General Growth Conditions Research Articles

$L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions

We establish several existence, uniqueness and comparison results for $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under the assumptions that the generator $g$ satisfies a one-sided Osgood condition together with a very general growth condition in $y$, a uniform continuity condition and/or a sub-linear growth condition in $z$, and a generalized Mokobodzki condition for reflected BSDEs which relates the growth of $g$ and that of the barriers. This generalized Mokobodzki condition is proved to be necessary for existence of $L^{1}$ solutions of the reflected BSDEs. We also prove that the $L^{1}$ solutions of reflected BSDEs can be approximated by a penalization method and by some sequences of $L^{1}$ solutions of reflected BSDEs. The approach is based on a combination between existing methods, their refinement and perfection, but also on some novel ideas and techniques. These results strengthen some existing work on the $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs.

Global output feedback control of feedforward nonlinear time-delay systems with unknown output function

ABSTRACT This paper studies the problem of global output feedback control for a class of feedforward nonlinear systems with unknown multiple time-varying delays and unknown output function under more general uncertain growth condition. We allow that there exists unknown constant, input function and polynomial-of-output growth rates imposed on nonlinearities. By skilfully introducing dynamic gain method, backstepping method and Lyapunov-Krasovskii theorem, a delay-independent output feedback controller can be developed with only one dynamic gain to guarantee the boundedness of closed-loop system signals and the convergence of original system states and their corresponding observer states.

A general low-order partial regularity theory for asymptotically convex functionals with asymptotic dependence on the minimizer

We consider a minimizer $$u\in W^{1,p}(\Omega )$$ , where $$\Omega \subseteq \mathbb {R}^{n}$$ is a bounded open set, of the integral functional $$\begin{aligned} u\mapsto \int _{\Omega }f(x,u,Du)\ dx \end{aligned}$$ in the case when $$f : \Omega \subseteq \mathbb {R}^{n}\times \mathbb {R}^{N}\times \mathbb {R}^{N\times n}\rightarrow \mathbb {R}$$ is asymptotically related to a more regular function; since we assume that $$N\ge 2$$ , we study here the vectorial case. The asymptotical relatedness condition is such that dependence on u is allowed even as $$|\xi |\rightarrow +\infty $$ . Unlike in previous work f is allowed to satisfy a more general growth condition, which permits a coupling between the x and u variables. Due to the generality of the asymptotical relatedness condition this coupling induces a subtle restriction on the regularity required of the various functions in the growth condition, and we study these restrictions in this paper. In addition, here we do not obtain the full spectrum of Holder continuity for the minimizer, but rather a restricted range of Holder exponents that depend on the initial data.

Existence of a ground state solution for Choquard equation with the upper critical exponent

In this paper, we investigate the following Choquard equation −Δu+u=(Iα∗|u|N+αN−2)|u|N+αN−2−2u+g(u)inRN,where N⩾3,α∈(0,N), Iα is the Riesz potential and g is a given function. If g satisfies the general subcritical growth conditions, we obtain the existence of a positive ground state solution.

Nonuniformly elliptic energy integrals with [formula omitted]-growth

We study the local boundedness of minimizers of a nonuniformly energy integral of the form ∫Ωf(x,Dv)dx under p,q-growth conditions of the type λ(x)|ξ|p≤f(x,ξ)≤μ(x)1+|ξ|qfor some exponents q≥p>1 and with nonnegative functions λ,μ satisfying some summability conditions. We use here the original notation introduced in 1971 by Trudinger [26], where λ(x) and μ(x) had the role of the minimum and the maximum eigenvalues of an n×n symmetric matrix aijx and f(x,ξ)=∑i,j=1naijxξiξjwas the energy integrand associated to a linear nonuniformly elliptic equation in divergence form. In this paper we consider a class of energy integrals, associated to nonlinear nonuniformly elliptic equations and systems, with integrands f(x,ξ) satisfying the general growth conditions above.

$$\mathbb {L}^2$$ L 2 -solutions for reflected BSDEs with jumps under monotonicity and general growth conditions: a penalization method

In this paper, we study generalized reflected backward stochastic differential equations with a cadlag barrier, in a general filtration that supports a Brownian motion and an independent Poisson random measure. We give necessary and sufficient conditions for existence and uniqueness of \(\mathbb {L}^2\)-solutions for equations with generators monotone in y. We also prove that the solutions can be approximated via the penalization method. Furthermore, a comparison theorem is provided for such equations.

Generalized conditions of spherical carbonate concretion formation around decaying organic matter in early diagenesis

Isolated spherical carbonate concretions observed in marine sediments are fascinating natural objet trouve because of their rounded shapes and distinct sharp boundaries. They occur in varied matrices and often contain well preserved fossils. The formation process of such concretions has been explained by diffusion and rapid syn-depositional reactions with organic solutes and other pore water constituents. However, the rates, conditions and formation process of syngenetic spherical concretions are still not fully clear. Based on the examination of different kinds of spherical concretions from several locations in Japan, a diffusion based growth diagram was applied to define the generalized growth conditions of spherical concretions formed around decaying organic matter. All analytical data imply that the spherical concretions formed very rapidly, at least three to four orders of magnitude faster than previously estimated timescales. The values indicate that spherical concretions are preferentially grown within clay- to silt-grade marine sediments deposited in relatively deep (a few tens of metres) environments dominated by diffusive solute transport, very early in diagenesis.

Stochastic suppression and stabilisation of non‐linear hybrid delay systems with general one‐sided polynomial growth condition and decay rate

This study investigates the stochastic suppression and stabilisation of non-linear hybrid delay systems with general one-sided polynomial growth condition and decay rate. Given an unstable non-linear hybrid delay system x ˙ ( t ) = f ( x ( t ) , x ( t − τ ( t ) ) , r ( t ) , t ) with general one-sided polynomial growth condition, the authors introduce two independent Brownian noise and perturb this system into stochastic hybrid delay system d x ( t ) = f ( x ( t ) , x ( t − τ ( t ) ) , r ( t ) , t ) d t + h ( x ( t ) , r ( t ) , t ) d B 1 ( t ) + g ( x ( t ) , r ( t ) , t ) d B 2 ( t ) . It is shown that the non-linear diffusion term g ( x ( t ) , r ( t ) , t ) may suppress the potential explosion of hybrid delay system. Under a stronger condition, another linear diffusion term h ( x ( t ) , r ( t ) , t ) will make the perturbed stochastic hybrid delay system is almost surely stable with general decay rate.

Doubly Nonlinear Equations of Porous Medium Type

In this paper we prove the existence of solutions to doubly nonlinear equations whose prototype is given by $$\partial_t u^m- {\rm div}\, D_{\xi}\, f(x,Du) =0,$$ with $${m\in (0,\infty )}$$ , or more generally with an increasing and piecewise C1 nonlinearity b and a function f depending on u $$\partial_{t}b(u) - {\rm div}\, D_{\xi}\, f(x,u,Du)= -D_u f(x,u,Du).$$ For the function f we merely assume convexity and coercivity, so that, for instance, $${f(x,u,\xi)=\alpha(x)|\xi|^p + \beta(x)|\xi|^q}$$ with 1 < p < q and non-negative coefficients α, β with $${\alpha(x)+\beta(x)\geqq \nu > 0}$$ , and $${f(\xi)=\exp(\tfrac12|\xi|^2)}$$ are covered. Thus, for functions $${f(x,u,\xi )}$$ satisfying only a coercivity assumption from below but very general growth conditions from above, we prove the existence of variational solutions; mean while, if $${f(x,u,\xi )}$$ grows naturally when $${\left\vert \xi \right\vert \rightarrow +\infty }$$ as a polynomial of order p (for instance in the case of the p-Laplacian operator), then we obtain the existence of solutions in the sense of distributions as well as the existence of weak solutions. Our technique is purely variational and we treat both the cases of bounded and unbounded domains. We introduce a nonlinear version of the minimizing movement approach that could also be useful for the numerics of doubly nonlinear equations.

Lp (1 < p ⩽ 2) solutions of one-dimensional BSDEs whose generator is weakly monotonic in y and non-Lipschitz in z

ABSTRACTIn this paper, we start with establishing the existence of a minimal (maximal) Lp (1 < p ⩽ 2) solution to a one-dimensional backward stochastic differential equation (BSDE), where the generator g satisfies a p-order weak monotonicity condition together with a general growth condition in y and a linear growth condition in z. Then, we propose and prove a comparison theorem of Lp (1 < p ⩽ 2) solutions to one-dimensional BSDEs with q-order (1 ⩽ q < p) weak monotonicity and uniform continuity generators. As a consequence, an existence and uniqueness result of Lp (1 < p ⩽ 2) solutions is also given for BSDEs whose generator g is q-order (1 ⩽ q < p) weakly monotonic with a general growth in y and uniformly continuous in z.

Adaptive output feedback control for a class of large‐scale output‐constrained non‐linear time‐delay systems

This study is concerned with the adaptive stabilisation problem for a class of large-scale output-constrained non-linear time-delay systems in strict-feedback form. The investigative system satisfies a more general growth condition and the precise knowledge of the time delay is not required to be known a priori. Firstly, an observer is designed by the dynamic gain control design technique. Then, based on a novel Lyapunov-Krasovskii functional, the output constraints are satisfied and the stability of the closed-loop control system is achieved. Finally, a numerical example is given to demonstrate the effectiveness of the proposed design procedure.

A note on explosion suppression for nonlinear delay differential systems by polynomial noise

Liu and Shen discussed the role of stochastic suppression on the explosive solution by a polynomial noise for a deterministic differential system satisfying a general polynomial growth condition. They further showed that the global solution of the corresponding perturbed system grows at most polynomially. However, the estimation of the asymptotic property of polynomial growth is rough, and we see the necessity to develop a more accurate estimation which is the main motivation of the present paper. As to the existence of time delays, we aim to discuss the stochastic roles of the polynomial noise for a deterministic delay differential system with the general polynomial growth condition. We show that a properly chosen polynomial stochastic noise not only can guarantee the existence and uniqueness of the global solution of the stochastically perturbed delay differential system, but also can make almost every sample path of the global solution grow at most with polynomial rate and even decay to the zero solution exponentially.

Relaxation of nonconvex unbounded integrals with general growth conditions in Cheeger–Sobolev spaces

We study relaxation of nonconvex integrals of the calculus of variations in the setting of Cheeger–Sobolev spaces when the integrand does not have polynomial growth and can take infinite values.

A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

We study radial solutions of the semilinear elliptic equationΔu+f(u)=0 under rather general growth conditions on f. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole–Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., Δu+up=0 and Δu+eu=0.

Local higher integrability of the gradient of a quasiminimizer under generalized Orlicz growth conditions

We study local quasiminimizers of the Dirichlet energy under generalized growth conditions. Special cases include standard, variable exponent and double phase growths. We show that the gradient of a local quasiminimizer has local higher integrability.

Lp solutions of anticipated BSDEs with weak monotonicity and general growth generators

ABSTRACTIn this article, we establish the stability theorems of Lp(1 < p ⩽ 2) solutions for multidimensional anticipated backward stochastic differential equations (BSDEs), in which the generator g is p-order weak monotonic in y and Lipschitz continuous in (z, η, ϑ). Moreover, we present the existence and uniqueness of Lp solutions for this kind of anticipated BSDEs with the help of stability theorems when generator g also satisfies general growth condition in y.

Controlling Growth High Uniformity Indium Selenide (In2Se3) Nanowires via the Rapid Thermal Annealing Process at Low Temperature

High uniformity Au-catalyzed indium selenide (In2Se3) nanowires are grown with the rapid thermal annealing (RTA) treatment via the vapor-liquid-solid (VLS) mechanism. The diameters of Au-catalyzed In2Se3 nanowires could be controlled with varied thicknesses of Au films, and the uniformity of nanowires is improved via a fast pre-annealing rate, 100 °C/s. Comparing with the slower heating rate, 0.1 °C/s, the average diameters and distributions (standard deviation, SD) of In2Se3 nanowires with and without the RTA process are 97.14 ± 22.95 nm (23.63%) and 119.06 ± 48.75 nm (40.95%), respectively. The in situ annealing TEM is used to study the effect of heating rate on the formation of Au nanoparticles from the as-deposited Au film. The results demonstrate that the average diameters and distributions of Au nanoparticles with and without the RTA process are 19.84 ± 5.96 nm (30.00%) and about 22.06 ± 9.00 nm (40.80%), respectively. It proves that the diameter size, distribution, and uniformity of Au-catalyzed In2Se3 nanowires are reduced and improved via the RTA pre-treated. The systemic study could help to control the size distribution of other nanomaterials through tuning the annealing rate, temperatures of precursor, and growth substrate to control the size distribution of other nanomaterials.Graphical Rapid thermal annealing (RTA) process proved that it can uniform the size distribution of Au nanoparticles, and then it can be used to grow the high uniformity Au-catalyzed In2Se3 nanowires via the vapor-liquid-solid (VLS) mechanism. Comparing with the general growth condition, the heating rate is slow, 0.1 °C/s, and the growth temperature is a relatively high growth temperature, > 650 °C. RTA pre-treated growth substrate can form smaller and uniform Au nanoparticles to react with the In2Se3 vapor and produce the high uniformity In2Se3 nanowires. The in situ annealing TEM is used to realize the effect of heating rate on Au nanoparticle formation from the as-deposited Au film. The byproduct of self-catalyzed In2Se3 nanoplates can be inhibited by lowering the precursors and growth temperatures.

Stochastic suppression and almost surely stabilization of non-autonomous hybrid system with a new general one-sided polynomial growth condition

This paper investigates the stochastic suppression and almost surely stabilization of non-autonomous hybrid system with a new general one-sided polynomial growth condition. Given an unstable non-autonomous hybrid system dy(t)=f(y(t),r(t),t)dt with a new general one-sided polynomial growth condition, we introduce two independent Brownian noise and perturb this system into stochastic hybrid system dx(t)=f(x(t),r(t),t)dt+q(r(t))δ(t)x(t)dB1(t)+σ(r(t))δ(t)|x(t)|βx(t)dB2(t). It shows that the polynomial form of Brownian noise may suppress the potential explosion of hybrid system. Under a stronger condition, another linear Brownian noise will make the perturbed stochastic hybrid system is almost surely stable with general decay rate.

Hölder continuity of parabolic quasi-minimizers

We are concerned with scalar valued local parabolic quasi-minimizers u associated to a Carathéodory integrand f obeying p-growth assumptions for p>1. For such parabolic quasi-minimizers we prove local Hölder continuity provided they are already locally bounded. The superquadratic case p≥2 has already been considered in [17,22], whereas the subquadratic case could, at least to our knowledge, not be treated in the past. In this paper we are able to handle both the sub- and superquadratic case under more general growth conditions.

A representation theorem for generators of BSDEs with general growth generators in [formula omitted] and its applications

In this paper we first prove a general representation theorem for generators of backward stochastic differential equations (BSDEs for short) by utilizing a localization method involved with stopping time tools and approximation techniques, where the generators only need to satisfy a weak monotonicity condition and a general growth condition in y and a Lipschitz condition in z. This result basically solves the problem of representation theorems for generators of BSDEs with general growth generators in y. Then, such representation theorem is adopted to prove a probabilistic formula, in viscosity sense, of semilinear parabolic PDEs of second order. The representation theorem approach seems to be a potential tool to the research of viscosity solutions of PDEs.