General Growth Conditions Research Articles

Partial Lipschitz regularity of minimizers of asymptotically convex functionals with general growth conditions

We consider the partial Lipschitz continuity of minimizers of functionals of the formv↦∫Ωf(x,v,Dv)dx, where Ω⊆Rn is open and bounded. The map (x,u,ξ)↦f(x,u,ξ) appearing in the above functional is assumed to be Hölder continuous with respect to its first two arguments, of class C2 in its third argument, and asymptotically related to the more regular map (x,u,ξ)↦a(x,u)G(ξ) as |ξ|→+∞. The main novelty is the allowance that, even as |ξ|→+∞, the map f may retain full dependence on the minimizer u, not only its gradient Du and the spatial coordinate x.

Global [formula omitted]-estimates and Hölder continuity of weak solutions to elliptic equations with the general nonstandard growth conditions

We study the elliptic equations in divergence form with the general nonstandard growth conditions involving Lebesgue measurable functions on Ω. In this paper we prove global L∞-estimates and Hölder continuity of weak solutions for these equations. Our results improve the one in Giusti (2003) even for the case when p(x)≡p and q(x)≡q are constants.

Existence, Uniqueness and Stability of $$L^1$$ L 1 Solutions for Multidimensional Backward Stochastic Differential Equations with Generators of One-Sided Osgood Type

We establish a general existence and uniqueness result of $$L^1$$ solution for a multidimensional backward stochastic differential equation (BSDE for short) with generator g satisfying a one-sided Osgood condition as well as a general growth condition in y, and a Lipschitz condition together with a sublinear growth condition in z, which improves some existing results. In particular, we put forward and prove a stability theorem of the $$L^1$$ solutions for the first time. A new type of $$L^1$$ solution is also investigated. Some delicate techniques involved in the relationship between convergence in $$L^1$$ and in probability and dividing appropriately the time interval play crucial roles in our proofs.

Hölder regularity of quasiminimizers under generalized growth conditions

We prove Harnack’s inequality for local (quasi)minimizers in generalized Orlicz spaces without polynomial growth or coercivity conditions. As a consequence, we obtain the local Holder continuity of local (quasi)minimizers. The results include as special cases standard, variable exponent and double phase growth.

Ground state solution for a class of Schrödinger equations involving general critical growth term

In this paper, we study a class of Schrödinger equations−△u=k(u),x∈RN,where and k satisfies very general critical growth conditions. By using the Pohozaev constraint, we obtain a positive ground state solution which is radially symmetric.

Hölder Continuity of Homeomorphisms with Controlled Growth of Their Spherical Means

We continue the study of homeomorphisms preserving integrally controlled weighted p-module of the ring domains. It was established earlier that under appropriate growth condition for the spherical mean of the weight such mappings are locally Holder continuous with respect to logarithms of the distances. In this paper, we consider much more general growth conditions and derive the differentiability almost everywhere, local Lipschitz and Holder continuity. The sharpness of these results is illustrated by several examples. The distortion estimates for measures under such mappings are also established.

Existence, uniqueness and approximation for L p solutions of reflected BSDEs with generators of one-sided Osgood type

We prove several existence and uniqueness results for Lp (p > 1) solutions of reflected BSDEs with continuous barriers and generators satisfying a one-sided Osgood condition together with a general growth condition in y and a uniform continuity condition or a linear growth condition in z. A necessary and sufficient condition with respect to the growth of barrier is also explored to ensure the existence of a solution. And, we show that the solutions may be approximated by the penalization method and by some sequences of solutions of reflected BSDEs. These results are obtained due to the development of those existing ideas and methods together with the application of new ideas and techniques, and they unify and improve some known works.

A Harnack inequality for the quasi-minima of scalar integral functionals with general growth conditions

In this paper we proof a Harnack inequality and a regularity theorem for quasi-minima of scalar integral functionals with general growth conditions.

Stabilization and destabilizationviatime-varying noise for uncertain nonlinear systems

This paper considers the stochastic stabilization and destabilization for uncertain nonlinear systems. Remarkably, the systems in question allow serious parameter unknowns (which don’t belong to any known constant set) and serious time-variations, and possess more general growth conditions than those in the related existing literature. The former feature makes the time-invariant scheme inapplicable, and a time-varying one is proposed, mainly to compensate the serious parameter unknowns, as well as serious time-variations. First, a time-varying stochastic noise is successfully constructed to super-exponentially stabilize the special but representative case without adverse serious time-variations. Then, for the general case and general decay rate, it suffices to find a fast enough time-varying gain for the stochastic noise. Moreover, by a time-varying method, the stochastic destabilization with general growth rate is also achieved for uncertain nonlinear systems.

Subdifferentiation of Regularized Functions

We study the Moreau regularization process for functions satisfying a general growth condition on general Banach spaces. We give differentiability criteria and we study the relationships between the subdifferentials of the function and the subdifferentials of its approximations. We also consider the Lasry-Lions process.

Existence of solutions to one-dimensional BSDEs with semi-linear growth and general growth generators

This paper is devoted to solving a one-dimensional backward stochastic differential equation (BSDE in short) when the generator g has a semi-linear growth and a general growth in (y,z). This condition is not only strictly weaker than the linear growth condition of g in (y,z), but also the (weak) monotonicity and general growth condition of g in y together with the linear growth condition of g in z. We establish, in this setting, three existence results on a solution and the minimal (maximal) solution to the BSDE, where the generator g may be discontinuous in y. These results virtually unify and improve some existing results.

Positive Solutions for Nonlinear Nonhomogeneous Robin Problems

We consider a nonlinear, nonhomogeneous Robin problem with a Carathéodory reaction which satisfies certain general growth conditions near 0^+ and near + \infty . We show the existence and regularity of positive solutions, the existence of a smallest positive solution and under an additional condition on the reaction, we show the uniqueness of the positive solutions. We then show that our setting incorporates certain parametric Robin equations of interest such as nonlinear equidiffusive logistic equations.

Organic Research and Development in Denmark (1996-2010) – Effects on the Organic Sector and Society

The ICROFS (International Centre for Research in Organic Food Systems) has conducted an analysis of the effects of organic research in Denmark (1996-2010) on the Danish organic sector and on society in general. Over these 15 years, three national programs and one program with European collaboration have been implemented in Denmark, financed via special government grants that amounted to just over 500 million DKK (approx. € 67 million—or approximately $ 80 million). The analysis itself was carried out as a compilation of information from three perspectives, each of which has been independently documented: • Interviews with (representatives of) end-users of results from research and development (R&D) investigating their assessment of the challenges in the sector and solutions developed from 1996-2010 • Assessment of the R&D endeavours in different thematic areas (dairy/milk, pigs, crops, etc.) as they related to end-users and the stated challenges at that time • Documentation of the dissemination of R&D results in relation to themes and challenges in the sector The results showed very good correspondence between end-users’ perceptions of the challenges overcome in the sector, the R&D initiated in the research programmes, and the dissemination of research results and other forms of knowledge transfer. The analysis documented direct effects of the research initiatives targetting the challenges in the sector such as higher yields, weed and pest control, animal health and welfare, the potential for phasing out the use of antibiotics in Danish dairy herds and reducing the problems caused by seedborne diseases. It also describes where research did not contribute as much to overcoming challenges. In contrast, the analysis showed that the effects of the research in the organic processing industry and among relevant governmental and non-governmental organisations were of a more indirect character. Research has helped stabilize the supply and quality of raw materials at a time of growing demand and sales. Organic research also generates new knowledge and leads to new opportunities that can provide inspiration for a green conversion, product diversification and growth also in conventional agriculture. The analysis showed that research under the national research programs overall have been very applied and directed at the barriers in the sector in order to support the general market and growth conditions for the organic sector. Having laid a solid foundation, the private sector has been able to take advantage of commercial opportunities when demand grew, while adhering to the organic policy objectives of the market-driven growth in the organic sector.

Formula omitted] solutions of multidimensional BSDEs with weak monotonicity and general growth generators

In this paper, we first establish the existence and uniqueness of Lp(p>1) solutions for multidimensional backward stochastic differential equations (BSDEs) under a weak monotonicity condition together with a general growth condition in y for the generator g. Then, we overview several conditions related closely to the weak monotonicity condition and compare them in an effective way. Finally, we put forward and prove a stability theorem and a comparison theorem of Lp(p>1) solutions for this kind of BSDEs.

Noise suppresses explosive solutions of differential systems: A new general polynomial growth condition

Recently, Liu and Shen (2012) have discussed the problem of suppression explosive solutions by noise for nonlinear deterministic differential system with coefficients satisfying a general one-sided polynomial growth condition. Liu and Shen introduced the Brownian noise feedback |x(t)|β∑x(t)dB(t) to the nonlinear differential system, and they showed that the corresponding stochastic system has a unique global solution. However, Liu and Shen's condition is still very restrictive and many more general nonlinear differential systems fail. This paper aims to introduce a new more general polynomial growth condition which can be applied to more nonlinear differential systems. Under the new polynomial growth condition, we show that the considered stochastic system has a unique global solution although the corresponding deterministic system may explode in a finite time. Moreover, we show that the flexible stochastic feedback g(x(t)) can guarantee the solution not only be limited in the sense of moment but also grows at most polynomially. Finally, we provide two examples to illustrate superiority of the new general polynomial growth condition.

Local Boundedness of Minimizers with Limit Growth Conditions

The energy integral of the calculus of variations, which we consider in this paper, has a limit behavior when the maximum exponent $$q$$q, in the growth estimate of the integrand, reaches a threshold. In fact, if $$q$$q is larger than this threshold, counterexamples to the local boundedness and regularity of minimizers are known. In this paper, we prove the local boundedness of minimizers (and also of quasi-minimizers) under this stated limit condition. Some other general and limit growth conditions are also considered.

Boundedness of local minimizers of anisotropic scalar integral functionals with general growth conditions

Boundedness of local minimizers of anisotropic scalar integral functionals with general growth conditions

A dual‐observer approach for global output feedback stabilization of planar nonlinear systems with output‐dependent growth rates

SummaryIn this paper, we investigate the problem of global output feedback stabilization for a class of planar nonlinear systems under a more general growth condition, which encompasses both lower‐order and higher‐order state growths with output‐dependent rates. For more accurate estimation, two new observers with nonlinear gains are constructed to estimate the states on the lower‐order and higher‐order scales, respectively. The estimates produced from the dual‐observer are used delicately in the output feedback control law with both lower‐order and higher‐order modes. The overall stability of the system is guaranteed by rigorously choosing these nonlinear gains in the control law and the dual‐observer.

Positive solutions of quasi-linear elliptic equations with dependence on the gradient

In the present paper we prove a multiplicity theorem for a quasi-linear elliptic problem with dependence on the gradient ensuring the existence of a positive solution and of a negative solution. In addition, we show the existence of the extremal constant-sign solutions: the smallest positive solution and the biggest negative solution. Our approach relies on extremal solutions for an auxiliary parametric problem. Other basic tools used in our paper are sub-supersolution techniques, Schaefer’s fixed point theorem, regularity results and strong maximum principle. In our hypotheses we only require a general growth condition with respect to the solution and its gradient, and an assumption near zero involving the first eigenvalue of the negative $$p$$ -Laplacian operator.

Weak Solutions to Fokker–Planck Equations and Mean Field Games

We deal with systems of PDEs, arising in mean field games theory, where viscous Hamilton–Jacobi and Fokker–Planck equations are coupled in a forward-backward structure. We consider the case of local coupling, when the running cost depends on the pointwise value of the distribution density of the agents, in which case the smoothness of solutions is mostly unknown. We develop a complete weak theory, proving that those systems are well-posed in the class of weak solutions for monotone couplings under general growth conditions, and for superlinear convex Hamiltonians. As a key tool, we prove new results for Fokker–Planck equations under minimal assumptions on the drift, through a characterization of weak and renormalized solutions. The results obtained give new perspectives even for the case of uncoupled equations as far as the uniqueness of weak solutions is concerned.