Positive solutions for discontinuous problems with applications to $$\phi $$-Laplacian equations

We establish existence and localization of positive solutions for general discontinuous problems for which a Harnack-type inequality holds. In this way, a wide range of ordinary differential problems such as higher order boundary value problems or \(\phi \)-Laplacian equations can be treated. In particular, we study the Dirichlet–Neumann problem involving the \(\phi \)-Laplacian. Our results rely on Bohnenblust–Karlin fixed point theorem which is applied to a multivalued operator defined in a product space.     

Global gradient estimate on graph and its applications

Continuing our previous work(ar Xiv:1509.07981v1),we derive another global gradient estimate for positive functions,particularly for positive solutions to the heat equation on finite or locally finite graphs.In general,the gradient estimate in the present paper is independent of our previous one.As applications,it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs.These global gradient estimates can be compared with the Li–Yau inequality on graphs contributed by Bauer et al.[J.Differential Geom.,99,359–409(2015)].In many topics,such as eigenvalue estimate and heat kernel estimate(not including the Liouville type theorems),replacing the Li–Yau inequality by the global gradient estimate,we can get similar results.     

On quasi $$\epsilon $$-solution for robust convex optimization problems

This paper devotes to the quasi \(\epsilon \)-solution (one sort of approximate solutions) for a robust convex optimization problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we establish approximate optimality theorem and approximate duality theorems in term of Wolfe type on quasi \(\epsilon \)-solution for the robust convex optimization problem. Moreover, some examples are given to illustrate the obtained results.     

Quasilinear elliptic equations on $$\mathbb{R}^N $$with infinitely many solutions

We give verifiable conditions ensuring that second order quasilinear elliptic equations on have infinitely many solutions in the Sobolev space for generic right-hand sides. This amounts to translating in concrete terms the more elusive hypotheses of an abstract theorem. Salient points include the proof that a key denseness property is equivalent to the existence of nontrivial solutions to an auxiliary problem, and an estimate of the size of the set of critical points of nonlinear Schrödinger operators. Conditions for the real-analyticity of Nemytskii operators are also discussed.     

Classification and refined singularity of positive solutions for nonlinear Maxwell equations arising in mesoscopic electromagnetism

This paper is a continuation of [13], where we studied the existence and other analytic properties of positive radial solutions for a system of nonlinear Maxwell equations in the plane R²${\mathbb{R}}^2$, which arises in the modelling of mesoscopic scale electromagnetic phenomena. In this paper we derive local estimates of singular positive solutions, based on which a classification theorem of general positive solutions is established. The refined singularity of general positive solutions is also investigated by employing the theory of infinite dimensional dynamical systems.     

A Korovkin type approximation theorems via $$\mathcal{I}$$ -convergence

Using the concept of -convergence we provide a Korovkin type approximation theorem by means of positive linear operators defined on an appropriate weighted space given with any interval of the real line. We also study rates of convergence by means of the modulus of continuity and the elements of the Lipschitz class.     

Hamilton's gradient estimate for the heat kernel on complete manifolds

In this paper we extend a gradient estimate of R. Hamilton for positive solutions to the heat equation on closed manifolds to bounded positive solutions on complete, non-compact manifolds with . We accomplish this extension via a maximum principle of L. Karp and P. Li and a Berstein-type estimate on the gradient of the solution. An application of our result, together with the bounds of P. Li and S.T. Yau, yields an estimate on the gradient of the heat kernel for complete manifolds with non-negative Ricci curvature that is sharp in the order of for the heat kernel on .

     

Gradient estimates on the weighted p-Laplace heat equation

In this paper, by a regularization process we derive new gradient estimates for positive solutions to the weighted p-Laplace heat equation when the m-Bakry–Émery curvature is bounded from below by ?K for some constant $K\ge 0$. When the potential function is constant, which reduce to the gradient estimate established by Ni and Kotschwar for positive solutions to the p-Laplace heat equation on closed manifolds with nonnegative Ricci curvature if $K\searrow 0$, and reduce to the Davies, Hamilton and Li–Xu's gradient estimates for positive solutions to the heat equation on closed manifolds with Ricci curvature bounded from below if $p=2$.     

New Characterizations of the Clifford Torus as a Lagrangian Self-Shrinker

In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus \({\mathbb {S}}^1(1)\times {\mathbb {S}}^1(1)\) is the unique compact orientable Lagrangian self-shrinker in \({\mathbb {C}}^2\) with \(|A|^2\le 2\), which gives an affirmative answer to Castro–Lerma’s conjecture in Castro and Lerma (Int Math Res Not 6:1515–1527; 2014). We also prove that the Clifford torus is the unique compact orientable embedded Lagrangian self-shrinker with nonnegative or nonpositive Gauss curvature in \({\mathbb {C}}^2\).     

The Raising Steps Method: Applications to the $$\bar{\partial }$$ Equation in Stein Manifolds

In order to have estimates on the solutions of the equation \(\bar{\partial }u=\omega \) on a Stein manifold, we introduce a new method, the “raising steps method”, to get global results from local ones. In particular, it allows us to transfer results from open sets in \({\mathbb {C}}^{n}\) to open sets in a Stein manifold. Using it, we get \(\displaystyle L^{r}-L^{s}\) results for solutions of the equation \(\bar{\partial }u=\omega \) with a gain, \(\displaystyle s>r\), in strictly pseudo convex domains in Stein manifolds. We also get \(\displaystyle L^{r}-L^{s}\) results for domains in \({\mathbb {C}}^{n}\) locally biholomorphic to convex domains of finite type.     

Complement of gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds

In this paper, along the idea of Souplet and Zhang, we deduce a local elliptic‐type gradient estimates for positive solutions of the nonlinear parabolic equation: on for α ≥ 1 and α ≤ 0. As applications, related Liouville‐type theorem is exported. Our results are complement of known results. Copyright © 2016 John Wiley & Sons, Ltd.     

Gradient estimates for ut=ΔF(u) on manifolds and some Liouville-type theorems

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations:$u_t=\mathrm{\Delta }F(u),$ with $F^{\prime }(u)>0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME):$u_t=\mathrm{\Delta }\left(u^p\right),p>0,$ and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of p than that for Aronson–Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yau?s celebrated Liouville theorem for positive harmonic functions.     

Existence of periodic solutions for a class of first order delay differential equations dealing with vectors

By the weak linking theorem and the local linking theorem, we study the existence of periodic solutions for the following delay non-autonomous systems

(1)     

A Fixed Point Theorem for Systems of Nonlinear Operator Equations and Applications to $$(p_1, p_2)$$-Laplacian System

A new fixed point theorem for systems of nonlinear operator equations is established by means of topological degree theory and positively 1-homogeneous operator, where the components has a positively 1-homogeneous majorant or minorant. As applications, the existence of positive solutions for \((p_1, p_2)\)-Laplacian system is considered under some conditions concerning the positive eigenvalues corresponding to the relevant positively 1-homogeneous operators.     

Actions of the groups $${\mathbb C}$$ and $${\mathbb C^*}$$ on Stein varieties

In this paper we present recent results concerning global aspects of and -actions on Stein surfaces. Our approach is based on a byproduct of techniques from Geometric Theory of Foliations (holonomy, stability), Potential theory (parabolic Riemann surfaces, Riemann-Koebe Uniformization theorem) and Several Complex Variables (Hartogs’ extension theorems, Theory of Stein spaces). Our main motivation comes from the original works of M. Suzuki and Orlik-Wagreich. Some of their results are extended to a more general framework. In particular, we prove some linearization theorems for holomorphic actions of and on normal Stein analytic spaces of dimension two. We also add a list of questions and open problems in the subject. The underlying idea is to present the state of the art of this research field.      

Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds

In this paper, we study the local gradient estimate for the positive solution to the following equation:

     

$$\text{ Pin }^-(2)$$-monopole equations and intersection forms with local coefficients of four-manifolds

We introduce a variant of the Seiberg-Witten equations, \(\text{ Pin }^-(2)\)-monopole equations, and give its applications to intersection forms with local coefficients of four-manifolds. The first application is an analogue of Froyshov’s results on four-manifolds with definite intersection forms with local coefficients. The second is a local coefficient version of Furuta’s \(10/8\)-inequality. As a corollary, we construct nonsmoothable spin four-manifolds satisfying Rohlin’s theorem and the \(10/8\)-inequality.     

Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds

We derive a sharp, localized version of elliptic type gradientestimates for positive solutions (bounded or not) to the heatequation. These estimates are related to the Cheng–Yauestimate for the Laplace equation and Hamilton's estimate forbounded solutions to the heat equation on compact manifolds.As applications, we generalize Yau's celebrated Liouville theoremfor positive harmonic functions to positive ancient (includingeternal) solutions of the heat equation, under certain growthconditions. Surprisingly this Liouville theorem for the heatequation does not hold even in Rⁿ without such a condition.We also prove a sharpened long-time gradient estimate for thelog of the heat kernel on noncompact manifolds. 2000 MathematicsSubject Classification 35K05, 58J35.     

A generalized elastic net regularization with smoothed $$\ell _{q}$$ penalty for sparse vector recovery

In this paper, we propose an iterative algorithm for solving the generalized elastic net regularization problem with smoothed \(\ell _{q} (0<q \le 1)\) penalty for recovering sparse vectors. We prove the convergence result of the algorithm based on the algebraic method. Under certain conditions, we show that the iterative solutions converge to a local minimizer of the generalized elastic net regularization problem and we also present an error bound. Theoretical analysis and numerical results show that the proposed algorithm is promising.