Abstract. We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions.

Abstract. A highly nonlinear eigenvalue problem is studied in a Sobolev space with variable exponent. The Euler-Lagrange equation for the minimization of a Rayleigh quotient of two Luxemburg norms is derived. The asymptotic case with a "variable infinity" is treated. Local uniqueness is proved for the viscosity solutions.

Abstract. We study the Stekloff eigenvalue problem for the so-called pseudo p-Laplacian operator. After proving the existence of an unbounded sequence of eigenvalues, we focus on the first nontrivial eigenvalue σ2,p, providing various equivalent characterizations for it. We also prove an upper bound for σ2,p in terms of geometric quantities. The latter can be seen as the nonlinear analogue of the Brock-Weinstock inequality for the first nontrivial Stekloff eigenvalue of the (standard) Laplacian. Such an estimate is obtained by exploiting a family of sharp weighted Wulff inequalities, which are here derived and appear to be interesting in themselves.Mathematics Subject Classification (2010). 35P30, 47A75, 34B15.

We give a simple convexity-based proof of the following fact: the only eigenfunction of the p−Laplacian that does not change sign is the first one. The method of proof covers also more general nonlinear eigenvalue problems.

In this paper, we consider the isoperimetric problem in the space R N with density. Our result states that, if the density f is l.s.c. and converges to a limit a > 0 at infinity, being f ≤ a far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities posively answers a conjecture made in [10].The first interesting question in this setting is of course the existence of isoperimetric sets, that are sets E with the property that P f (E) = J(|E| f ) where, for any V ≥ 0,Depending on the assumptions on f , the answer to this question may be trivial or extremely complicate.Let us start with a very simple, yet fundamental, observation. Fix a volume V > 0 and let {E i } be an isoperimetric sequence of volume V : this means that |E i | f = V for every i ∈ N, and P f (E i ) → J(V ). Thus, possibly up to a subsequence, the sets E i converge to some set E in the L 1 loc sense. As a consequence, standard lower semi-continuity results in BV ensure that P f (E) ≤ lim inf P f (E i ) = J(V ) (at least, for instance, if f > 0. . . ); therefore, if actually |E| f = V , then obviously E is an isoperimetric set. Unfortunately, this simple observation is

Abstract.Given an open set Ω, we consider the problem of providing sharp lower bounds for λ2(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 λ2 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. IntroductionIn this paper, we are concerned with Dirichlet eigenvalues of the p−Laplace operatorwhere 1 < p < ∞. For every open set Ω ⊂ R N having finite measure, these are defined as the real numbers λ such that the boundary value problemhas non trivial (weak) solutions. In particular, we are mainly focused on the following spectral optimization problemwhere c > 0 is a given number, λ 2 (·) is the second Dirichlet eigenvalue of the p−Laplacian and | · | stands for the N −dimensional Lebesgue measure. We will go back on the question of the well-posedness of this problem in a while, for the moment let us focus on the particular case p = 2: in this case we are facing the eigenvalue problem for the usual Laplace operator and, as it is well known, Dirichlet eigenvalues form a discrete nondecreasing sequence of positive real numbers 0 < λ 1 (Ω) ≤ λ 2 (Ω) ≤ λ 3 (Ω) ≤ . . . , going to ∞. In particular, it is meaningful to speak of a second eigenvalue so that problem (1.1) is well-posed and we know that its solution is given by any disjoint union of two balls having measure c/2. Moreover, these are the only sets which minimize λ 2 under volume constraint. Using the scaling properties both of the eigenvalues of −∆ and of the Lebesgue measure, we can reformulate the previous result in scaling invariant form as follows

We consider the Schrödinger operator −∆ + V for negative potentials V , on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We show that the spectrum of −∆ + V is positive, provided that V is greater than a negative multiple of the logarithmic gradient of the solution to the Lane-Emden equation −∆u = u q−1 (for some 1 ≤ q < 2). In this case, the ground state energy of −∆ + V is greater than the first eigenvalue of the Dirichlet-Laplacian, up to an explicit multiplicative factor. This is achieved by means of suitable Hardy-type inequalities, that we prove in this paper. D(H2010 Mathematics Subject Classification. 35P15, 47A75, 49S05.

We construct an open set Ω ⊂ R N on which an eigenvalue problem for the p−Laplacian has not isolated first eigenvalue and the spectrum is not discrete. The same example shows that the usual Lusternik-Schnirelmann minimax construction does not exhaust the whole spectrum of this eigenvalue problem.2010 Mathematics Subject Classification. 35P30, 47A75, 58E05.