Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds

Let M be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets D in M for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in $$L^2(D)$$ L 2 ( D ) , and the supremum of the torsion function for D are comparable with the square of the capacitary width for D if the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale.

Optimal Control of a Nonlinear PDE Governed by Fractional Laplacian

We consider an optimal control problem containing a control system described by a partial nonlinear differential equation with the fractional Dirichlet–Laplacian, associated to an integral cost. We investigate the existence of optimal solutions for such a problem. In our study we use Filippov’s approach combined with a lower closure theorem for orientor fields.

A spectral shape optimization problem with a nonlocal competing term

We study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist.

On Efficiency and Localisation for the Torsion Function

We consider the torsion function for the Dirichlet Laplacian −Δ, and for the Schrödinger operator −Δ + V on an open set ${\Omega }\subset \mathbb {R}^{m}$ Ω ⊂ ℝ m of finite Lebesgue measure $0<|{\Omega }|<\infty $ 0 < | Ω | < ∞ with a real-valued, non-negative, measurable potential V. We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.

A Note on the Three Dimensional Dirac Operator with Zigzag Type Boundary Conditions

In this note the three dimensional Dirac operator $$A_m$$ A m with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that $$A_m$$ A m is self-adjoint in $$L^2(\Omega ;{\mathbb {C}}^4)$$ L 2 ( Ω ; C 4 ) for any open set $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 and its spectrum is described explicitly in terms of the spectrum of the Dirichlet Laplacian in $$\Omega $$ Ω . In particular, whenever the spectrum of the Dirichlet Laplacian is purely discrete, then also the spectrum of $$A_m$$ A m consists of discrete eigenvalues that accumulate at $$\pm \infty $$ ± ∞ and one additional eigenvalue of infinite multiplicity.

Regularity of the free boundary for the two-phase Bernoulli problem

We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.