Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

In this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (−Δ) s in ℝ n , for n ≥ 2, such as (0.1) ( − Δ ) s u + E ( x ) u + V ( x ) u q − 1 = K ( x ) f ( u ) + u 2 s ⋆ − 1 . $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\star}-1}.$$ Here, s ∈ (0, 1), q ∈ 2 , 2 s ⋆ $q \in\left[2,2_{s}^{\star}\right)$ with 2 s ⋆ := 2 n n − 2 s $2_{s}^{\star}:=\frac{2 n}{n-2 s}$ being the fractional critical Sobolev exponent, E(x), K(x), V(x) > 0 : ℝ n → ℝ are measurable functions which satisfy joint “vanishing at infinity” conditions in a measure-theoretic sense, and f (u) is a continuous function on ℝ of quasi-critical, super-q-linear growth with f (u) ≥ 0 if u ≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝ n such as (0.2) ( − Δ ) s u + E ( x ) u + V ( x ) u q − 1 = λ K ( x ) u r − 1 , $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=\lambda K(x) u^{r-1},$$ where 2 < r < q < ∞(both possibly (super-)critical), E(x), K(x), V(x) > 0 : ℝ n → ℝ are measurable functions satisfying joint integrability conditions, and λ > 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spaces Ms ;q,p (ℝ n ) as well as their associated compact embedding results.

High-Dimensional Separability for One- And Few-Shot Learning

This work is driven by a practical question, corrections of Artificial Intelligence (AI) errors. Systematic re-training of a large AI system is hardly possible. To solve this problem, special external devices, correctors, are developed. They should provide quick and non-iterative system fix without modification of a legacy AI system. A common universal part of the AI corrector is a classifier that should separate undesired and erroneous behavior from normal operation. Training of such classifiers is a grand challenge at the heart of the one- and few-shot learning methods. Effectiveness of one- and few-short methods is based on either significant dimensionality reductions or the blessing of dimensionality effects. Stochastic separability is a blessing of dimensionality phenomenon that allows one-and few-shot error correction: in high-dimensional datasets under broad assumptions each point can be separated from the rest of the set by simple and robust linear discriminant. The hierarchical structure of data universe is introduced where each data cluster has a granular internal structure, etc. New stochastic separation theorems for the data distributions with fine-grained structure are formulated and proved. Separation theorems in infinite-dimensional limits are proven under assumptions of compact embedding of patterns into data space. New multi-correctors of AI systems are presented and illustrated with examples of predicting errors and learning new classes of objects by a deep convolutional neural network.

A New Upper Bound for Sampling Numbers

We provide a new upper bound for sampling numbers $$(g_n)_{n\in \mathbb {N}}$$ ( g n ) n ∈ N associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants $$C,c>0$$ C , c > 0 (which are specified in the paper) such that $$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$ g n 2 ≤ C log ( n ) n ∑ k ≥ ⌊ c n ⌋ σ k 2 , n ≥ 2 , where $$(\sigma _k)_{k\in \mathbb {N}}$$ ( σ k ) k ∈ N is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding $$\mathrm {Id}:H(K) \rightarrow L_2(D,\varrho _D)$$ Id : H ( K ) → L 2 ( D , ϱ D ) . The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of $$H^s_{\text {mix}}(\mathbb {T}^d)$$ H mix s ( T d ) in $$L_2(\mathbb {T}^d)$$ L 2 ( T d ) with $$s>1/2$$ s > 1 / 2 . We obtain the asymptotic bound $$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$ g n ≤ C s , d n - s log ( n ) ( d - 1 ) s + 1 / 2 , which improves on very recent results by shortening the gap between upper and lower bound to $$\sqrt{\log (n)}$$ log ( n ) . The result implies that for dimensions $$d>2$$ d > 2 any sparse grid sampling recovery method does not perform asymptotically optimal.

Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations

In this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involving q(x)-Laplacian parabolic equation ∂ρt,x/∂t=divxρt,x∇xG′ρ+Vqx−2∇xG′ρ+V. The potential V is not necessarily smooth but belongs to a Sobolev space W1,∞Ω. Given the initial datum ρ0 as a probability density on Ω, we use a descent algorithm in the probability space to discretize the q(x)-Laplacian parabolic equation in time. Then, we use compact embedding W1,q.Ω↪↪Lq.Ω established by Fan and Zhao to study the convergence of our algorithm to a weak solution of the q(x)-Laplacian parabolic equation. Finally, we establish the convergence of solutions of the q(x)-Laplacian parabolic equation to equilibrium in the p(.)-variable exponent Wasserstein space.

Continuous Dependence on a Parameter of Exponential Attractors for Nonclassical Diffusion Equations

In this paper, a new abstract result is given to verify the continuity of exponential attractors with respect to a parameter for the underlying semigroup. We do not impose any compact embedding on the main assumptions in the abstract result which is different from the corresponding result established by Efendiev et al. in 2004. Consequently, it can be used for equations whose solutions have no higher regularity. As an application, we prove the continuity of exponential attractors in H01 for a class of nonclassical diffusion equations with initial datum in H01.

Generalized linking theorem with applications to nonautonomous Hamiltonian systems and Dirac equations on compact spin manifold

Let [Formula: see text] be a Riemannian manifold with finite volume and [Formula: see text] be a linear topological space. We consider the strongly indefinite superlinear problem [Formula: see text] where [Formula: see text] is a self-adjoint linear operator, [Formula: see text] is a real Hilbert space with the compact embedding [Formula: see text] if [Formula: see text] for some [Formula: see text], and [Formula: see text]. We obtain the existence of two solutions provided that [Formula: see text] and [Formula: see text] for a certain choice of [Formula: see text], [Formula: see text]. Moreover, we prove that, if [Formula: see text] and [Formula: see text] small enough, there exist prescribed number of nontrivial solutions. As applications, the corresponding results hold true for nonautonomous Hamiltonian systems and Dirac equations on compact spin manifold.