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This paper is devoted to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L = --$\Delta$+V (x)\cdot \nabla with Dirichlet boundary conditions, where V is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue $\lambda$\_1($\Omega$, V) for a bounded quasi-open set $\Omega$ which enjoys similar properties to the case of open sets. Then, given m > 0 and $\tau$ $\ge$ 0, we show that the minimum of the following non-variational problem min $\lambda$\_1($\Omega$, V) : $\Omega$ $\subset$ D quasi-open, |$\Omega$| $\le$ m, |V|\_{\infty} $\le$ $\tau$. is achieved, where the box D $\subset$ R^d 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 $\Omega$ * solving the minimization problem min $\lambda$\_1($\Omega$, $\Phi$) : $\Omega$ $\subset$ D quasi-open, |$\Omega$| $\le$ m , where $\Phi$ is a given Lipschitz function on D. We prove that the topological boundary $\partial$$\Omega$ * is composed of a regular part which is locally the graph of a C ^{1,$\alpha$} function and a singular part which is empty if d < d * , discrete if d = d * and of locally finite H^{d--d *} Hausdorff measure if d > d * , where d * $\in$ {5, 6, 7} is the smallest dimension at which there exists a global solution to the one-phase free boundary problem with singularities. Moreover, if D is smooth, we prove that, for each x $\in$ $\partial$$\Omega$ * $\cap$ $\partial$D, $\partial$$\Omega$ * is C^{ 1,$\alpha$} in a neighborhood of x, for some $\alpha$ $\le$ 1 /2. This last result is optimal in the sense that C ^{1,1/2} is the best regularity that one can expect. |
