The nonstationary venttsel' problem with quadratic growth with respect to the gradient

1996 ◽  
Vol 80 (6) ◽  
pp. 2197-2207 ◽  
Author(s):  
D. E. Apushkinskaya ◽  
A. I. Nazarov
Keyword(s):  
Quadratic Growth ◽  
Venttsel Problem

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.

scholarly journals Quantification of leaf greenness and leaf spectral profile in plant diagnosis using an optical scanner

2012 ◽  
Vol 36 (3) ◽  
pp. 309-317 ◽  
Author(s):  
Ryoichi Doi
Keyword(s):  
Spectral Data ◽  
Crop Production ◽  
Response Pattern ◽  
Quadratic Growth ◽  
Spectral Profile ◽  
Growth Data ◽  
Data Set ◽  
Optical Scanner ◽  
Color Scale ◽  
Management Measures

Observation of leaf spectral profile (color) enables suitable management measures to be taken for crop production. An optical scanner was used: 1) to obtain an equation to determine the greenness of plant leaves and 2) to examine the power to discriminate among plants grown under different nutritional conditions. Sweet basil seedlings grown on vermiculite were supplemented with one-fifth-strength Hoagland solutions containing 0, 0.2, 1, 5, 20, and 50 mM NH4+. The 5 mM treatment resulted in the greatest leaf and shoot weights, indicating a quadratic growth response pattern to the NH4+ gradient. An equation involving b*, black and green to describe the greenness of leaves was provided by the spectral profiling of a color scale for rice leaves as the standard. The color scale values for the basil leaves subjected to 0.2 and 1 mM NH4+ treatments were 1.00 and 1.12, respectively. The other treatments resulted in significantly greater values of 2.25 to 2.42, again indicating a quadratic response pattern. Based on the spectral data set consisting of variables of red-green-blue and other color models and color scale values, in discriminant analysis, 81% of the plants were correctly classified into the six NH4+ treatment groups. Combining the spectral data set with the growth data set consisting of leaf and shoot weights, 92% of the plant samples were correctly classified whereas, using the growth data set, only 53% of plants were correctly classified. Therefore, the optical scanning of leaves and the use of spectral profiles helped plant diagnosis when biomass measurements were not effective.

The Psychological Well-Being of Taiwanese Youth

2009 ◽  
Vol 24 (3) ◽  
pp. 397-429 ◽  
Author(s):  
Chin-Chun Yi ◽  
Chyi-In Wu ◽  
Ying-Hwa Chang ◽  
Ming-Yi Chang
Keyword(s):  
Depressive Symptoms ◽  
Growth Curve ◽  
Panel Study ◽  
School Context ◽  
Subsequent Development ◽  
Well Being ◽  
Quadratic Growth ◽  
School Factors ◽  
Psychological Well Being ◽  
Over Time

This study examines the growth trajectory of the psychological well-being of Taiwanese adolescents from early to late adolescence. Under the competitive educational system in Taiwan, family and school context are two major loci accounting for the developmental outcome. Data are taken from the Taiwan Youth Project, which is a longitudinal panel study of 2696 students since the year 2000. The study uses individual depressive symptoms as the dependent variable. Family cohesion, family educational strategy as well as classroom effects at school are chosen to indicate the potential contextual influence. Using the latent growth curve method, the analysis confirms that family and school factors do produce different effects over time. Family context is salient at the initial status, but not for subsequent development. Class cohesion as well as adolescents' perceptions of unfairness by teachers determine the depressive level, the linear slope and the non-linear quadratic growth curve. In other words, once the adolescent gets used to junior high school, the school context tends to exert more pronounced effects. Further analysis on gender comparisons indicates that selective family and school effects are more pronounced among females, with a greater degree of depressive symptoms over time. The article concludes that while family and school have different impacts on the growth curve of individual depressive symptoms, the school context exerts salient effects over an adolescent's life course.

scholarly journals PARALLEL DEVELOPMENT OF MEMORY, DISABILITY, AND COMORBIDITY IN U.S. ADULTS AGE 65 AND OLDER

2019 ◽  
Vol 3 (Supplement_1) ◽  
pp. S659-S659
Author(s):  
Nicholas J Bishop ◽  
Kaipeng Wang
Keyword(s):  
Health Outcomes ◽  
Chronic Conditions ◽  
Developmental Process ◽  
Developmental Change ◽  
Survey Design ◽  
Word Recall ◽  
Quadratic Growth ◽  
Mobility Limitations ◽  
Older Adulthood ◽  
Baseline Health

Abstract Cognitive health, physical function, and chronic disease represent interdependent health outcomes that may exert influence on the course of each other’s development. To investigate the association between baseline health in each domain and developmental change across domains, we estimated trajectories of working memory, mobility limitations, and comorbidity among US adults age 65 and older over 18 years. We drew observations from the nationally-representative Health and Retirement Study with an analytic sample consisting of 5,963 adults age 65 and over in 1998. Immediate word recall, an 11-item Nagi scale of mobility limitations, and a summary count of eight doctor-diagnosed chronic conditions were measured biennially from 1998 to 2016. Parallel-process quadratic growth models with individually-varying time scores were used to estimate non-linear trajectories of each health measure, allowing identification of associations between baseline health and developmental change in each health process at both earlier and later stages of older adulthood. All estimates adjusted for covariates, complex survey design, and missing data. Greater baseline immediate word recall was associated with less rapid increase in mobility limitations at earlier ages. More baseline mobility limitations were associated with faster increase in comorbidity at earlier ages. Greater baseline chronic conditions were associated with more rapid increase in mobility limitations at later ages. These results highlight the importance of conceptualizing health among older adults as an interdependent and developmental process and should help clinicians recognize that single-domain health status may influence the progression of other health outcomes at different stages of older adulthood.

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