Blow-up solutions concentrated along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds

Let (M, g) and \({(K, \kappa)}\) be two Riemannian manifolds of dimensions m and k, respectively. Let \({\omega \in C^{2} (N), \omega > 0}\) . The warped product \({M \times_\omega K}\) is the (m +  k)-dimensional product manifold \({M \times K}\) furnished with metric \({g + \omega^{2} \kappa}\) . We prove that the supercritical problem $$- \Delta_{g + \omega^{2} \kappa} u + hu = u^{\frac{m+2}{m-2} \pm \varepsilon} ,\quad u > 0,\quad {\rm in}\,\, (M \times_{\omega} K, g + \omega^{2} \kappa)$$ has a solution concentrated along a k-dimensional minimal submanifold \({\Gamma}\) of \({M \times_{\omega } N}\) as the real parameter \({\varepsilon}\) goes to zero, provided the function h and the sectional curvatures along \({\Gamma}\) satisfy a suitable condition.     

The Degree of Weakly Discretely Generated Spaces

A space X is discretely generated at a point \({x \in X}\) if for any \({A \subseteqq X}\) with \({x \in \textsf{cl}(A)}\) , there exists a discrete set \({D \subseteqq A}\) such that \({x \in \textsf{cl}(D)}\) . The space X is discretely generated if it is discretely generated at every point \({x \in X}\) . We say that X is weakly discretely generated if for any non-closed set \({A \subseteqq X}\) , there exists a discrete set \({D \subseteqq A}\) such that \({\textsf{cl}(D) \setminus A \neq \emptyset}\) . New results about these properties in the classes of pseudocompact and ?ech-complete spaces are obtained and a theorem of Ivanov and Osipov concerning the ordinal function idc is generalized to the class of ?ech-complete spaces.     

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

The overlap, \({\mathcal{D}_N}\) , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\) . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\) . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\) , and \({\tilde{\gamma}}\) . In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.     

Multiplicity of solutions to a nonlinear elliptic problem with nonlinear boundary conditions

We study the problem $$\left\{\begin{array}{ll}\Delta_p u = |u|^{q-2}u, & \quad x \in \Omega ,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu}= \lambda |u|^{p-2}u, &\quad x \in \partial \Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N}\) is a bounded smooth domain, \({\nu}\) is the outward unit normal at \({\partial \Omega}\) and \({\lambda > 0}\) is regarded as a bifurcation parameter. When p = 2 and in the superlinear regime q > 2, we show existence of n nontrivial solutions for all \({\lambda > \lambda_n}\) , \({\lambda_n}\) being the n-th Steklov eigenvalue. It is proved in addition that bifurcation from the trivial solution takes place at all \({\lambda_n}\) ’s. Similar results are obtained in the sublinear case 1 < q < 2. In this case, bifurcation from infinity takes place in those \({\lambda_n}\) with odd multiplicity. Partial extensions of these features are shown in the nonlinear diffusion case \({p \neq 2}\) and related problems under spatially heterogeneous reactions are also addressed.     

Around a problem of Nicole Brillouët–Belluot

We determine nontrivial intervals \({I \subset(0,+\infty)}\) , numbers \({\alpha\in\mathbb R}\) and continuous bijections \({f \colon I \to I}\) such that f(x)f ^?1(x) = x ^α for every \({x\in I}\) .     

Variants of the Kakeya problem over an algebraically closed field

First, we study constructible subsets of \({\mathbb{A}^n_k}\) which contain a line in any direction. We classify the smallest such subsets in \({\mathbb{A}^3}\) of the type \({R \cup \{g \neq 0\},}\) where \({g \in k[x_1,\ldots, x_n]}\) is irreducible of degree d and \({R \subset V(g)}\) is closed. Next, we study subvarieties \({X \subset \mathbb{A}^N}\) for which the set of directions of lines contained in X has the maximal possible dimension. These are variants of the Kakeya problem in an algebraic geometry context.     

A Geometric Uncertainty Principle with an Application to Pleijel’s Estimate

Let \({\Omega \subset \mathbb{R}^2}\) be an open, bounded domain and \({\Omega = \bigcup_{i = 1}^{N} \Omega_{i}}\) be a partition. Denote the Fraenkel asymmetry by \({0 \leq \mathcal{A}(\Omega_i) \leq 2}\) and write $$D(\Omega_i) := \frac{|\Omega_{i}| - {\rm min}_{1 \leq j \leq N}{|\Omega_{j}|}}{|\Omega_{i}|}$$ with \({0 \leq D(\Omega_{i}) \leq 1}\) . For N sufficiently large depending only on \({\Omega}\) , there is an uncertainty principle $$\left(\sum_{i=1}^{N}{\frac{|\Omega_{i}|}{|\Omega|}{\mathcal{A}}(\Omega_i)}\right) + \left(\sum_{i=1}^{N}{\frac{|\Omega_i|}{|\Omega|}D(\Omega_i)}\right) \geq \frac{1}{60000}.$$ The statement remains true in dimensions \({n \geq 3}\) for some constant \({c_{n} > 0}\) . As an application, we give an (unspecified) improvement of Pleijel’s estimate on the number of nodal domains of a Laplacian eigenfunction and an improved inequality for a spectral partition problem.     

The Cubic Complex Moment Problem

Let \({s = \{s_{jk}\}_{0 \leq j+k \leq 3}}\) be a given complex-valued sequence. The cubic complex moment problem involves determining necessary and sufficient conditions for the existence of a positive Borel measure \({\sigma}\) on \({\mathbb{C}}\) (called a representing measure for s) such that \({s_{jk} = \int_{\mathbb{C}}\bar{z}^j z^k d\sigma(z)}\) for \({0 \leq j + k \leq 3}\) . Put $$\Phi = \left(\begin{array}{lll} s_{00} & s_{01} & s_{10} \\s_{10} & s_{11} & s_{20} \\s_{01} & s_{02} & s_{11}\end{array}\right), \quad \Phi_z = \left(\begin{array}{lll}s_{01} & s_{02} & s_{11} \\s_{10} & s_{12} & s_{21} \\s_{02} & s_{03} & s_{12}\end{array} \right)\quad {\rm and}\quad\Phi_{\bar{z}} = (\Phi_z)^*.$$ If \({\Phi \succ 0}\) , then the commutativity of \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) is necessary and sufficient for the existence a 3-atomic representing measure for s. If \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) do not commute, then we show that s has a 4-atomic representing measure. The proof is constructive in nature and yields a concrete parametrization of all 4-atomic representing measures of s. Consequently, given a set \({K \subseteq \mathbb{C}}\) necessary and sufficient conditions are obtained for s to have a 4-atomic representing measure \({\sigma}\) which satisfies \({{\rm supp} \sigma \cap K \neq \emptyset}\) or \({{\rm supp} \sigma \subseteq K}\) . The cases when \({K = \overline{\mathbb{D}}}\) and \({K = \mathbb{T}}\) are considered in detail.     

The error term in the Sato–Tate conjecture

Let \({f(z) = \sum_{n=1}^\infty a(n)e^{2\pi i nz} \in S_k^{\mathrm{new}}(\Gamma_0(N))}\) be a newform of even weight \({k \geq 2}\) that does not have complex multiplication. Then \({a(n) \in \mathbb{R}}\) for all n; so for any prime p, there exists \({\theta_p \in [0, \pi]}\) such that \({a(p) = 2p^{(k-1)/2} {\rm cos} (\theta_p)}\) . Let \({\pi(x) = \#\{p \leq x\}}\) . For a given subinterval \({[\alpha, \beta]\subset[0, \pi]}\) , the now-proven Sato–Tate conjecture tells us that as \({x \to \infty}\) , $$ \#\{p \leq x: \theta_p \in I\} \sim \mu_{ST} ([\alpha, \beta])\pi(x),\quad \mu_{ST} ([\alpha, \beta]) = \int\limits_{\alpha}^\beta \frac{2}{\pi}{\rm sin}^2(\theta) d\theta. $$ Let \({\epsilon > 0}\) . Assuming that the symmetric power L-functions of f are automorphic, we prove that as \({x \to \infty}\) , $$ \#\{p \leq x: \theta_p \in I\} = \mu_{ST} ([\alpha, \beta])\pi(x) + O\left(\frac{x}{(\log x)^{9/8-\epsilon}} \right), $$ where the implied constant is effectively computable and depends only on k,N, and \({\epsilon}\) .     

Flat bundles on G/Γ and stability

Let G be a connected complex Lie group and Γ a cocompact lattice in G. Let H be a connected reductive complex affine algebraic group and \({\rho\, : \Gamma\, \longrightarrow H}\) a homomorphism such that \({\rho(\Gamma)}\) is not contained in some proper parabolic subgroup of H. Let \({E^\rho_H}\) be the holomorphic principal H–bundle on G/Γ associated to ρ. We prove that \({E^\rho_H}\) is polystable. If ρ satisfies the further condition that \({\rho(\Gamma)}\) is contained in a compact subgroup of H, then we prove that \({E^\rho_H}\) is stable.     

Singular integral operators with operator-valued kernels,and extrapolation of maximal regularity into rearrangement invariant Banach function spaces

We prove two extrapolation results for singular integral operators with operator-valued kernels, and we apply these results in order to obtain the following extrapolation of L ^p -maximal regularity: if an autonomous Cauchy problem on a Banach space has L ^p -maximal regularity for some \({p \in (1,\infty )}\) , then it has \({\mathbb{E}_w}\) -maximal regularity for every rearrangement invariant Banach function space \({\mathbb{E}}\) with Boyd indices \({1 < p_\mathbb{E} \leq q_\mathbb{E} < \infty}\) and every Muckenhoupt weight \({w \in A_{p \mathbb{E}}}\) . We prove a similar result for nonautonomous Cauchy problems on the line.     

A Theory of Stationary Trees and the Balanced Baumgartner–Hajnal–Todorcevic Theorem for Trees

Building on early work by Stevo Todorcevic, we develop a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem. Let \({\kappa}\) be any infinite regular cardinal, let ξ be any ordinal such that \({2^{|\xi|} < \kappa}\) , and let k be any natural number. Then $$non-(2^{<\kappa})-special\, tree \rightarrow (\kappa + \xi)^{2}_k.$$ This is a generalization to trees of the Balanced Baumgartner–Hajnal–Todorcevic Theorem, which we recover by applying the above to the cardinal \({(2^{< \kappa})^{+}}\) , the simplest example of a non- \({(2^{< \kappa})}\) -special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem. Let \({\kappa}\) be any infinite regular cardinal, let ξ be any ordinal such that \({2^{|\xi|} < \kappa}\) , and let k be any natural number. Let P be a partially ordered set such that \({P \rightarrow (2^{< \kappa})^{1}_{2^{< \kappa}} }\) . Then $$P \rightarrow (\kappa + \xi)^{2}_{k}.$$      

Supercritical Holes for the Doubling Map

For a map \({S : X \to X}\) and an open connected set (= a hole) \({H \subset X}\) we define \({\mathcal{J}_H(S)}\) to be the set of points in X whose S-orbit avoids H. We say that a hole H ₀ is supercritical if

1. for any hole H such that \({\overline{H}_0 \subset H}\) the set \({\mathcal{J}_H(S)}\) is either empty or contains only fixed points of S;

1. for any hole H such that \({\overline{H} \subset H_0}\) the Hausdorff dimension of \({\mathcal{J}_H(S)}\) is positive.

The purpose of this note is to completely characterize all supercritical holes for the doubling map Tx =  2x mod 1.     

Stability of a conditional Cauchy equation on a set of measure zero

In this paper, we prove the Hyers–Ulam stability theorem when \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy $$|f(x + y) - g(x) - h(y)| \leq \epsilon$$ in a set \({\Gamma \subset \mathbb{R}^{2}}\) of measure \({m(\Gamma) = 0}\) , which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy the Pexider equation $$f(x + y) - g(x) - h(y) = 0$$ in \({\Gamma}\) , then the equation holds for all \({x, y \in \mathbb{R}}\) . Using our method of construction of the set, we can find a set \({\Gamma \subset \mathbb{R}^{2n}}\) of 2n-dimensional measure 0 and obtain the above result for the functions \({f, g, h : \mathbb{R}^{n} \to \mathbb{C}}\) .     

The limitations of the Poincaré inequality for Grušin type operators

We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operators H in divergence form on \({L_2(\mathbf{R}^{n}\times \mathbf{R}^{m})}\) . We assume the coefficients are real symmetric and \({a_1H_\delta\geq H\geq a_2H_\delta}\) for some \({a_1,a_2>0}\) where H _δ is a generalized Gru?in operator, $$H_\delta=-\nabla_{x_1}\,|x_1|^{\left(2\delta_1,2\delta_1'\right)} \,\nabla_{x_1}-|x_1|^{\left(2\delta_2,2\delta_2'\right)} \,\nabla_{x_2}^2.$$ Here \({x_1 \in \mathbf{R}^n,\; x_2 \in \mathbf{R}^m,\;\delta_1,\delta_1'\in[0,1\rangle,\;\delta_2,\delta_2'\geq0}\) and \({|x_1|^{\left(2\delta,2\delta'\right)}=|x_1|^{2\delta}}\) if \({|x_1|\leq 1}\) and \({|x_1|^{\left(2\delta,2\delta'\right)}=|x_1|^{2\delta'}}\) if \({|x_1|\geq 1}\) . We prove that the Poincaré inequality, formulated in terms of the geometry corresponding to the control distance of H, is valid if n ≥ 2, or if n = 1 and \({\delta_1\vee\delta_1'\in[0,1/2\rangle}\) but it fails if n = 1 and \({\delta_1\vee\delta_1'\in[1/2,1\rangle}\) . The failure is caused by the leading term. If \({\delta_1\in[1/2, 1\rangle}\) , it is an effect of the local degeneracy \({|x_1|^{2\delta_1}}\) , but if \({\delta_1\in[0, 1/2\rangle}\) and \({\delta_1'\in [1/2,1\rangle}\) , it is an effect of the growth at infinity of \({|x_1|^{2\delta_1'}}\) . If n = 1 and \({\delta_1\in[1/2, 1\rangle}\) , then the semigroup S generated by the Friedrichs’ extension of H is not ergodic. The subspaces \({x_1\geq 0}\) and \({x_1\leq 0}\) are S-invariant, and the Poincaré inequality is valid on each of these subspaces. If, however, \({n=1,\; \delta_1\in[0, 1/2\rangle}\) and \({\delta_1'\in [1/2,1\rangle}\) , then the semigroup S is ergodic, but the Poincaré inequality is only valid locally. Finally, we discuss the implication of these results for the Gaussian and non-Gaussian behaviour of the semigroup S.     

On the number of reducible polynomials of bounded naive height

We prove an asymptotical formula for the number of reducible integer polynomials of degree d and of naive height at most T when \({T \to \infty}\) . The main term turns out to be of the form \({\kappa_d T^d}\) for each \({d \geq 3}\) , where the constant \({\kappa_d}\) is given in terms of some infinite Dirichlet series involving the volumes of symmetric convex bodies in \({\mathbb{R}^d}\) . For d = 2, we prove that there are asymptotically \({\kappa_2 T^2 \,\text{log} T}\) of such polynomials, where \({\kappa_2:=6(3\sqrt{5}+2\,\text{log} (1+\sqrt{5}) -2 \,\text{log}\, 2)/\pi^2}\) . Earlier results in this direction were given by van der Waerden, Pólya and Szegö, Dörge, Chela, and Kuba.     

On the Relaxed Colouring Game and the Unilateral Colouring Game

In the paper we introduce the new game—the unilateral \({\mathcal{P}}\) -colouring game which can be used as a tool to study the r-colouring game and the (r, d)-relaxed colouring game. Let be given a graph G, an additive hereditary property \({\mathcal {P}}\) and a set C of r colours. In the unilateral \({\mathcal {P}}\) -colouring game similarly as in the r-colouring game, two players, Alice and Bob, colour the uncoloured vertices of the graph G, but in the unilateral \({\mathcal {P}}\) -colouring game Bob is more powerful than Alice. Alice starts the game, the players play alternately, but Bob can miss his move. Bob can colour the vertex with an arbitrary colour from C, while Alice must colour the vertex with a colour from C in such a way that she cannot create a monochromatic minimal forbidden subgraph for the property \({\mathcal {P}}\) . If after |V(G)| moves the graph G is coloured, then Alice wins the game, otherwise Bob wins. The \({\mathcal {P}}\) -unilateral game chromatic number, denoted by \({\chi_{ug}^\mathcal {P}(G)}\) , is the least number r for which Alice has a winning strategy for the unilateral \({\mathcal {P}}\) -colouring game with r colours on G. We prove that the \({\mathcal {P}}\) -unilateral game chromatic number is monotone and is the upper bound for the game chromatic number and the relaxed game chromatic number. We give the winning strategy for Alice to play the unilateral \({\mathcal {P}}\) -colouring game. Moreover, for k ≥  2 we define a class of graphs \({\mathcal {H}_k =\{G|{\rm every \;block \;of\;}G \; {\rm has \;at \;most}\; k \;{\rm vertices}\}}\) . The class \({\mathcal {H}_k }\) contains, e.g., forests, Husimi trees, line graphs of forests, cactus graphs. Let \({\mathcal {S}_d}\) be the class of graphs with maximum degree at most d. We find the upper bound for the \({\mathcal {S}_2}\) -unilateral game chromatic number for graphs from \({\mathcal {H}_3}\) and we study the \({\mathcal {S}_d}\) -unilateral game chromatic number for graphs from \({\mathcal {H}_4}\) for \({d \in \{2,3\}}\) . As the conclusion from these results we obtain the result for the d-relaxed game chromatic number: if \({G \in \mathcal {H}_k}\) , then \({\chi_g^{(d)}(G) \leq k + 2-d}\) , for \({k \in \{3, 4\}}\) and \({d \in \{0, \ldots, k-1\}}\) . This generalizes a known result for trees.     

Sign-definiteness of q-eigenfunctions for a super-linear p-Laplacian eigenvalue problem

We consider the following q-eigenvalue problem for the p-Laplacian $$\left\{\begin{array}{ll}-{\rm div}\big( |\nabla u|^{p-2}\nabla u\big) = \lambda \|u\|_{L^{q}(\Omega)}^{p-q}|u|^{q-2}u \quad \quad\, {\rm in} \,\,\,\, \Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,\,{\rm on } \,\,\,\, \partial\Omega,\end{array}\right.$$ where \({\lambda\in\mathbb{R},}\) p > 1, Ω is a bounded and smooth domain of \({\mathbb{R}^{N},}\) N > 1, \({1\leq q < p^{\star}}\) , \({p^{\star}=\frac{Np}{N-p}}\) if p < N and \({p^{\star}=\infty}\) if \({p\geq N.}\) Let λ _q denote the first q-eigenvalue. We prove that in the super-linear case, \({p < q < p^{\star},}\) there exists \({\epsilon_{q}>0}\) such that if \({\lambda\in(\lambda_{q},\lambda _{q}+\epsilon_{q})}\) is a q-eigenvalue, then any corresponding q-eigenfunction does not change sign in Ω. As a consequence of this result we obtain, in the super-linear case, the isolatedness of λ _q for those Ω such that the Lane–Emden problem $$\left\{\begin{array}{ll}-{\rm div}\big(|\nabla u|^{p-2}\nabla u\big) = |u|^{q-2}u \qquad\quad\quad\quad \,\,{\rm in}\,\,\,\Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,{\rm on } \,\,\, \partial\Omega,\end{array}\right.$$ has exactly one positive solution.     

Converting homotopies to isotopies and dividing homotopies in half in an effective way

We prove two theorems about homotopies of curves on two-dimensional Riemannian manifolds. We show that, for any \({\epsilon > 0}\) , if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by \({L + \epsilon}\) . If the manifold is orientable, then for any \({\epsilon > 0}\) we show that, if we can contract a curve \({\gamma}\) traversed twice through curves of length bounded by L, then we can also contract \({\gamma}\) through curves bounded in length by \({L + \epsilon}\) . Our method involves cutting curves at their self-intersection points and reconnecting them in a prescribed way. We consider the space of all curves obtained in this way from the original homotopy, and use a novel approach to show that this space contains a path which yields the desired homotopy.