Curl and tangential trace, are to be understood row-wise. For compatible tensor fields T = ∇v, (0.1) reduces to a non-standard variant of the well known Korn's first inequality in R 3 , namelyfor all vector fields v ∈ H 1 (Ω, R 3 ), for which ∇v n , n = 1, . . . , 3, are normal at Γ t . On the other hand, identifying vector fields v ∈ H 1 (Ω, R 3 ) (having the proper boundary conditions) with skew-symmetric tensor fields T , (0.1) turns to Poincaré's inequalityTherefore, (0.1) may be viewed as a natural common generalization of Korn's first and Poincaré's inequality. From another point of view, (0.1) states that one can omit compatibility of the tensor field T at the expense of measuring the deviation from compatibility through Curl T . Decisive tools for this unexpected estimate are the classical Korn's first inequality, Helmholtz decompositions for mixed boundary conditions and the Maxwell estimate.

Let Ω ⊂ R 3 be a bounded weak Lipschitz domain with boundary Γ := ∂ Ω divided into two weak Lipschitz submanifolds Γτ and Γν and let ε denote an L ∞ -matrix field inducing an inner product in L 2 (Ω). The main result of this contribution is the so called 'Maxwell compactness property', that is, the Hilbert spaceis compactly embedded into L 2 (Ω). We will also prove some canonical applications, such as Maxwell estimates, Helmholtz decompositions and a static solution theory. Furthermore, a Fredholm alternative for the underlying time-harmonic Maxwell problem and all corresponding and related results for exterior domains formulated in weighted Sobolev spaces are straightforward. 7 3.

Communicated by Hans-Dieter AlberFor a bounded domain R N with connected Lipschitz boundary, we prove the existence of some c > 0, such thatholds for all square-integrable tensor fields P : ! R N N , having square-integrable generalized "rotation" Curl P : ! R N .N 1/N=2 and vanishing tangential trace on @ , where both operations are to be understood row-wise. Here, in each row, the operator curl is the vector analytical reincarnation of the exterior derivative d in R N . For compatible tensor fields P, that is, P D rv, the latter estimate reduces to a non-standard variant of Korn's first inequality infor all vector fields v 2 H 1 . , R N /, for which rv n , n D 1, : : : , N, are normal at @ .

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Let Ω ⊂ R n , n ≥ 2, be a bounded Lipschitz domain and 1 < q < ∞. We prove the inequalitybeing valid for tensor fields T : Ω → R n×n with a normal boundary condition on some open and non-empty part Γν of the boundary ∂Ω. Here dev T = T − 1 n tr (T ) · Id denotes the deviatoric part of the tensor T and Div is the divergence row-wise. Furthermore, we provebeing valid for tensor fields T with a tangential boundary condition on some open and non-empty part Γτ of ∂Ω. Here, sym T = 1 2 (T + T ⊤ ) denotes the symmetric part of T and Curl is the rotation row-wise.