By Bouchitte G., Buttazzo G., De Pascale L.

We exhibit that the matter of discovering the simplest mass distribution, either in conductivity and elasticity situations, may be approximated by way of suggestions of a p-Laplace equation, as p→+S. This turns out to supply a range criterion whilst the optimum options are nonunique.

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**Example text**

4) for any X, Y, Z ∈ Γ (T M ). 4) is a linear connection on D that satisﬁes the conditions (i) and (ii). Next, suppose that ∇ is another linear connection on D satisfying (i) and (ii). 6) and for any X, Y ∈ Γ (T M ). 7) − 2g(∇QX QY, QZ). 3) respectively, we conclude that ∇ = D, which proves the uniqueness of D. 14)) ∇X Y − ∇Y X − [X, Y ] = 0, ∀ X, Y ∈ Γ (T M ). 8) If (M, g) is a semi–Riemannian manifold then we say that g is parallel with respect to ∇ if we have (∇X g)(Y, Z) = X(g(Y, Z)) − g(∇X Y, Z) − g(Y, ∇X Z) = 0, ∀ X, Y, Z ∈ Γ (T M ).

3) for any X, Y, Z, U ∈ Γ (T M ), and call it the Vr˘ anceanu curvature tensor ﬁeld of (D, g). Some of the most important properties of R∗ are stated in the next lemma. 1. Let (M, g, D) be a non–holonomic manifold such that g is Vr˘ anceanu–parallel on D. 6) (QZ,QX,QY ) for any X, Y, Z, U ∈ Γ (T M ). Proof. 4) is a well known property of the curvature tensor ﬁeld of any linear connection on M . 14) we deduce that 42 1 GEOMETRY OF DISTRIBUTIONS ON A MANIFOLD R◦ (QX, QY )QZ = R∗ (QX, QY )QZ + h (QZ, Q T ∗ (QX, QY )) = R∗ (QX, QY )QZ + h (QZ, h(QY, QX) − h(QX, QY )).

Let (M, g, D) be a semi–Riemannian non–holonomic manifold such that g is Vr˘ anceanu–parallel on D. 19) where {QX, QY } is an arbitrary basis of the non–degenerate D–plane W . Proof. 17). 21) = −2g(h(QX, QY ), h(QX, QY )). 18). 17). 1. 19) for the particular case of Riemannian submersions. 38c) in Tondeur [Ton97]. Now, suppose that (M, g, D) is a Riemannian non–holonomic manifold. This means that (D, g) is a Riemannian distribution, but (M, g) might be proper semi–Riemannian manifold. 7. Let (M, g, D) be a Riemannian non–holonomic manifold such that g is Vr˘ anceanu–parallel on D and h(QX, QY ) is a space–like or light–like vector ﬁeld for any two linearly independent vector ﬁelds {QX, QY }.

### A p-Laplacian Approximation for Some Mass Optimization Problems by Bouchitte G., Buttazzo G., De Pascale L.

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