Higher differentiability for a class of obstacle problems with nearly linear growth conditions

2021 ◽  
Vol 31 (4) ◽  
pp. 767-789
Author(s):  
Chiara Gavioli
Keyword(s):  
Linear Growth ◽  
Growth Conditions ◽  
Obstacle Problems ◽  
Higher Differentiability

scholarly journals Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrea Gentile
Keyword(s):  
Besov Space ◽  
Lipschitz Space ◽  
Growth Conditions ◽  
Obstacle Problems ◽  
Quadratic Growth ◽  
Regularity Assumption ◽  
Lipschitz Regularity ◽  
Fixed Boundary ◽  
Higher Differentiability ◽  
Admissible Functions

Abstract We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form min ⁡ { ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) : v ∈ K ψ ⁢ ( Ω ) } , \min\biggl{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where the function 𝑓 satisfies 𝑝-growth conditions with respect to the gradient variable, for 1 < p < 2 1<p<2 , and K ψ ⁢ ( Ω ) \mathcal{K}_{\psi}(\Omega) is the class of admissible functions v ∈ u 0 + W 0 1 , p ⁢ ( Ω ) v\in u_{0}+W^{1,p}_{0}(\Omega) such that v ≥ ψ v\geq\psi a.e. in Ω, where u 0 ∈ W 1 , p ⁢ ( Ω ) u_{0}\in W^{1,p}(\Omega) is a fixed boundary datum. Here we show that a Sobolev or Besov–Lipschitz regularity assumption on the gradient of the obstacle 𝜓 transfers to the gradient of the solution, provided the partial map x ↦ D ξ ⁢ f ⁢ ( x , ξ ) x\mapsto D_{\xi}f(x,\xi) belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with sub-quadratic growth conditions with respect to the gradient variable, i.e. f ⁢ ( x , ξ ) ≈ a ⁢ ( x ) ⁢ | ξ | p f(x,\xi)\approx a(x)\lvert\xi\rvert^{p} with 1 < p < 2 1<p<2 , and where the map 𝑎 belongs to a Sobolev or Besov–Lipschitz space.

Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions

2019 ◽  
Vol 31 (6) ◽  
pp. 1501-1516 ◽  
Author(s):  
Chiara Gavioli
Keyword(s):  
Variational Inequality ◽  
Growth Conditions ◽  
Obstacle Problems ◽  
Differentiability Property ◽  
Integer Order ◽  
Differentiability Of Solutions ◽  
Higher Differentiability ◽  
Admissible Functions ◽  
Fixed Function

AbstractWe establish the higher differentiability of integer order of solutions to a class of obstacle problems assuming that the gradient of the obstacle possesses an extra integer differentiability property. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form\int_{\Omega}\langle\mathcal{A}(x,Du),D(\varphi-u)\rangle\,dx\geq 0\quad\text{% for all }\varphi\in\mathcal{K}_{\psi}(\Omega).The main novelty is that the operator {\mathcal{A}} satisfies the so-called {p,q}-growth conditions with p and q linked by the relation\frac{q}{p}<1+\frac{1}{n}-\frac{1}{r},for {r>n}. Here {\psi\in W^{1,p}(\Omega)} is a fixed function, called obstacle, for which we assume {D\psi\in W^{1,2q-p}_{\mathrm{loc}}(\Omega)}, and {\mathcal{K}_{\psi}=\{w\in W^{1,p}(\Omega):w\geq\psi\text{ a.e. in }\Omega\}} is the class of admissible functions. We require for the partial map {x\mapsto\mathcal{A}(x,\xi\/)} a higher differentiability of Sobolev order in the space {W^{1,r}}, with {r>n} satisfying the condition above.

scholarly journals <sup>210</sup>Pb-<sup>226</sup>Ra chronology reveals rapid growth rate of <i>Madrepora oculata</I> and <i>Lophelia pertusa</i> on world's largest cold-water coral reef

2011 ◽  
Vol 8 (6) ◽  
pp. 12247-12283
Author(s):  
P. Sabatier ◽  
J.-L. Reyss ◽  
J. M. Hall-Spencer ◽  
C. Colin ◽  
N. Frank ◽  
...  
Keyword(s):  
Growth Rate ◽  
Coral Reef ◽  
Linear Growth ◽  
Cold Water ◽  
Optimal Growth ◽  
Growth Conditions ◽  
Age And Growth ◽  
Linear Growth Rate ◽  
Lophelia Pertusa ◽  
Water Coral

Abstract. Here we show the use of the 210Pb-226Ra excess method to determine the growth rate of corals from one of the world's largest known cold-water coral reef, the Røst Reef off Norway. Two large branching framework-forming cold-water coral specimens, one Lophelia pertusa and one Madrepora oculata were collected alive at 350 m water depth from the Røst Reef at ~67° N and ~9° E. Pb and Ra isotopes were measured along the major growth axis of both specimens using low level alpha and gamma spectrometry and the corals trace element compositions were studied using ICP-QMS. Due to the different chemical behaviors of Pb and Ra in the marine environment, 210Pb and 226Ra were not incorporated the same way into the aragonite skeleton of those two cold-water corals. Thus to assess of the growth rates of both specimens we have here taken in consideration the exponential decrease of initially incorporated 210Pb as well as the ingrowth of 210Pb from the decay of 226Ra. Moreover a~post-depositional 210Pb incorporation is found in relation to the Mn-Fe coatings that could not be entirely removed from the oldest parts of the skeletons. The 226Ra activities in both corals were fairly constant, then assuming constant uptake of 210Pb through time the 210Pb-226Ra chronology can be applied to calculate linear growth rate. The 45.5 cm long branch of M. oculata reveals an age of 31 yr and a~linear growth rate of 14.4 ± 1.1 mm yr−1, i.e. 2.6 polyps per year. However, a correction regarding a remaining post-depositional Mn-Fe oxide coating is needed for the base of the specimen. The corrected age tend to confirm the radiocarbon derived basal age of 40 yr (using 14C bomb peak) with a mean growth rate of 2 polyps yr−1. This rate is similar to the one obtained in Aquaria experiments under optimal growth conditions. For the 80 cm-long specimen of L. pertusa a remaining contamination of metal-oxides is observed for the middle and basal part of the coral skeleton, inhibiting similar accurate age and growth rate estimates. However, the youngest branch was free of Mn enrichment and this 15 cm section reveals a growth rate of 8 mm yr−1 (~1 polyp every two to three years). However, the 210Pb growth rate estimate is within the lowermost ranges of previous growth rate estimates and may thus reflect that the coral was not developing at optimal growth conditions. Overall, 210Pb-226Ra dating can be successfully applied to determine the age and growth rate of framework-forming cold-water corals, however, removal of post-depositional Mn-Fe oxide deposits is a prerequisite. If successful, large branching M. oculata and L. pertusa coral skeletons provide unique oceanographic archive for studies of intermediate water environmentals with an up to annual time resolution and spanning over many decades.

scholarly journals On the approximations of solutions to stochastic differential equations under polynomial condition

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 11-25
Author(s):  
Dusan Djordjevic ◽  
Miljana Jovanovic
Keyword(s):  
Differential Equations ◽  
Theoretical Result ◽  
Stochastic Differential Equations ◽  
Approximate Method ◽  
Linear Growth ◽  
Growth Conditions ◽  
Time Interval ◽  
Initial Equation ◽  
The Subject ◽  
Approximate Equations

The subject of this paper is an analytic approximate method for a class of stochastic differential equations with coefficients that do not necessarily satisfy the Lipschitz and linear growth conditions but behave like a polynomials. More precisely, equations from the observed class have unique solutions with bounded moments and their coefficients satisfy polynomial condition. Approximate equations are defined on partitions of a time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. The rate of Lp convergence increases when degrees in Taylor approximations of coefficients increase. At the end of the paper, an example is provided to support the main theoretical result.

scholarly journals Existence and stability results for random impulsive fractional pantograph equations

Filomat ◽  
2016 ◽  
Vol 30 (14) ◽  
pp. 3839-3854 ◽  
Author(s):  
A. Anguraj ◽  
A. Vinodkumar ◽  
K. Malar
Keyword(s):  
Continuous Dependence ◽  
Linear Growth ◽  
Initial Conditions ◽  
Growth Conditions ◽  
Differential Systems ◽  
Existence And Stability ◽  
Stability Results ◽  
Rassias Stability

In this paper, we study the existence, uniqueness, stability through continuous dependence on initial conditions and Hyers-Ulam-Rassias stability results for random impulsive fractional pantograph differential systems by relaxing the linear growth conditions. Finally examples are given to illustrate the applications of the abstract results.

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