Lower order eigenvalues of a system of equations of the drifting Laplacian on the metric measure spaces

Let \(\Omega \) be a bounded domain with smooth boundary in an n-dimensional metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and let \(\mathbf {u}=(u^1, \ldots , u^n)\) be a vector-valued function from \(\Omega \) to \(\mathbb {R}^n\). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of the drifting Laplacian: \(\mathbb {L}_{\phi } \mathbf {u} + \alpha [ \nabla (\mathrm {div}\mathbf { u}) -\nabla \phi \mathrm {div} \mathbf {u}]= - \widetilde{\sigma } \mathbf {u}\), in \( \Omega \), and \(u|_{\partial \Omega }=0,\) where \(\mathbb {L}_{\phi } = \Delta - \nabla \phi \cdot \nabla \) is the drifting Laplacian and \(\alpha \) is a nonnegative constant. We establish some universal inequalities for lower order eigenvalues of this problem on the metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and the Gaussian shrinking soliton \((\mathbb {R}^n, \langle ,\rangle _{\mathrm {can}}, e^{-\frac{|x|^2}{4}}dv, \frac{1}{2})\). Moreover, we give an estimate for the upper bound of the second eigenvalue of this problem in terms of its first eigenvalue on the gradient product Ricci soliton \((\Sigma \times \mathbb {R}, \langle ,\rangle , e^{-\frac{\kappa t^2}{2}}dv, \kappa )\), where \( \Sigma \) is an Einstein manifold with constant Ricci curvature \(\kappa \).     

Global $$C^{1,\al pha }$$ regularity and e xistence of multi ple solutions for singular p( x)-La placian equations

In this paper we study the following singular p(x)-Laplacian problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} - \text{ div } \left( |\nabla u|^{p(x)-2} \nabla u\right) =\frac{ \lambda }{u^{\beta (x)}}+u^{q(x)}, &{} \text{ in }\quad \Omega , \\ u>0, &{} \text{ in }\quad \Omega , \\ u=0, &{} \text{ on }\quad \partial \Omega , \end{array}\right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 2\), with smooth boundary \(\partial \Omega \), \(\beta \in C^1(\bar{\Omega })\) with \( 0< \beta (x) <1\), \(p\in C^1(\bar{\Omega })\), \(q \in C(\bar{\Omega })\) with \(p(x)>1\), \(p(x)< q(x) +1 <p^*(x)\) for \(x \in \bar{\Omega }\), where \( p^*(x)= \frac{Np(x)}{N-p(x)} \) for \(p(x) <N\) and \( p^*(x)= \infty \) for \( p(x) \ge N\). We establish \(C^{1,\alpha }\) regularity of weak solutions of the problem and strong comparison principle. Based on these two results, we prove the existence of multiple (at least two) positive solutions for a certain range of \(\lambda \).

     

The Transition Density of Brownian Motion Killed on a Bounded Set

We study the transition density of a standard two-dimensional Brownian motion killed when hitting a bounded Borel set A. We derive the asymptotic form of the density, say \(p^A_t(\mathbf{x},\mathbf{y})\), for large times t and for \(\mathbf{x}\) and \(\mathbf{y}\) in the exterior of A valid uniformly under the constraint \(|\mathbf{x}|\vee |\mathbf{y}| =O(t)\). Within the parabolic regime \(|\mathbf{x}|\vee |\mathbf{y}| = O(\sqrt{t})\) in particular \(p^A_t(\mathbf{x},\mathbf{y})\) is shown to behave like \(4e_A(\mathbf{x})e_A(\mathbf{y}) (\lg t)^{-2} p_t(\mathbf{y}-\mathbf{x})\) for large t, where \(p_t(\mathbf{y}-\mathbf{x})\) is the transition kernel of the Brownian motion (without killing) and \(e_A\) is the Green function for the ‘exterior of A’ with a pole at infinity normalized so that \(e_A(\mathbf{x}) \sim \lg |\mathbf{x}|\). We also provide fairly accurate upper and lower bounds of \(p^A_t(\mathbf{x},\mathbf{y})\) for the case \(|\mathbf{x}|\vee |\mathbf{y}|>t\) as well as corresponding results for the higher dimensions.     

The finite basis problem for infinite involution semigroups of triangular 2 $$\times $$ 2 matrices

Let \(T_n(\mathbb {F})\) and \(UT_n(\mathbb {F})\) be the semigroups of all upper triangular \(n\times n\) matrices and all upper triangular \(n\times n\) matrices with 0s and/or 1s on the main diagonal over a field \(\mathbb {F}\) with \(\mathsf {char}(\mathbb {F})=0\), respectively. In this paper, we address the finite basis problem for \(T_2(\mathbb {F})\) and \(UT_2(\mathbb {F})\) as involution semigroups under the skew transposition. By giving a sufficient condition under which an involution semigroup is nonfinitely based, we show that both \(T_2(\mathbb {F})\) and \(UT_2(\mathbb {F})\) are nonfinitely based, and that there is a continuum of nonfinitely based involution monoid varieties between the involution monoid variety \(\mathsf {var} UT_2(\mathbb {F})\) generated by \(UT_2(\mathbb {F})\) and the involution monoid variety \(\mathsf {var} T_2(\mathbb {F})\) generated by \(T_2(\mathbb {F})\). Moreover, \(\mathsf {var} UT_2(\mathbb {F})\) cannot be defined within \(\mathsf {var} T_2(\mathbb {F})\) by any finite set of identities.     

Jump and Variational Inequalities for Rough Operators

In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones et al. More precisely, we show some jump and variational inequalities for the families \(\mathcal T:=\{T_\varepsilon \}_{\varepsilon >0}\) of truncated singular integrals and \(\mathcal M:=\{M_t\}_{t>0}\) of averaging operators with rough kernels, which are defined respectively by

$$\begin{aligned} T_\varepsilon f(x)=\int _{|y|>\varepsilon }\frac{\Omega (y')}{|y|^n}f(x-y)dy \end{aligned}$$

and

$$\begin{aligned} M_t f(x)=\frac{1}{t^n}\int _{|y|<t}\Omega (y')f(x-y)dy, \end{aligned}$$

where the kernel \(\Omega \) belongs to \(L\log ^+\!\!L(\mathbf S^{n-1})\) or \(H^1(\mathbf S^{n-1})\) or \(\mathcal {G}_\alpha (\mathbf S^{n-1})\) (the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.

     

Korn’s inequality and John domains

Let \(\Omega \subset \mathbb {R}^n\), \(n\ge 2\), be a bounded domain satisfying the separation property. We show that the following conditions are equivalent:

1. (i)
   \(\Omega \) is a John domain;
    
2. (ii)
   for a fixed \(p\in (1,\infty )\), the Korn inequality holds for each \(\mathbf {u}\in W^{1,p}(\Omega ,\mathbb {R}^n)\) satisfying \(\int _\Omega \frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\,dx=0\), \(1\le i,j\le n\),

   $$\begin{aligned} \Vert D\mathbf {u}\Vert _{L^p(\Omega )}\le C_K(\Omega , p)\Vert \epsilon (\mathbf {u})\Vert _{L^p(\Omega )}; \qquad (K_{p}) \end{aligned}$$
    
3. (ii’)
   for all \(p\in (1,\infty )\), \((K_p)\) holds on \(\Omega \);
    
4. (iii)
   for a fixed \(p\in (1,\infty )\), for each \(f\in L^p(\Omega )\) with vanishing mean value on \(\Omega \), there exists a solution \(\mathbf {v}\in W^{1,p}_0(\Omega ,\mathbb {R}^n)\) to the equation \(\mathrm {div}\,\mathbf {v}=f\) with

   $$\begin{aligned} \Vert \mathbf {v}\Vert _{W^{1,p}(\Omega ,\mathbb {R}^n)}\le C(\Omega , p)\Vert f\Vert _{L^p(\Omega )};\qquad (DE_p) \end{aligned}$$
    
5. (iii’)
   for all \(p\in (1,\infty )\), \((DE_p)\) holds on \(\Omega \).
    

For domains satisfying the separation property, in particular, for finitely connected domains in the plane, our result provides a geometric characterization of the Korn inequality, and gives positive answers to a question raised by Costabel and Dauge (Arch Ration Mech Anal 217(3):873–898, 2015) and a question raised by Russ (Vietnam J Math 41:369–381, 2013). For the plane, our result is best possible in the sense that, there exist infinitely connected domains which are not John but support Korn’s inequality.

     

Optimal bounds for convergence of expected spectral distributions to the semi-circular law

Let \(\mathbf {X}=(X_{jk})_{j,k=1}^n\) denote a Hermitian random matrix with entries \(X_{jk}\), which are independent for \(1\le j\le k\le n\). We consider the rate of convergence of the empirical spectral distribution function of the matrix \(\mathbf {X}\) to the semi-circular law assuming that \(\mathbf{E}X_{jk}=0\), \(\mathbf{E}X_{jk}^2=1\) and that

$$\begin{aligned} \sup _{n\ge 1}\sup _{1\le j,k\le n}\mathbf{E}|X_{jk}|^4=:\mu _4<\infty , \end{aligned}$$

and

$$\begin{aligned} \sup _{1\le j,k\le n}|X_{jk}|\le D_0n^{\frac{1}{4}}. \end{aligned}$$

By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix \(\mathbf {W}=\frac{1}{\sqrt{n}}\mathbf {X}\) and the semicircular law is of order \(O(n^{-1})\).

     

Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two complex variables

The main object of study in this paper is the double holomorphic Eisenstein series \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) having two complex variables \(\mathbf{s}=(s_1,s_2)\) and two parameters \(\mathbf{z}= (z_1,z_2)\) which satisfies either \(\mathbf{z}\in (\mathfrak {H}^+)^2\) or \(\mathbf{z}\in (\mathfrak {H}^-)^2\), where \(\mathfrak {H}^{\pm }\) denotes the complex upper and lower half-planes, respectively. For \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\), its transformation properties and asymptotic aspects are studied when the distance \(|z_2-z_1|\) becomes both small and large under certain natural settings on the movement of \(\mathbf{z}\in (\mathfrak {H}^{\pm })^2\). Prior to the proofs our main results, a new parameter \(\eta \), which plays a pivotal role in describing our results, is introduced in connection with the difference \(z_2-z_1\). We then establish complete asymptotic expansions for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) when \(\mathbf{z}\) moves within the poly-sector either \((\mathfrak {H}^+)^2\) or \((\mathfrak {H}^-)^2\), so as to \(\eta \rightarrow 0\) through \(|\arg \eta |<\pi /2\) in the ascending order of \(\eta \) (Theorem 1). This further leads us to show that counterpart expansions exist for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) in the descending order of \(\eta \) as \(\eta \rightarrow \infty \) through \(|\arg \eta |<\pi /2\) (Theorem 2). Our second main formula in Theorem 2 yields a functional equation for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) (Corollaries 2.1, 2.2), and also reduces naturally to various expressions of \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) in closed forms for integer lattice point \(\mathbf{s}\in \mathbb {Z}^2\) (Corollaries 2.3–2.17). Most of these results reveal that the particular values of \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) at \(\mathbf{s}\in \mathbb {Z}^2\) are closely linked to Weierstraß’ elliptic function, the classical Eisenstein series reformulated by Ramanujan, and the Jordan–Kronecker type functions, each associated with the bases \(2\pi (1, z_j)\), \(j=1,2\). The latter two functions were extensively utilized by Ramanujan in the course of developing his theories of Eisenstein series, elliptic functions, and theta functions. As for the methods used, crucial roles in the proofs are played by the Mellin–Barnes type integrals, manipulated with several properties of hypergeometric functions; the transference from Theorem 1 to Theorem 2 is, for instance, achieved by a connection formula for Kummer’s confluent hypergeometric functions.     

Sharp MSE Bounds for Proximal Denoising

Denoising has to do with estimating a signal \(\mathbf {x}_0\) from its noisy observations \(\mathbf {y}=\mathbf {x}_0+\mathbf {z}\). In this paper, we focus on the “structured denoising problem,” where the signal \(\mathbf {x}_0\) possesses a certain structure and \(\mathbf {z}\) has independent normally distributed entries with mean zero and variance \(\sigma ^2\). We employ a structure-inducing convex function \(f(\cdot )\) and solve \(\min _\mathbf {x}\{\frac{1}{2}\Vert \mathbf {y}-\mathbf {x}\Vert _2^2+\sigma {\lambda }f(\mathbf {x})\}\) to estimate \(\mathbf {x}_0\), for some \(\lambda >0\). Common choices for \(f(\cdot )\) include the \(\ell _1\) norm for sparse vectors, the \(\ell _1-\ell _2\) norm for block-sparse signals and the nuclear norm for low-rank matrices. The metric we use to evaluate the performance of an estimate \(\mathbf {x}^*\) is the normalized mean-squared error \(\text {NMSE}(\sigma )=\frac{{\mathbb {E}}\Vert \mathbf {x}^*-\mathbf {x}_0\Vert _2^2}{\sigma ^2}\). We show that NMSE is maximized as \(\sigma \rightarrow 0\) and we find the exact worst-case NMSE, which has a simple geometric interpretation: the mean-squared distance of a standard normal vector to the \({\lambda }\)-scaled subdifferential \({\lambda }\partial f(\mathbf {x}_0)\). When \({\lambda }\) is optimally tuned to minimize the worst-case NMSE, our results can be related to the constrained denoising problem \(\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-\mathbf {x}\Vert _2\}\). The paper also connects these results to the generalized LASSO problem, in which one solves \(\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-{\mathbf {A}}\mathbf {x}\Vert _2\}\) to estimate \(\mathbf {x}_0\) from noisy linear observations \(\mathbf {y}={\mathbf {A}}\mathbf {x}_0+\mathbf {z}\). We show that certain properties of the LASSO problem are closely related to the denoising problem. In particular, we characterize the normalized LASSO cost and show that it exhibits a “phase transition” as a function of number of observations. We also provide an order-optimal bound for the LASSO error in terms of the mean-squared distance. Our results are significant in two ways. First, we find a simple formula for the performance of a general convex estimator. Secondly, we establish a connection between the denoising and linear inverse problems.     

Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let \(\mathfrak{h}\) be an algebraic subalgebra of the tangent Lie algebra \(\mathfrak{g}\) of G. We find all subalgebras \(\mathfrak{h}\) that have no nontrivial characters and whose centralizers \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) and \(P(\mathfrak{g})^\mathfrak{h} \) in the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g})\) and in the associated graded algebra \(P(\mathfrak{g})\), respectively, are commutative. For all these subalgebras, we prove that \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g} \) and \(P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g} \). Furthermore, we obtain a criterion for the commutativity of \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) in terms of representation theory.     

Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers

We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form \(-\Delta _p u = |\nabla u|^p + \sigma \) in a bounded domain \(\Omega \subset \mathbb {R}^n\). Here \(\Delta _p\), \(p>1\), is the standard p-Laplacian operator defined by \(\Delta _p u=\mathrm{div}\, (|\nabla u|^{p-2}\nabla u)\), and the datum \(\sigma \) is a signed distribution in \(\Omega \). The class of solutions that we are interested in consists of functions \(u\in W^{1,p}_0(\Omega )\) such that \(|\nabla u|\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\), a space pointwise Sobolev multipliers consisting of functions \(f\in L^{p}(\Omega )\) such that

$$\begin{aligned} \int _{\Omega } |f|^{p} |\varphi |^p dx \le C \int _{\Omega } (|\nabla \varphi |^p + |\varphi |^p) dx \quad \forall \varphi \in C^\infty (\Omega ), \end{aligned}$$

for some \(C>0\). This is a natural class of solutions at least when the distribution \(\sigma \) is nonnegative and compactly supported in \(\Omega \). We show essentially that, with only a gap in the smallness constants, the above equation has a solution in this class if and only if one can write \(\sigma =\mathrm{div}\, F\) for a vector field F such that \(|F|^{\frac{1}{p-1}}\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\). As an important application, via the exponential transformation \(u\mapsto v=e^{\frac{u}{p-1}}\), we obtain an existence result for the quasilinear equation of Schrödinger type \(-\Delta _p v = \sigma \, v^{p-1}\), \(v\ge 0\) in \(\Omega \), and \(v=1\) on \(\partial \Omega \), which is interesting in its own right.

     

Regularity for Inhomogeneous Dirichlet Problems of Some Schrödinger Equations on Domains

Let \(n\ge 3, \Omega \) be a bounded, simply connected and semiconvex domain in \({\mathbb {R}}^n\) and \(L_{\Omega }:=-\Delta +V\) a Schrödinger operator on \(L^2 (\Omega )\) with the Dirichlet boundary condition, where \(\Delta \) denotes the Laplace operator and the potential \(0\le V\) belongs to the reverse Hölder class \(RH_{q_0}({\mathbb {R}}^n)\) for some \(q_0\in (\max \{n/2,2\},\infty ]\). Assume that the growth function \(\varphi :\,{\mathbb {R}}^n\times [0,\infty ) \rightarrow [0,\infty )\) satisfies that \(\varphi (x,\cdot )\) is an Orlicz function and \(\varphi (\cdot ,t)\in {\mathbb {A}}_{\infty }({\mathbb {R}}^n)\) (the class of uniformly Muckenhoupt weights). Let \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) be the Musielak–Orlicz–Hardy space whose elements are restrictions of elements of the Musielak–Orlicz–Hardy space, associated with \(L_{{\mathbb {R}}^n}:=-\Delta +V\) on \({\mathbb {R}}^n\), to \(\Omega \). In this article, the authors show that the operators \(VL^{-1}_\Omega \) and \(\nabla ^2L^{-1}_\Omega \) are bounded from \(L^1(\Omega )\) to weak-\(L^1(\Omega )\), from \(L^p(\Omega )\) to itself, with \(p\in (1,2]\), and also from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to the Musielak–Orlicz space \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself. As applications, the boundedness of \(\nabla ^2{\mathbb {G}}_D\) on \(L^p(\Omega )\), with \(p\in (1,2]\), and from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself is obtained, where \({\mathbb {G}}_D\) denotes the Dirichlet Green operator associated with \(L_\Omega \). All these results are new even for the Hardy space \(H^1_{L_{{\mathbb {R}}^n},\,r}(\Omega )\), which is just \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) with \(\varphi (x,t):=t\) for all \(x\in {\mathbb {R}}^n\) and \(t\in [0,\infty )\).     

Hölder continuity for continuous solutions of the singular minimal surface equation with arbitrary zero set

In this paper we prove the following theorem: Let \(\Omega \subset \mathbb {R}^{n}\) be a bounded open set, \(\psi \in C_{c}^{2}(\mathbb {R}^{n})\), \(\psi > 0\) on \(\partial \Omega \), be given boundary values and u a nonnegative solution to the problem

$$\begin{aligned}&u \in C^{0}(\overline{\Omega }) \cap C^{2}(\{u> 0\}) \\&u = \psi \quad \text { on } \; \partial \Omega \\&{\text {div}} \left( \frac{Du}{\sqrt{1 + |Du|^{2}}}\right) = \frac{\alpha }{u \sqrt{1 + |Du|^{2}}} \quad \text { in } \; \{u > 0\} \end{aligned}$$

where \(\alpha > 0\) is a given constant. Then \(u \in C^{0, \frac{1}{2}} (\overline{\Omega })\). Furthermore we prove strict mean convexity of the free boundary \(\partial \{u = 0\}\) provided \(\partial \{u = 0\}\) is assumed to be of class \(C^{2}\) and \(\alpha \ge 1\).

     

On the <Emphasis Type="Italic">p</Emphasis>-Laplacian with Robin boundary conditions and boundary trace theorems

Let \(\Omega \subset \mathbb {R}^\nu \), \(\nu \ge 2\), be a \(C^{1,1}\) domain whose boundary \(\partial \Omega \) is either compact or behaves suitably at infinity. For \(p\in (1,\infty )\) and \(\alpha >0\), define

$$\begin{aligned} \Lambda (\Omega ,p,\alpha ):=\inf _{\begin{array}{c} u\in W^{1,p}(\Omega )\\ u\not \equiv 0 \end{array}}\dfrac{\displaystyle \int _\Omega |\nabla u|^p \mathrm {d} x - \alpha \displaystyle \int _{\partial \Omega } |u|^p\mathrm {d}\sigma }{\displaystyle \int _\Omega |u|^p\mathrm {d} x}, \end{aligned}$$

where \(\mathrm {d}\sigma \) is the surface measure on \(\partial \Omega \). We show the asymptotics

$$\begin{aligned} \Lambda (\Omega ,p,\alpha )=-(p-1)\alpha ^{\frac{p}{p-1}} - (\nu -1)H_\mathrm {max}\, \alpha + o(\alpha ), \quad \alpha \rightarrow +\infty , \end{aligned}$$

where \(H_\mathrm {max}\) is the maximum mean curvature of \(\partial \Omega \). The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities.

     

On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients

The gradient descent method minimizes an unconstrained nonlinear optimization problem with \({\mathcal {O}}(1/\sqrt{K})\), where K is the number of iterations performed by the gradient method. Traditionally, this analysis is obtained for smooth objective functions having Lipschitz continuous gradients. This paper aims to consider a more general class of nonlinear programming problems in which functions have Hölder continuous gradients. More precisely, for any function f in this class, denoted by \({{\mathcal {C}}}^{1,\nu }_L\), there is a \(\nu \in (0,1]\) and \(L>0\) such that for all \(\mathbf{x,y}\in {{\mathbb {R}}}^n\) the relation \(\Vert \nabla f(\mathbf{x})-\nabla f(\mathbf{y})\Vert \le L \Vert \mathbf{x}-\mathbf{y}\Vert ^{\nu }\) holds. We prove that the gradient descent method converges globally to a stationary point and exhibits a convergence rate of \({\mathcal {O}}(1/K^{\frac{\nu }{\nu +1}})\) when the step-size is chosen properly, i.e., less than \([\frac{\nu +1}{L}]^{\frac{1}{\nu }}\Vert \nabla f(\mathbf{x}_k)\Vert ^{\frac{1}{\nu }-1}\). Moreover, the algorithm employs \({\mathcal {O}}(1/\epsilon ^{\frac{1}{\nu }+1})\) number of calls to an oracle to find \({\bar{\mathbf{x}}}\) such that \(\Vert \nabla f({{\bar{\mathbf{x}}}})\Vert <\epsilon \).     

Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions,

$$\begin{aligned} Q^\Omega _\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text { on } \partial \Omega , \end{aligned}$$

in a class of bounded smooth domains \(\Omega \in \mathbb {R}^\nu \); here \(n\) is the outward unit normal and \(\alpha >0\) is a constant. We show that for each \(j\in \mathbb {N}\) and \(\alpha \rightarrow +\infty \), the \(j\)th eigenvalue \(E_j(Q^\Omega _\alpha )\) has the asymptotics

$$\begin{aligned} E_j(Q^\Omega _\alpha )=-\alpha ^2 -(\nu -1)H_\mathrm {max}(\Omega )\,\alpha +{\mathcal O}(\alpha ^{2/3}), \end{aligned}$$

where \(H_\mathrm {max}(\Omega )\) is the maximum mean curvature at \(\partial \Omega \). The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of \(H_\mathrm {max}\). In particular, we show that the ball is the strict minimizer of \(H_\mathrm {max}\) among the smooth star-shaped domains of a given volume, which leads to the following result: if \(B\) is a ball and \(\Omega \) is any other star-shaped smooth domain of the same volume, then for any fixed \(j\in \mathbb {N}\) we have \(E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )\) for large \(\alpha \). An open question concerning a larger class of domains is formulated.

     

Improved Moser–Trudinger inequality of Tintarev type in dimension <Emphasis Type="Italic">n</Emphasis> and the existence of its extremal functions

Let \(\Omega \) be a smooth bounded domain in \(\mathbb R^n\) with \(n\ge 2\), \(W^{1,n}_0(\Omega )\) be the usual Sobolev space on \(\Omega \) and define \(\lambda _1(\Omega ) = \inf \nolimits _{u\in W^{1,n}_0(\Omega )\setminus \{0\}}\frac{\int _\Omega |\nabla u|^n \mathrm{d}x}{\int _\Omega |u|^n \mathrm{d}x}\). Based on the blow-up analysis method, we shall establish the following improved Moser–Trudinger inequality of Tintarev type

$$\begin{aligned} \sup _{u\in W^{1,n}_0(\Omega ), \int _\Omega |\nabla u|^n \mathrm{{d}}x-\alpha \int _\Omega |u|^n \mathrm{{d}}x \le 1} \int _\Omega \exp (\alpha _{n} |u|^{\frac{n}{n-1}}) \mathrm{{d}}x < \infty , \end{aligned}$$

for any \(0 \le \alpha < \lambda _1(\Omega )\), where \(\alpha _{n} = n \omega _{n-1}^{\frac{1}{n-1}}\) with \(\omega _{n-1}\) being the surface area of the unit sphere in \(\mathbb R^n\). This inequality is stronger than the improved Moser–Trudinger inequality obtained by Adimurthi and Druet (Differ Equ 29:295–322, 2004) in dimension 2 and by Yang (J Funct Anal 239:100–126, 2006) in higher dimension and extends a result of Tintarev (J Funct Anal 266:55–66, 2014) in dimension 2 to higher dimension. We also prove that the supremum above is attained for any \(0< \alpha < \lambda _{1}(\Omega )\). (The case \(\alpha =0\) corresponding to the Moser–Trudinger inequality is well known.)

     

Eigenvalue Counting Function for Robin Laplacians on Conical Domains

We study the discrete spectrum of the Robin Laplacian \(Q^{\Omega }_\alpha \) in \(L^2(\Omega )\), \(u\mapsto -\Delta u, \quad D_n u=\alpha u \text { on }\partial \Omega \), where \(D_n\) is the outer unit normal derivative and \(\Omega \subset {\mathbb {R}}^{3}\) is a conical domain with a regular cross-section \(\Theta \subset {\mathbb {S}}^2\), n is the outer unit normal, and \(\alpha >0\) is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of \(Q^{\Omega }_\alpha \) is \(-\alpha ^2\) and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of \(Q^\Omega _\alpha \) is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues of \(Q^{\Omega }_\alpha \) in \((-\infty ,-\alpha ^2-\lambda )\), with \(\lambda >0\), behaves for \(\lambda \rightarrow 0\) as

$$\begin{aligned} \dfrac{\alpha ^2}{8\pi \lambda } \int _{\partial \Theta } \kappa _+(s)^2\mathrm {d}s +o\left( \frac{1}{\lambda }\right) , \end{aligned}$$

where \(\kappa _+\) is the positive part of the geodesic curvature of the cross-section boundary.

     

Existence of positive solution for nonlocal singular fourth order Kirchhoff equation with Hardy potential

This paper is concerned with the existence of positive solution to a class of singular fourth order elliptic equation of Kirchhoff type

$$\begin{aligned} \triangle ^2 u-\lambda M(\Vert \nabla u\Vert ^2)\triangle u-\frac{\mu }{\vert x\vert ^4}u=\frac{h(x)}{u^\gamma }+k(x)u^\alpha , \end{aligned}$$

under Navier boundary conditions, \(u=\triangle u=0\). Here \(\varOmega \subset {\mathbf {R}}^N\), \(N\ge 1\) is a bounded \(C^4\)-domain, \(0\in \varOmega \), h(x) and k(x) are positive continuous functions, \(\gamma \in (0,1)\), \(\alpha \in (0,1)\) and \(M:{\mathbf {R}}^+\rightarrow {\mathbf {R}}^+\) is a continuous function. By using Galerkin method and sharp angle lemma, we will show that this problem has a positive solution for \(\lambda > \frac{\mu }{\mu ^*m_0}\) and \(0<\mu <\mu ^*\). Here \(\mu ^*=\Big (\frac{N(N-4)}{4}\Big )^2\) is the best constant in the Hardy inequality. Besides, if \(\mu =0\), \(\lambda >0\) and h, k are Lipschitz functions, we show that this problem has a positive smooth solution. If \(h,k\in C^{2,\,\theta _0}(\overline{\varOmega })\) for some \(\theta _0\in (0,1)\), then this problem has a positive classical solution.

     

An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process

Let \(\{X(t):t\in \mathbb R_+\}\) be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, \(\mathbb E X(t) = 0, \mathbb E X^2(t) = 1\) and correlation function satisfying (i) \(r(t) = 1 - C|t|^{\alpha } + o(|t|^{\alpha })\) as \(t\rightarrow 0\) for some \(0\le \alpha \le 2\) and \(C>0\); (ii) \(\sup _{t\ge s}|r(t)|<1\) for each \(s>0\) and (iii) \(r(t) = O(t^{-\lambda })\) as \(t\rightarrow \infty \) for some \(\lambda >0\). For any \(n\ge 1\), consider n mutually independent copies of X and denote by \(\{X_{r:n}(t):t\ge 0\}\) the rth smallest order statistics process, \(1\le r\le n\). We provide a tractable criterion for assessing whether, for any positive, non-decreasing function \(f, \mathbb P(\mathscr {E}_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text { i.o.})\) equals 0 or 1. Using this criterion we find, for a family of functions \(f_p(t)\) such that \(z_p(t)=\mathbb P(\sup _{s\in [0,1]}X_{r:n}(s)>f_p(t))=O((t\log ^{1-p} t)^{-1})\), that \(\mathbb P(\mathscr {E}_{f_p})= 1_{\{p\ge 0\}}\). Consequently, with \(\xi _p (t) = \sup \{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}\), for \(p\ge 0\) we have \(\lim _{t\rightarrow \infty }\xi _p(t)=\infty \) and \(\limsup _{t\rightarrow \infty }(\xi _p(t)-t)=0\) a.s. Complementarily, we prove an Erdös–Révész type law of the iterated logarithm lower bound on \(\xi _p(t)\), namely, that \(\liminf _{t\rightarrow \infty }(\xi _p(t)-t)/h_p(t) = -1\) a.s. for \(p>1\) and \(\liminf _{t\rightarrow \infty }\log (\xi _p(t)/t)/(h_p(t)/t) = -1\) a.s. for \(p\in (0,1]\), where \(h_p(t)=(1/z_p(t))p\log \log t\).