Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

On a variational problem involving critical Sobolev growth in dimension three

In this paper we consider the following nonlinear problem: \({{-\Delta u=Ku^{5}}}\), u > 0 in \({{\Omega}}\), u =  0 on \({{\partial \Omega}}\), where K > 0 in \({{\Omega}}\), K =  0 on \({{\partial \Omega}}\) and \({{\Omega}}\) is a bounded domain of \({{\mathbb{R}^{3}}}\). We prove a version of a Morse lemma at infinity for this problem, which allows us to describe the critical points at infinity of the associated variational functional. Using a topological argument, we prove an existence result.     

Extinction for a fast diffusion equation with a nonlinear nonlocal source

In this article, the authors establish conditions for the extinction of solutions, in finite time, of the fast diffusion equation \({u_t=\Delta u^m+a\int_\Omega u^p(y,t)\,{d}y,\ 0 < m < 1,}\) in a bounded domain \({\Omega\subset R^N}\) with N > 2. More precisely speaking, it is shown that if p > m, any solution with small initial data vanishes in finite time, and if p < m, the maximal solution is positive in Ω for all t > 0. For the critical case p = m, whether the solutions vanish in finite time or not depends on the value of aμ, where \({\mu=\int_{\Omega}\varphi(x)\,{d}x}\) and \({\varphi}\) is the unique positive solution of the elliptic problem \({-\Delta\varphi(x)=1,\ x\in \Omega; \varphi(x)=0,\ x\in\partial\Omega}\) .     

Sign-changing solutions for supercritical elliptic problems in domains with small holes

Let \(\mathcal {D}\) be a bounded, smooth domain in \(\mathbb {R}^N\) , N ≥ 3, \(P\in \mathcal {D}\) . We consider the boundary value problem in \(\Omega = \mathcal {D} \setminus B_\delta(P)\) ,

$\begin{aligned}\Delta u + |u|^{p-1} u = 0\, \quad in\, \Omega,\\u = 0\quad on\, \partial\Omega,\end{aligned}$

with p supercritical, namely \(p > \frac{N+2}{N-2}\) . Given any positive integer m, we find a sequence \(p_1 < p_2 < p_3 < \cdots , \quad with \lim_{k\to+\infty} p_k = +\infty \), such that if p is given, with p ≠ p _j for all j, then for all δ > 0 sufficiently small, this problem has a sign-changing solution which has exactly m + 1 nodal domains.

     

Multiplicity Results for the Fractional Laplacian in Expanding Domains

In this paper, we establish a multiplicity result of nontrivial weak solutions for the problem \((-\Delta )^{\alpha } u +u= h(u)\)    in \(\Omega _{\lambda }\), \(u=0\)    on \(\partial \Omega _{\lambda }\), where \(\Omega _{\lambda }=\lambda \Omega \), \(\Omega \) is a smooth and bounded domain in \({\mathbb {R}}^N, N>2\alpha \), \(\lambda \) is a positive parameter, \(\alpha \in (0,1)\), \((-\Delta )^{\alpha }\) is the fractional Laplacian and the nonlinear term h(u) has subcritical growth. We use minimax methods, the Ljusternick–Schnirelmann and Morse theories to get multiplicity results depending on the topology of \(\Omega \).     

Similarity Classification and Properties of Some Extended Holomorphic Curves

For \({\Omega\subseteq \mathbb{C}}\) a connected open set, and \({{\mathcal U}}\) a unital Banach algebra (or a unital C*-algebra), let \({{\xi (U)}}\) and \({ P({\mathcal U})}\) denote the sets of all idempotents and projections in \({{\mathcal U}}\), respectively. If \({e:\Omega\rightarrow \xi ({\mathcal U})}\) (resp.\({P({\mathcal U}))}\) is a holomorphic \({{\mathcal U}}\)-valued map, then e is called an extended holomorphic curve on \({ \xi ({\mathcal U})}\) (resp. \({P({\mathcal U})}\)). In this article, we focus on discussing the similarity classification problem of extended holomorphic curves. First, we introduce the definition of the commutant of extended holomorphic curves. By using K ₀-group of the commutant of the extended holomorphic curve, we characterize the curve which has unique finite (SI) decomposition up to similarity. Subsequently, we also obtain a similarity classification theorem. Second, we also discuss the unitary equivalence problem of some curves with respect to inductive limit C*-algebras.     

Semilinear elliptic equations with Hardy potential and subcritical source term

Let \(\Omega \) be a smooth bounded domain in \({\mathbb {R}}^N\) (\(N>2\)) and \(\delta (x):=\text {dist}\,(x,\partial \Omega )\). Assume \(\mu \in {\mathbb {R}}_+, \nu \) is a nonnegative finite measure on \(\partial \Omega \) and \(g \in C(\Omega \times {\mathbb {R}}_+)\). We study positive solutions of

$$\begin{aligned} -\Delta u - \frac{\mu }{\delta ^2} u = g(x,u) \text { in } \Omega , \qquad \text {tr}^*(u)=\nu . \end{aligned}$$

(P)

Here \(\text {tr}^*(u)\) denotes the normalized boundary trace of u which was recently introduced by Marcus and Nguyen (Ann Inst H Poincaré Anal Non Linéaire, 34, 69–88, 2017). We focus on the case \(0<\mu < C_H(\Omega )\) (the Hardy constant for \(\Omega \)) and provide qualitative properties of positive solutions of (P). When \(g(x,u)=u^q\) with \(q>0\), we prove that there is a critical value \(q^*\) (depending only on \(N, \mu \)) for (P) in the sense that if \(q<q^*\) then (P) possesses a solution under a smallness assumption on \(\nu \), but if \(q \ge q^*\) this problem admits no solution with isolated boundary singularity. Existence result is then extended to a more general setting where g is subcritical [see (1.28)]. We also investigate the case where g is linear or sublinear and give an existence result for (P).

     

Spectral multiplier theorems and averaged <Emphasis Type="Italic">R</Emphasis>-boundedness

Let A be a 0-sectorial operator with a bounded \(H^\infty (\Sigma _\sigma )\)-calculus for some \(\sigma \in (0,\pi ),\) e.g. a Laplace type operator on \(L^p(\Omega ),\, 1< p < \infty ,\) where \(\Omega \) is a manifold or a graph. We show that A has a \(\mathcal {H}^\alpha _2(\mathbb {R}_+)\) Hörmander functional calculus if and only if certain operator families derived from the resolvent \((\lambda - A)^{-1},\) the semigroup \(e^{-zA},\) the wave operators \(e^{itA}\) or the imaginary powers \(A^{it}\) of A are R-bounded in an \(L^2\)-averaged sense. If X is an \(L^p(\Omega )\) space with \(1 \le p < \infty \), R-boundedness reduces to well-known estimates of square sums.     

The fractional Neumann and Robin type boundary conditions for the regional fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian

Let \({p \in (1,\infty)}\), \({s \in (0,1)}\) and \({\Omega \subset {\mathbb{R}^{N}}}\) a bounded open set with boundary \({\partial\Omega}\) of class C ^1,1. In the first part of the article we prove an integration by parts formula for the fractional p-Laplace operator \({(-\Delta)_{p}^{s}}\) defined on \({\Omega \subset {\mathbb{R}^{N}}}\) and acting on functions that do not necessarily vanish at the boundary \({\partial\Omega}\). In the second part of the article we use the above mentioned integration by parts formula to clarify the fractional Neumann and Robin boundary conditions associated with the fractional p-Laplacian on open sets.     

Generating Hyperbolic Singularities in Semitoric Systems Via Hopf Bifurcations

Let \((M,\Omega )\) be a connected symplectic 4-manifold and let \(F=(J,H) :M\rightarrow \mathbb {R}^2\) be a completely integrable system on M with only non-degenerate singularities. Assume that F does not have singularities with hyperbolic blocks and that \(p_1,\ldots ,p_n\) are the focus–focus singularities of F. For each subset \(S=\{i_1,\ldots ,i_j\}\), we will show how to modify F locally around any \(p_i, i \in S\), in order to create a new integrable system \(\widetilde{F}=(J, \widetilde{H}) :M \rightarrow \mathbb {R}^2\) such that its classical spectrum \(\widetilde{F}(M)\) contains j smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of \(\widetilde{F}\). Moreover the focus–focus singularities of \(\widetilde{F}\) are precisely \(p_i\), \(i \in \{1,\ldots ,n\} \setminus S\). The proof is based on Eliasson’s linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.     

Homotopically invisible singular curves

Given a smooth manifold M and a totally nonholonomic distribution \(\Delta \subset TM \) of rank \(d\ge 3\), we study the effect of singular curves on the topology of the space of horizontal paths joining two points on M. Singular curves are critical points of the endpoint map \(F\,{:}\,\gamma \mapsto \gamma (1)\) defined on the space \(\Omega \) of horizontal paths starting at a fixed point x. We consider a sub-Riemannian energy \(J\,{:}\,\Omega (y)\rightarrow \mathbb R\), where \(\Omega (y)=F^{-1}(y)\) is the space of horizontal paths connecting x with y, and study those singular paths that do not influence the homotopy type of the Lebesgue sets \(\{\gamma \in \Omega (y)\,|\,J(\gamma )\le E\}\). We call them homotopically invisible. It turns out that for \(d\ge 3\) generic sub-Riemannian structures in the sense of Chitour et al. (J Differ Geom 73(1):45–73, 2006) have only homotopically invisible singular curves. Our results can be seen as a first step for developing the calculus of variations on the singular space of horizontal curves (in this direction we prove a sub-Riemannian minimax principle and discuss some applications).     

Generalized Bernstein Operators on the Classical Polynomial Spaces

We study generalizations of the classical Bernstein operators on the polynomial spaces \(\mathbb {P}_{n}[a,b]\), where instead of fixing \(\mathbf {1}\) and x, we reproduce exactly \(\mathbf {1}\) and a polynomial \(f_1\), strictly increasing on [a, b]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixing \(\mathbf {1}\) and \(f_1\). These operators are defined by non-decreasing sequences of nodes precisely when \(f_1^\prime > 0\) on (a, b), but even if \(f_1^\prime \) vanishes somewhere inside (a, b), they converge to the identity.     

The Regularity Problems with Data in Hardy–Sobolev Spaces for Singular Schrödinger Equation in Lipschitz Domains

Let the nonnegative singular potential V belong to the reverse Hölder class \({\mathcal B}_n\) on \({\mathbb R}^n\), and let (n???1)/n?p?≤?2, we establish the solvability and derivative estimates for the solutions to the Neumann problem and the regularity problem of the Schrödinger equation ??Δu?+?Vu?=?0 in a connected Lipschitz domain Ω, with boundary data in the Hardy space \(H^p(\partial \Omega)\) and the modified Hardy–Sobolev space \(H_{1, V}^p(\partial \Omega)\) related to the potential V. To deal with the H ^p regularity problem, we construct a new characterization of the atomic decomposition for \(H_{1, V}^p(\partial \Omega)\) space. The invertibility of the boundary layer potentials on Hardy spaces and Hölder spaces are shown in this paper.     

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

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

     

Removability of the logarithmic singularity for the elliptic PDEs with measurable coefficients and its consequences

This paper introduces the notion of log-regularity (or log-irregularity) of the boundary point \(\zeta \) (possibly \(\zeta =\infty \)) of the arbitrary open subset \(\Omega \) of the Greenian deleted neigborhood of \(\zeta \) in \({\mathbb {R}}^2\) concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the log-harmonic measure of \(\zeta \) is null (or positive). A necessary and sufficient condition for the removability of the logarithmic singularity, that is to say for the existence of a unique solution to the Dirichlet problem in \(\Omega \) in a class \(O(\log |\cdot - \zeta |)\) is established in terms of the Wiener test for the log-regularity of \(\zeta \). From a topological point of view, the Wiener test at \(\zeta \) presents the minimal thinness criteria of sets near \(\zeta \) in minimal fine topology. Precisely, the open set \(\Omega \) is a deleted neigborhood of \(\zeta \) in minimal fine topology if and only if \(\zeta \) is log-irregular. From the probabilistic point of view, the Wiener test presents asymptotic law for the log-Brownian motion near \(\zeta \) conditioned on the logarithmic kernel with pole at \(\zeta \).     

Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: existence

We apply the compactness results obtained in the first part of this work, to prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation

$$-\triangle u + V \left(\left|x\right|\right) u = g \left(\left|x\right|, u\right) \quad {\rm in} \Omega \subseteq \mathbb{R}^{N},\,N \geq 3,$$

where \({\Omega}\) is a radial domain (bounded or unbounded) and u satisfies u =  0 on \({\partial\Omega}\) if \({\Omega \neq\mathbb{R}^{N}}\) and \({u \rightarrow 0}\) as \({\left|x\right| \rightarrow \infty}\) if \({\Omega}\) is unbounded. The potential V may be vanishing or unbounded at zero or at infinity and the nonlinearity g may be superlinear or sublinear. If g is sublinear, the case with a forcing term \({g\left(\left|\cdot\right|, 0\right) \neq 0}\) is also considered. Our results allow to deal with V and g exhibiting behaviours at zero or at infinity which are new in the literature and, when \({\Omega = \mathbb{R}^{N}}\), do not need to be compatible with each other.

     

Joint universality for Dirichlet L-functions and zeros of their linear combinations in the $$\chi $$-aspect

We establish a joint universality theorem for Dirichlet L-functions in the character aspect. This is an extension of a result obtained by Bagchi and Gonek independently, and is an analogue of the joint universality for Dirichlet L-functions in the t-aspect. Zeros of linear combinations of Dirichlet L-functions in the t-aspect have been investigated by various authors. Using our joint universality theorem, we investigate zeros of such combinations from a new viewpoint. More precisely, we show that for any region \(\Omega \) in the strip \(1/2< \mathrm {Re}\,s <1\), any non-zero meromorphic functions \(H_1 (s), \dots , H_r(s)\) on \(\Omega \) with \(r \ge 2\) and any positive integer N, there exist a positive integer m and Dirichlet characters \(\varphi _1, \dots , \varphi _r \bmod m\) such that \(\sum _{j=1}^r H_j (s) L(s, \varphi _r)\) has at least N distinct zeros in \(\Omega \).     

Morrey Spaces on Domains: Different Approaches and Growth Envelopes

We deal with Morrey spaces on bounded domains \(\Omega \) obtained by different approaches. In particular, we consider three settings \(\mathcal {M}_{u,p}(\Omega )\), \(\mathbb {M}_{u,p}(\Omega )\) and \(\mathfrak {M}_{u,p}(\Omega )\), where \(0<p\le u<\infty \), commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes \(\mathfrak {E}_{\mathsf {G}}(\mathcal {M}_{u,p}(\Omega ))\) as well as \(\mathfrak {E}_{\mathsf {G}}(\mathfrak {M}_{u,p}(\Omega ))\), and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between p and u in the sense of whether \(\frac{n}{u}\ge \frac{1}{p}\) or \(\frac{n}{u} < \frac{1}{p}\) plays a decisive role when it comes to the behaviour of these spaces.     

Toroidal Geometry Stabilizing a Latitudinal Ring of Point Vortices on a Torus

We carry out the linear stability analysis of a polygonal ring configuration of N point vortices, called an N-ring, along the line of latitude \(\theta _0\) on a torus with the aspect ratio \(\alpha \). Deriving a criterion for the stability depending on the parameters N, \(\theta _0\) and \(\alpha \), we reveal how the aspect ratio \(\alpha \) contributes to the stability of the N-ring. While the N-ring necessarily becomes unstable when N is sufficiently large for fixed \(\alpha \), the stability is closely associated with the geometric property of the torus for variable \(\alpha \); for low aspect ratio \(\alpha \sim 1\), \(N=7\) is a critical number determining the stability of the N-ring when it is located along a certain range of latitudes, which is an analogous result to those in a plane and on a sphere. On the other hand, the stability is determined by the sign of curvature for high aspect ratio \(\alpha \gg 1\). That is to say, the N-ring is neutrally stable if it is located on the inner side of the toroidal surface with a negative curvature, while the N-ring on its outer side with a positive curvature is unstable. Furthermore, based on the linear stability analysis, we describe nonlinear evolution of the N-ring when it becomes unstable. It is difficult to deal with this problem, since the evolution equation of the N point vortices is formulated as a Hamiltonian system with N degrees of freedom, which is in general non-integrable. Thus, we reduce the Hamiltonian system to a simple integrable system by introducing a cyclic symmetry. Owing to this reduction, we successfully find some periodic orbits in the reduced system, whose local bifurcations and global transitions for variable \(\alpha \) are characterized in terms of the fundamental group of the torus.