This paper is dedicated to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L=-Δ+V(x)·∇\documentclass[12pt]{minimal}
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\begin{document}$$L = -\Delta + V(x) \cdot \nabla $$\end{document} with Dirichlet boundary conditions, where V is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue λ1(Ω,V)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _1(\Omega ,V)$$\end{document} for a bounded quasi-open set Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} which enjoys similar properties to the case of open sets. Then, given m>0\documentclass[12pt]{minimal}
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\begin{document}$$m>0$$\end{document} and τ≥0\documentclass[12pt]{minimal}
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\begin{document}$$\tau \ge 0$$\end{document}, we show that the minimum of the following non-variational problem min{λ1(Ω,V):Ω⊂Dquasi-open,|Ω|≤m,‖V‖L∞≤τ}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \min \Big \{\lambda _1(\Omega ,V)\ :\ \Omega \subset D\ \text {quasi-open},\ |\Omega |\le m,\ \Vert V\Vert _{L^\infty }\le \tau \Big \}. \end{aligned}$$\end{document}is achieved, where the box D⊂Rd\documentclass[12pt]{minimal}
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\begin{document}$$D\subset {\mathbb {R}}^d$$\end{document} is a bounded open set. The existence when V is fixed, as well as when V varies among all the vector fields which are the gradient of a Lipschitz function, are also proved. The second interest and main result of this paper is the regularity of the optimal shape Ω∗\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^*$$\end{document} solving the minimization problem min{λ1(Ω,∇Φ):Ω⊂Dquasi-open,|Ω|≤m},\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \min \Big \{\lambda _1(\Omega ,\nabla \Phi )\ :\ \Omega \subset D\ \text {quasi-open},\ |\Omega |\le m\Big \}, \end{aligned}$$\end{document}where Φ\documentclass[12pt]{minimal}
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\begin{document}$$\Phi $$\end{document} is a given Lipschitz function on D. We prove that the optimal set Ω∗\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^*$$\end{document} is open and that its topological boundary ∂Ω∗\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Omega ^*$$\end{document} is composed of a regular part, which is locally the graph of a C1,α\documentclass[12pt]{minimal}
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\begin{document}$$C^{1,\alpha }$$\end{document} function, and a singular part, which is empty if d<d∗\documentclass[12pt]{minimal}
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\begin{document}$$d<d^*$$\end{document}, discrete if d=d∗\documentclass[12pt]{minimal}
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\begin{document}$$d=d^*$$\end{document} and of locally finite Hd-d∗\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}^{d-d^*}$$\end{document} Hausdorff measure if d>d∗\documentclass[12pt]{minimal}
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\begin{document}$$d>d^*$$\end{document}, where d∗∈{5,6,7}\documentclass[12pt]{minimal}
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\begin{document}$$d^*\in \{5,6,7\}$$\end{document} is the smallest dimension at which there exists a global solution to the one-phase free boundary problem with singularities. Moreover, if D is smooth, we prove that, for each x∈∂Ω∗∩∂D\documentclass[12pt]{minimal}
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\begin{document}$$x\in \partial \Omega ^{*}\cap \partial D$$\end{document}, ∂Ω∗\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Omega ^*$$\end{document} is C1,1/2\documentclass[12pt]{minimal}
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\begin{document}$$C^{1,1/2}$$\end{document} in a neighborhood of x.
