An alternative approach to the Faber–Krahn inequality for Robin problems

We give a simple proof of the Faber–Krahn inequality for the first eigenvalue of the p-Laplace operator with Robin boundary conditions. The techniques introduced allow to work with much less regular domains by using test function arguments. We substantially simplify earlier proofs, and establish the sharpness of the inequality for a larger class of domains at the same time.     

A quantitative form of Faber–Krahn inequality

The Faber–Krahn inequality states that balls are the unique minimizers of the first eigenvalue of the p-Laplacian among all sets with fixed volume. In this paper we prove a sharp quantitative form of this inequality. This extends to the case \(p>1\) a recent result proved by Brasco et al. (Duke Math J 164:1777–1831, 2015) for the Laplacian.     

On the Hong–Krahn–Szego inequality for the p-Laplace operator

Given an open set Ω, we consider the problem of providing sharp lower bounds for λ ₂(Ω), i.e. its second Dirichlet eigenvalue of the p-Laplace operator. After presenting the nonlinear analogue of the Hong–Krahn–Szego inequality, asserting that the disjoint unions of two equal balls minimize λ ₂ among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ∞ are considered as well.     

Spectral optimization for the Stekloff–Laplacian: The stability issue

We consider the problem of minimizing the first non-trivial Stekloff eigenvalue of the Laplacian, among sets with given measure. We prove that the Brock–Weinstock inequality, asserting that optimal shapes for this spectral optimization problem are balls, can be improved by means of a (sharp) quantitative stability estimate. This result is based on the analysis of a certain class of weighted isoperimetric inequalities already proved in Betta et al. (1999) [2]: we provide some new (sharp) quantitative versions of these, achieved by means of a suitable calibration technique.     

The dual Orlicz Brunn–Minkowski inequality for the intersection body

Acta Mathematica Hungarica - Let $$q geq 2$$ be an integer and let $$s_{q}(n)$$ be the sum-of-digits function in base q of the positive integer n. In this paper we obtain asymptotic formulae for...     

The Brunn–Minkowski–Firey inequality for nonconvex sets

The definition of Minkowski–Firey Lp$L_p$-combinations is extended from convex bodies to arbitrary subsets of Euclidean space. The Brunn–Minkowski–Firey inequality (along with its equality conditions), previously established only for convex bodies, is shown to hold for compact sets.     

A Faber–Krahn Inequality for Solutions of Schrödinger’s Equation on Riemannian Manifolds

We consider a bounded open set with smooth boundary \(\Omega \subset M\) in a Riemannian manifold (M, g), and suppose that there exists a non-trivial function \(u\in C({\overline{\Omega }})\) solving the problem

$$\begin{aligned} -\Delta u=V(x)u, \,\, \text{ in }\,\,\Omega , \end{aligned}$$

in the distributional sense, with \(V\in L^\infty (\Omega )\), where \(u\equiv 0\) on \(\partial \Omega .\) We prove a sharp inequality involving \(||V||_{L^{\infty }(\Omega )}\) and the first eigenvalue of the Laplacian on geodesic balls in simply connected spaces with constant curvature, which slightly generalises the well-known Faber–Krahn isoperimetric inequality. Moreover, in a Riemannian manifold which is not necessarily simply connected, we obtain a lower bound for \(||V||_{L^{\infty }(\Omega )}\) in terms of its isoperimetric or Cheeger constant. As an application, we show that if \(\Omega \) is a domain on a m-dimensional minimal submanifold of \({\mathbb {R}}^n\) which lies in a ball of radius R, then

$$\begin{aligned} ||V||_{L^{\infty }(\Omega )}\ge \left( \frac{m}{2R}\right) ^{2}. \end{aligned}$$

     

On the Aleksandrov–Bakelman–Pucci estimate for the infinity Laplacian

We prove L ^∞ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and p-Laplacian, namely $$\begin{array}{ll} -\Delta_p^N u=f\quad{\rm for } \; n < p \leq\infty.\end{array}$$ We are able to provide a stable family of results depending continuously on the parameter p. We also prove the failure of the classical Alexandrov–Bakelman–Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.     

The Lewy–Stampacchia inequality for the obstacle problem with quadratic growth in the gradient

In this paper we prove that at least one solution of the obstacle problem for a linear elliptic operator perturbed by a nonlinearity having quadratic growth in the gradient satisfies the Lewy–Stampacchia inequality.

---

Résumé Nous démontrons dans cet article qu’au moins une solution du problème de l’obstacle pour un opérateur elliptique linéaire perturbé par une non linéarité à croissance quadratique par rapport au gradient vérifie l’inégalité de Lewy–Stampacchia.

---

---

Mathematics Subject Classification (2000) 35J85, 47J20     

The Fejér-Riesz inequality for the Besov spaces

A Fejér-Riesz inequality for the weighted Besov spaces B _p,q is given. Some characterizations of functions in B _p,q in terms of their Taylor coefficients are obtained.     

The hyperbolic meaning of the Milnor–Wood inequality

We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce various properties of the twist function, and use it to give new proofs of several well-known results, including the Milnor–Wood inequality, using purely hyperbolic-geometric methods. Our methods express inequalities in Milnor’s function as equalities, with the deficiency from equality given by an area in the hyperbolic plane. We find that the twist of certain products found in surface group presentations is equal to the area of certain hyperbolic polygons arising as their fundamental domains.     

Robin–Robin domain decomposition methods for the steady-state Stokes–Darcy system with the Beavers–Joseph interface condition

Domain decomposition methods for solving the coupled Stokes–Darcy system with the Beavers–Joseph interface condition are proposed and analyzed. Robin boundary conditions are used to decouple the Stokes and Darcy parts of the system. Then, parallel and serial domain decomposition methods are constructed based on the two decoupled sub-problems. Convergence of the two methods is demonstrated and the results of computational experiments are presented to illustrate the convergence.     

The Moore–Penrose inverse of the normalized graph Laplacian

We prove a formula that relates the Moore–Penrose inverses of two matrices A,B$A,B$ such that A=N⁻¹BM⁻¹$A=N^{-1}B{M}^{-1}$ and discuss some applications, in particular to the representation of the Moore–Penrose inverse of the normalized Laplacian of a graph. The Laplacian matrix of an undirected graph is symmetric and is strictly related to its connectivity properties. However, our formula applies to asymmetric matrices, so that we can generalize our results for asymmetric Laplacians, whose importance for the study of directed graphs is increasing.