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The Isoperimetric Problem for the Curl Operator
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In the last decades, many mathematicians have studied the curl operator in compact three-manifolds , mainly the structure of its spectrum and some isoperimetric problems associated with it. In this paper, we will see that all the compact three-manifolds (both closed and and with non-empty boundary) have always optimal lower bounds for the absolute value of their non-null eigenvalues of curl. We will also show that these bounds are always attained and compute the optimal domains and the multiplicities of their associated eigenvalues. So, we have solved the isoperimetric problem associated to the curl operator, and, by the way, we have solved and old Cantarella, de Turck, Gluck and Teytel conjecture in the negative.
Forward citations
Cited by 2 Pith papers
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The sharp curl-Sobolev inequality
For n ≡ 3 (mod 4), the sharp constant of the conformal curl–Sobolev quotient on S^n is (n+1)/2 ω_n^{1/n}, attained exactly by conformal images of positive Killing forms.
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On the sharp constants in curl-Sobolev inequalities on $\mathbb{S}^n$
Killing forms are strict local minimizers of the J1 curl-Sobolev quotient on S^n, while the same forms are unstable for J2, disproving the Frank-Loss conjecture.
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