Mean Value Inequality Research Articles

Gradient estimates for fundamental solutions of a Schrödinger operator on stratified Lie groups

Let L = − Δ G + Υ \mathcal L=-\Delta _{\mathbb {G}}+\Upsilon be a Schrödinger operator with a nonnegative potential Υ \Upsilon belonging to the reverse Hölder class B Q / 2 B_{Q/2} , where Q Q is the homogeneous dimension of the stratified Lie group G \mathbb {G} . Inspired by Shen’s pioneer work and Li’s work, we study fundamental solutions of the Schrödinger operator L \mathcal L on the stratified Lie group G \mathbb {G} in this paper. By proving an exponential decreasing variant of mean value inequality, we obtain the exponential decreasing upper estimates, the local Hölder estimates and the gradient estimates of the fundamental solutions of the Schrödinger operator L \mathcal L on the stratified Lie group. As two applications, we obtain the De Giorgi-Nash-Moser theory on the improved Hölder estimate for the weak solutions of the Schrödinger equation and a Liouville-type lemma for L \mathcal {L} -harmonic functions on G \mathbb {G} .

ON TWO-SIDED UNIDIRECTIONAL MEAN VALUE INEQUALITY IN A FRÉCHET SMOOTH SPACE

The paper is devoted to a new unidirectional mean value inequality for the Fréchet subdifferential of a continuous function. This mean value inequality finds an intermediate point and localizes its value both from above and from below; for this reason, the inequality is called two-sided. The inequality is considered for a continuous function defined on a Fréchet smooth space. This class of Banach spaces includes the case of a reflexive space and the case of a separable Asplund space. As some application of these inequalities, we give an upper estimate for the Fréchet subdifferential of the upper limit of continuous functions defined on a reflexive space.

On Nonlinear Boundary Value Problems for Differential Inclusions

We consider autonomous differential inclusions with nonlinear boundary conditions. Sufficient conditions for the existence of solutions in the class of absolutely continuous functions are obtained for these inclusions. It is shown that the corresponding existence theorem applies to the Cauchy problem and the antiperiodic boundary value problem. The result is used to derive a new mean value inequality for continuously differentiable functions.

Equivalence of minimax and viscosity solutions of path-dependent Hamilton–Jacobi equations

In the paper, we consider a path-dependent Hamilton–Jacobi equation with coinvariant derivatives over the space of continuous functions. Such equations arise from optimal control problems and differential games for time-delay systems. We study generalized solutions of the considered Hamilton–Jacobi equation both in the minimax and in the viscosity sense. A minimax solution is defined as a functional which epigraph and subgraph satisfy certain conditions of weak invariance, while a viscosity solution is defined in terms of a pair of inequalities for coinvariant sub- and supergradients. We prove that these two notions are equivalent, which is the main result of the paper. As a corollary, we obtain comparison and uniqueness results for viscosity solutions of a Cauchy problem for the considered Hamilton–Jacobi equation and a right-end boundary condition. The proof of the main result is based on a certain property of the coinvariant subdifferential. To establish this property, we develop a technique going back to the proofs of multidirectional mean-value inequalities. In particular, the absence of the local compactness property of the underlying continuous function space is overcome by using Borwein–Preiss variational principle with an appropriate gauge-type functional.

Parabolic mean value inequality and on-diagonal upper bound of the heat kernel on doubling spaces

Parabolic mean value inequality and on-diagonal upper bound of the heat kernel on doubling spaces

Uniqueness of subsolutions to the heat equation on Finsler measure spaces

AbstractLet $(M, F, m)$ be a forward complete Finsler measure space. In this paper, we prove that any nonnegative global subsolution in $L^p(M)(p>1)$ to the heat equation on $\mathbb R^+\times M$ is uniquely determined by the initial data. Moreover, we give an $L^p(0<p\leq 1)$ Liouville-type theorem for nonnegative subsolutions u to the heat equation on $\mathbb R\times M$ by establishing the local $L^p$ mean value inequality for u on M with Ric $_N\geq -K(K\geq 0)$ .

Global asymptotic synchronization of inertial memristive Cohen–Grossberg neural networks with proportional delays

In this article, the global asymptotic synchronization (GAS) using inertial memristive Cohen-Grossberg neural networks (IMCGNNs) with proportional delays (PDs) as drive–response systems is studied. By utilizing differential inclusion theory (DIT) and appropriate variable transformation, the studied systems can be converted to first-order differential systems. Both the feedback controller and the adaptive controller are designed, which are easy to implement in hardware. By constructing Lyapunov functionals and using mean value inequality analysis skills, two GAS criteria of the studied systems are gained, which are embodied in the form of algebraic inequalities and are easy to verify. Ultimately, the numerical examples with simulations are applied to sustain the obtained consequences.

Mean Value Inequalities for the Digamma Function

Mean Value Inequalities for the Digamma Function

New Masjed Jamei–Type Inequalities for Inverse Trigonometric and Inverse Hyperbolic Functions

In this paper, we establish two new inequalities of the Masjed Jamei type for inverse trigonometric and inverse hyperbolic functions and apply them to obtain some refinement and extension of Mitrinović–Adamović and Lazarević inequalities. The inequalities obtained in this paper go beyond the conclusions and conjectures in the previous literature. Finally, we apply the main results of this paper to the field of mean value inequality and obtain two new inequalities on Seiffert-like means and classical means.

Liouville type theorems and Hessian estimates for special Lagrangian equations

In this paper, we get a Liouville type theorem for the special Lagrangian equation with a certain ’convexity’ condition, where Warren–Yuan first studied the condition in (Comm Partial Differ Equ 33(4–6):922–932, 2008). Based on Warren–Yuan’s work, our strategy is to show a global Hessian estimate of solutions via the Neumann–Poincar\(\acute{\text {e}}\) inequality on special Lagrangian graphs, and mean value inequality for superharmonic functions on these graphs, where we need geometric measure theory. Moreover, we derive interior Hessian estimates on the gradient of the solutions to the equation with this ’convexity’ condition or with supercritical phase.

The dimensional estimates of exponential growth solutions to uniformly elliptic equations of non-divergence form

<p style='text-indent:20px;'>In this paper, we demonstrate that, for general second order uniformly elliptic equations of non-divergence form in unbounded cylinders with zero boundary condition, the space of solutions with a given exponential growth is of finite dimension. In particular, we generalize a result of Hang and Lin (1999) [<xref ref-type="bibr" rid="b4">4</xref>] in the non-divergence's setting by using a different approach. This allows us to consider the general lower order terms. Furthermore, a sharp estimate of the dimension is established by using the mean value inequality.</p>

A mean value inequality for the cyclic functions

Let be the cyclic function of order n and let be Stolarsky's one-parameter mean value family. We prove that for each n the inequality holds for all with 0<x<y if and only if .

Schrödinger heat kernel upper bounds on gradient shrinking Ricci solitons

In this paper we give new Gaussian type upper bounds for the Schrödinger heat kernel on complete gradient shrinking Ricci solitons with the scalar curvature bounded above. This result is a little broader than our earlier paper at some cases. The proof uses on a Davies type integral estimate and a local mean value inequality on gradient shrinking Ricci solitons.

Functional estimates and integral inequalities for the Fox–Wright function

In this paper, our aim is to show some mean value inequalities for the Fox–Wright function. In order to prove our main results, we present some monotonicity, convexity and concavity properties for some classes of functions related to the Fox–Wright function, which are in fact equivalents to the corresponding Turan type inequalities for this function. As a direct consequences, it deduces some new results involving some special functions, such as the generalized hypergeometric function, the four parameters Wright function and three parameter Mittag–Leffler type function. At the end of this paper, several integrals inequalities associated to the Fox–Wright function are established. As applications, new functional inequalities (such as Turan-type inequalities) for the incomplete Fox–Wright and incomplete generalized hypergeometric functions are established.

Hessian estimate for semiconvex solutions to the sigma-2 equation

We derive a priori interior Hessian estimates for semiconvex solutions to the sigma-2 equation. An elusive Jacobi inequality, a transformation rule under the Legendre-Lewy transform, and a mean value inequality for the still nonuniformly elliptic equation without area structure are the key to our arguments. Previously, this result was known for almost convex solutions.

Bubble tree convergence of conformally cross product preserving maps

We study a class of weakly conformal $3$-harmonic maps, called associative Smith maps, from $3$-manifolds into $7$-manifolds that parametrize associative $3$-folds in Riemannian $7$-manifolds equipped with $\mathrm{G}_2$-structures. Associative Smith maps are solutions of a conformally invariant nonlinear first order PDE system, called the Smith equation, that may be viewed as a $\mathrm{G}_2$-analogue of the Cauchy-Riemann system for $J$-holomorphic curves. In this paper, we show that associative Smith maps enjoy many of the same analytic properties as $J$-holomorphic curves in symplectic geometry. In particular, we prove: (i) an interior regularity theorem, (ii) a removable singularity result, (iii) an energy gap result, and (iv) a mean-value inequality. While our approach is informed by the holomorphic curve case, a number of nontrivial extensions are involved, primarily due to the degeneracy of the Smith equation. At the heart of above results is an $\varepsilon$-regularity theorem that gives quantitative $C^{1,\beta}$-regularity of $W^{1,3}$ associative Smith maps under a smallness assumption on the $3$-energy. The proof combines previous work on weakly $3$-harmonic maps and the observation that the associative Smith equation demonstrates a certain "compensation phenomenon" that shows up in many other geometric PDEs. Combining these analytical properties and the conformal invariance of the Smith equation, we explain how sequences of associative Smith maps with bounded $3$-energy may be conformally rescaled to yield bubble trees of such maps. When the $\mathrm{G}_2$-structure is closed, we prove that both the $3$-energy and the homotopy are preserved in the bubble tree limit. This result may be regarded as an associative analogue of Gromov's Compactness Theorem in symplectic geometry.

A chain of mean value inequalities

G = G(x, y) = ?xy, L = L(x,y) = x?y/log(x)?log(y)'' I=I(x,y)= 1/e(xx/yy) 1/(x-y), A=A(x.y)=x+y/2, be the geometric, logarithmic, identric, and arithmetic means of x and y. We prove that the inequalities L(G2,A2) &lt; G(L2,I2) &lt; A(L2,I2) &lt; I(G2,A2) are valid for all x, y &gt; 0 with x ? y. This refines a result of Seiffert.

Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below

ABSTRACT Given a smooth positive measure μ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on -forms. Thanks to an appropriate version of the Bochner–Kodaira–Nakano basic identity, we can give explicit geometric sufficient conditions for such coercivity to hold. Our results extend several known bounds in the literature to the case in which the manifold is neither assumed to be Kähler nor of ‘bounded geometry’. The key ingredients of our proof are a localization formula for the complex Laplacian (of the kind used in the theory of Schrödinger operators) and a mean value inequality for subsolutions of the heat equation on Riemannian manifolds due to Li, Schoen, and Tam. We also show in an appendix that the ‘twisted basic identities’, e.g. [McNeal JD and Varolin D. estimates for the operator. Bull Math Sci. 2015;5(2):179–249] are standard basic identities with respect to conformally Kähler metrics.

A generalized multidirectional mean value inequality and dynamic optimization

We consider a general calculus of variations problem with lower semicontinuous data which includes an optimal control problem with a dynamic differential inclusion constraint as a particular case. We provide a new approach to the derivation of necessary optimality conditions for such a problem, based on a new variant of the multidirectional mean-value inequality for smooth Banach spaces. The multidirectional mean-value inequality is a result in nonlinear and nonsmooth analysis which provides estimates for the minimum difference in function values between a closed convex subset of a Banach space E and a point .

Generalized Geodesic Convexity on Riemannian Manifolds

We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and relationships of these functions are discussed. A characterization for the existence of a global minimum point of a mathematical programming problem is presented. Moreover, a mean value inequality under geodesic log-preinvexity is extended to Cartan-Hadamard manifolds.