Elliptic Measures and Square Function Estimates on 1-Sided Chord-Arc Domains

In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $$L^p$$ L p , for some finite p, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $$A_\infty $$ A ∞ . In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors–David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. In addition, we obtain that for two given elliptic operators $$L_1$$ L 1 and $$L_2$$ L 2 , the absolute continuity of the surface measure with respect to the elliptic measure of $$L_1$$ L 1 is equivalent to the same property for $$L_2$$ L 2 provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally, for the case on which $$L_2$$ L 2 is either the transpose of $$L_1$$ L 1 or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex.

A Feature Track Correction (FTC) Observation Operator applied to Aeolus-AMV Collocations

<p>In this study we propose and test a feature track correction (FTC) observation operator for atmospheric motion vectors (AMVs).  The FTC has four degrees of freedom corresponding to wind speed multiplicative and additive corrections (γ and δ<em><strong>V</strong></em>), a vertical height assignment correction (<em>h</em>), and an estimate of the depth of the layer that contributes to the AMV (Δ<em>z</em>).  Since the effect of the FTC observation operator is to add a bias correction to a weighted average of the profile of background winds an alternate formulation is in terms of a profile of weights (<em>w<sub>k</sub></em>) and δ<em><strong>V </strong></em>.</p><p>The FTC observation operator is tested in the context of a collocation study between AMVs projected onto the collocated Aeolus horizontal line-of-sight (HLOS) and the Aeolus HLOS wind profiles.  This is a prototype for an implementation in a variational data assimilation system and here the Aeolus profiles act as the background in the FTC observation operator.  Results were obtained for ten days of data using modest QC.  The overall OMB or collocation difference SD for a global solution applied to the independent sample is 5.49 m/s with negligible mean.  For comparison the corresponding simple (or pure) collocation SD is 7.85 m/s, and the null solution, which only interpolates the Aeolus profile to the reported height of the AMV and removes the overall bias, has an OMB SD of 7.23 m/s. These values correspond to reductions of variance of 51.0% and 42.3%, due to the FTC observation operator in comparison to the simple collocation and null solution, respectively.</p><p>These preliminary tests demonstrate the potential for the FTC observation operator for </p><ul><li>Improving AMV collocations (including triple collocation) with profile wind data.</li> <li>Characterizing AMVs. For example, summary results for the HLOS winds show that AMVs compare best with wind profiles averaged over a 4.5 km layer centered 0.5 km above the reported AMV height.</li> <li>Improving AMV observation usage within data assimilation (DA) systems. Lower estimated error and more realistic representation of AMVs with variational FTC (VarFTC) should result in greater information extracted.  The FTC observation operator accomplishes this by accounting for the effects of <em>h</em> and Δ<em>z</em>. </li> </ul>

Local and Global Stability Analysis for Gilpin-Ayala Competition Model Involved in Harmful Species Via LMI Approach and Variational Methods

Firstly, the author do dynamic analysis for reaction-diffusion Gilpin-Ayala competition model with Dirichlet boundary value, involved in harmful species. Existence of multiple stationary solutions is verified by way of Mountain Pass lemma, and the local stability result of the null solution is obtained by employing linear approximation principle. Secondly, the author utilize variational methods and LMI technique to deduce the LMI-based global exponential stability criterion on the null solution which becomes the unique stationary solution of the ecosystem with delayed feedback under a reasonable boundedness assumption on population densities. Particularly, LMI criterion is involved in free weight coefficient matrix, which reduces the conservatism of the algorithm. In addition, a new impulse control stabilization criterion is also derived. Finally, two numerical examples show the effectiveness of the proposed methods. It is worth mentioning that the obtained stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria.

Local and Global Stability Analysis for Gilpin-Ayala Competition Model Involved in Harmful Species Via LMI Approach and Variational Methods

In this paper, stability of reaction-diffusion Gilpin-Ayala competition model with Dirichlet boundary value, involved in harmful species, was investigated. Employing Mountain Pass Lemma and linear approximation principle results in the local stability criterion of the null solution of the ecosystem which owns at least three stationary solutions. On the other hand, globally asymptotical stability criterion for the null solution of the ecosystem was derived by variational methods and LMI approach. It is worth mentioning that the stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria. Finally, two numerical examples show the effectiveness of the proposed methods.

On The Uniqueness of Solutions for Kawahara Type Equations

In this paper we extend to Kawahara type equations a uniqueness result obtained by C. E. Kenig, G. Ponce, and L. Vega for KdV type equations. We prove that, under certain decay's conditions, the null solution is the unique solution.

On the Stability of Volterra Difference Equations of Convolution Type

In \cite{Elaydi-10}, S.\ Elaydi obtained a characterization of the stability ofthe null solution of the Volterra difference equation\beqaex_n=\sum_{i=0}^{n-1} a_{n-i} x_i\textrm{,}\quad n\geq 1\textrm{,}\eeqaeby localizing the roots of its characteristic equation\beqae1-\sum_{n=1}^{\infty}a_nz^n=0\textrm{.}\eeqaeThe assumption that $(a_n)\in\ell^1$ was the single hypothesis considered for the validity of that characterization, which is an insufficient condition if theratio $R$ of convergence of the power series of the previous equation equals one. In fact, when $R=1$, this characterization conflicts with a result obtainedby Erd\"os et al in \cite{Erdos}. Here, we analyze the $R=1$ case and show thatsome parts of that characterization still hold. Furthermore, studies on stability for the $R<1$ case are presented. Finally, we state some new results related to stability via finite approximation.

Probabilistic Basin of Attraction and Its Estimation Using Two Lyapunov Functions

We study stability for dynamical systems specified by autonomous stochastic differential equations of the form dX(t)=f(X(t))dt+g(X(t))dW(t), with (X(t))t≥0 an Rd-valued Itô process and (W(t))t≥0 an RQ-valued Wiener process, and the functions f:Rd→Rd and g:Rd→Rd×Q are Lipschitz and vanish at the origin, making it an equilibrium for the system. The concept of asymptotic stability in probability of the null solution is well known and implies that solutions started arbitrarily close to the origin remain close and converge to it. The concept therefore pertains exclusively to system properties local to the origin. We wish to address the matter in a more practical manner: Allowing for a (small) probability that solutions escape from the origin, how far away can they then be started? To this end we define a probabilistic version of the basin of attraction, the γ-BOA, with the property that any solution started within it stays close and converges to the origin with probability at least γ. We then develop a method using a local Lyapunov function and a nonlocal one to obtain rigid lower bounds on γ-BOA.

Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over {\mathbb{R}^{d}} and in {L^{p}} -spaces with respect to tight evolution systems of measures. Here, the linear part of the equation is a nonautonomous second-order elliptic operator with unbounded coefficients defined in {I\times\mathbb{R}^{d}} , (I being a right-halfline). To the above Cauchy problem we associate a nonlinear evolution operator, which we study in detail, proving some summability improving properties. We also study the stability of the null solution to the Cauchy problem.

Uniqueness of the null solution to a nonlinear partial differential equation satisfied by the explosion probability of a branching diffusion

The explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.