We study a Hamiltonian system of free boundary type.We first establish uniform bounds and prove the existence of solutions as well as of the free boundary.Next, for any smooth and bounded domain, we prove uniqueness of positive solutions within a suitable interval and show that the associated energies and boundary values have a monotonic behavior.Some consequences are discussed about the parametrization of the unbounded Rabinowitz continuum for a class of superlinear elliptic systems.
It follows from recent results of V. Bakhtin, R. Oleinik, and the second named author that, given a metric space 𝒳, a continuous map γ [a,b] →𝒳 is a map of bounded variation if and only if f ∘ γ is a function of bounded variation for every Lipschitz function f𝒳→ℝ. In this note, we show that the continuity assumption is of crucial importance: for many interesting examples of metric spaces there are no analogs of that characterization without the continuity assumption on γ. The interesting examples are: ℓ_2, infinite metric trees, and Laakso-type spaces. However, for ultrametric spaces the said characterization holds without any continuity assumptions.
In this paper we prove that the S^1-invariance of the Poincaré action functional associated to the Lorentz force equation gives the existence of multiple critical points which are periodic solutions with a fixed period. To do this, we prove an abstract multiplicity result which is based upon the Lusternik-Schnirelman method with the S^1-index. The corresponding result in the context of the Fadell-Rabinowitz index is proved in Ekeland and Lasry (Ann. Math., 112 (1980)). The main feature of our abstract result is that it allows us to consider nonsmooth functionals satisfying only a weak compactness condition well adapted to the Poincaré functional.
In this paper, we study that the nearly critical nonlocal problem { -Δu=(|x|^-(n-2)∗ u^p-ε)u^p-1-ε Ω, u>0 Ω, u=0 , . where Ω is a smooth bounded domain in ℝ^n for n=3,4,5, ∗ denotes the standard convolution, ε>0 is a small parameter and p=n+2/n-2 is energy-critical exponent. We study the asymptotic behavior of least energy solutions as ε→0. These solutions are shown to blow-up at exactly one point x_0 and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied. Finally, in order to further locate the blowing-up point x_0, we prove that x_0 is a global maximum point of the Robin's function of Ω.
We prove a quantitative version of a sharp integral inequality by Hang, Wang, and Yan for both the Poisson operator and its adjoint. Our result has the strongest possible norm and the optimal stability exponent. This stability exponent is not necessarily equal to 2, displaying the same phenomenon that Figalli and Zhang observed for the p-Sobolev inequality.
In this paper we study qualitative properties of initial traces of solutions to the porous medium equation with power nonlinearity, and obtain necessary conditions for the existence of solutions to the corresponding Cauchy problem. Furthermore, we establish sharp sufficient conditions for the existence of solutions to the Cauchy problem using uniformly local Morrey spaces and their variations, and identify the optimal singularities of the initial data for the solvability of the Cauchy problem.
We construct an explicit family of stable proper weak biharmonic maps from the unit ball B^m, m≥ 5, to Euclidean spheres. To the best of the authors knowledge this is the first example of a stable proper weak biharmonic map from at compact domain. To achieve our result we first establish the second variation formula of the bienergy for maps from the unit ball into a Euclidean sphere. Employing this result, we examine the stability of the proper weak biharmonic maps q:B^m→𝕊^m^ℓ, m,ℓ∈ℕ with ℓ≤ m, which we recently constructed in and thus deduce the existence of an explicit family of stable proper biharmonic maps to Euclidean spheres.
We study invariant pseudo-Kähler structures on a solvmanifold G such that the Lie algebra 𝔤 is almost abelian, that is 𝔤=𝔥⋊ℝ, with 𝔥 abelian; comparing with the positive-definite case, an additional situation occurs, corresponding to the ideal 𝔥 being degenerate. We obtain a classification up to unitary isomorphism in all dimensions. We deduce that every nilpotent almost abelian Lie algebra endowed with a complex structure also admits a compatible pseudo-Kähler structure, and prove that this is no longer true for general almost abelian Lie algebras; indeed, we classify all the almost abelian Lie algebras that admit a complex structure and a symplectic structure but no compatible pseudo-Kähler metric. We study the curvature of the metrics we have obtained, and use some of them to construct Einstein pseudo-Kähler metrics in two dimensions higher.
We prove that the restricted normal holonomy group of a Kähler submanifold of the complex hyperbolic space ℂH^n is always transitive, provided the index of relative nullity is zero. This contrasts with the case of ℂP^n, where a Berger type result was proved by Console, Di Scala, and the second author. The proof is based on lifting the submanifold to the pseudo-Riemannian space ℂ^n,1 and developing new tools to handle the difficulties arising from possible degeneracies in holonomy tubes and associated distributions. In particular, we introduce the notion of weakly polar actions and a framework for dealing with degenerate submanifolds. These techniques could contribute to a broader understanding of submanifold geometry in spaces with indefinite signature, offering new insight into submanifolds in the dual setting of complex projective geometry.
Using a Quot scheme compactification, we calculate the virtual count of maps of degree d from a smooth projective curve of genus g to a hypersurface in a Grassmannian, sending specified points of the curve to special Schubert subvarieties restricted to the hypersurface. We study the question of whether this virtual count is in fact enumerative under suitable conditions on the hypersurface, in the regime when the map degree d is large.
The Hurwitz space ℋ_k,g is a compactification of the space of smooth genus-g curves with a simply-branched degree-k map to ℙ^1. In this paper, we initiate a study of the Chow rings of these spaces, proving in particular that when k=3 (which is the first case in which the Chow ring is not already known), the codimension-2 Chow group is generated by the fundamental classes of codimension-2 boundary strata. The key tool is to realize the codimension-1 boundary strata of ℋ_3,g as the images of gluing maps whose domains are products of Hurwitz spaces ℋ_k',g'(μ) with a single marked fiber of prescribed (not necessarily simple) ramification profile μ, and to prove that the spaces ℋ_k',g'(μ) with k'=2,3 have trivial Chow ring.
The Andreotti-Mayer locus is a subset of the moduli space of principally polarized abelian varieties, defined by a condition on the dimension of the singular locus of the theta divisor. It is known that the Jacobian locus in the moduli space is an irreducible component of the Andreotti-Mayer locus. In this paper, we generalize the Andreotti-Mayer locus to the case of the moduli space of abelian varieties with non-principal polarization and prove that the Prym locus of branched double coverings is an irreducible component of the generalized Andreotti-Mayer locus.
Let (ϕ_t) be a continuous semigroup of holomorphic self-maps of the unit disk 𝔻 with Denjoy-Wolff point τ∈𝔻. We study the rate of convergence of the forward orbits of (ϕ_t) to the Denjoy-Wolff point by finding explicit bounds for the quantity |ϕ_t(z)-τ|, z∈𝔻, t > 0. We further discuss the corresponding rate of convergence for the backward orbits of (ϕ_t).
We develop a higher-dimensional extension of multifractal analysis for typical fiber-bunched linear cocycles. Our main result is a relative variational principle, which shows that the topological entropy of Lyapunov exponent level sets can be approximated by the metric entropy of ergodic measures fully concentrated on those level sets, addressing a question posed by Breuillard and Sert. We also establish a variational principle for the generalized singular value function. As an application to dynamically defined linear cocycles, we obtain a multifractal formalism for open sets of C^1+α repellers and Anosov diffeomorphisms.
