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

2018 ◽  
Vol 177 ◽  
pp. 543-552 ◽  
Author(s):  
Petteri Harjulehto ◽  
Peter Hästö ◽  
Arttu Karppinen
Keyword(s):  
Growth Conditions ◽  
Higher Integrability

scholarly journals Global higher integrability for minimisers of convex functionals with (p,q)-growth

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Lukas Koch
Keyword(s):  
Hölder Continuity ◽  
Growth Conditions ◽  
Holder Continuity ◽  
Higher Integrability ◽  
Convex Functionals ◽  
Global Higher Integrability ◽  
Continuity Assumption

AbstractWe prove global $$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) -regularity for minimisers of convex functionals of the form $${\mathscr {F}}(u)=\int _{\varOmega } F(x,Du)\,{\mathrm{d}}x$$ F ( u ) = ∫ Ω F ( x , D u ) d x .$$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform $$\alpha $$ α -Hölder continuity assumption in x and controlled (p, q)-growth conditions in z with $$q<\frac{(n+\alpha )p}{n}$$ q < ( n + α ) p n .

scholarly journals Higher integrability for variational integrals with non-standard growth

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Mathias Schäffner
Keyword(s):  
Partial Regularity ◽  
Growth Conditions ◽  
Integral Functionals ◽  
Restrictive Assumption ◽  
Higher Integrability ◽  
Variational Integrals

AbstractWe consider autonomous integral functionals of the form $$\begin{aligned} {\mathcal {F}}[u]:=\int _\varOmega f(D u)\,dx \quad \text{ where } u:\varOmega \rightarrow {\mathbb {R}}^N, N\ge 1, \end{aligned}$$ F [ u ] : = ∫ Ω f ( D u ) d x where u : Ω → R N , N ≥ 1 , where the convex integrand f satisfies controlled (p, q)-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of $${\mathcal {F}}$$ F assuming $$\frac{q}{p}<1+\frac{2}{n-1}$$ q p < 1 + 2 n - 1 , $$n\ge 3$$ n ≥ 3 . This improves earlier results valid under the more restrictive assumption $$\frac{q}{p}<1+\frac{2}{n}$$ q p < 1 + 2 n .

Regularity results for a priori bounded minimizers of non-autonomous functionals with discontinuous coefficients

2019 ◽  
Vol 12 (1) ◽  
pp. 85-110 ◽  
Author(s):  
Raffaella Giova ◽  
Antonia Passarelli di Napoli
Keyword(s):  
Sobolev Space ◽  
A Priori ◽  
Elliptic Systems ◽  
Discontinuous Coefficients ◽  
Growth Conditions ◽  
Integral Functionals ◽  
Local Minimizers ◽  
Higher Integrability ◽  
Regularity Results ◽  
Higher Differentiability

AbstractWe prove the higher differentiability and the higher integrability of the a priori bounded local minimizers of integral functionals of the form\mathcal{F}(v,\Omega)=\int_{\Omega}f(x,Dv(x))\,{\mathrm{d}}x,with convex integrand satisfyingp-growth conditions with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to thex-variable belongs to a suitable Sobolev space. The a priori boundedness of the minimizers allows us to obtain the higher differentiability under a Sobolev assumption which is independent on the dimensionnand that, in the case{p\leq n-2}, improves previous known results. We also deal with solutions of elliptic systems with discontinuous coefficients under the so-called Uhlenbeck structure. In this case, it is well known that the solutions are locally bounded and therefore we obtain analogous regularity results without the a priori boundedness assumption.

scholarly journals Global continuity and higher integrability of a minimizer of an obstacle problem under generalized Orlicz growth conditions

2019 ◽  
Author(s):  
Arttu Karppinen
Keyword(s):  
Obstacle Problem ◽  
Variable Exponent ◽  
Polynomial Growth ◽  
Growth Conditions ◽  
Phase Growth ◽  
Higher Integrability ◽  
Double Phase ◽  
Special Cases

AbstractWe prove continuity up to the boundary of the minimizer of an obstacle problem and higher integrability of its gradient under generalized Orlicz growth. The result recovers similar results obtained in the special cases of polynomial growth, variable exponent growth and produces new results for Orlicz and double phase growth.

scholarly journals Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrea Gentile
Keyword(s):  
Sobolev Space ◽  
Regularity Result ◽  
Growth Conditions ◽  
Quadratic Growth ◽  
Lipschitz Regularity ◽  
Higher Integrability ◽  
Higher Differentiability

AbstractWe consider functionals of the form\mathcal{F}(v,\Omega)=\int_{\Omega}f(x,Dv(x))\,dx,with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space {W^{1,q}}. We prove a higher differentiability result for the minimizers. We also infer a Lipschitz regularity result of minimizers if {q>n}, and a result of higher integrability for the gradient if {q=n}. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

Influence of MBE Growth Conditions on Dislocation Densities of GaAs/Ge Epilayers Grown on Silicon Substrates

1985 ◽  
Vol 43 ◽  
pp. 364-365
Author(s):  
K.M. Hones ◽  
P. Sheldon ◽  
B.G. Yacobi ◽  
A. Mason
Keyword(s):  
High Efficiency ◽  
Lattice Mismatch ◽  
Low Cost ◽  
Growth Conditions ◽  
Nonradiative Recombination ◽  
Device Structure ◽  
Si Substrates ◽  
Transmission Electron ◽  
Dislocation Densities ◽  
Dislocation Type

There is increasing interest in growing epitaxial GaAs on Si substrates. Such a device structure would allow low-cost substrates to be used for high-efficiency cascade- junction solar cells. However, high-defect densities may result from the large lattice mismatch (∼4%) between the GaAs epilayer and the silicon substrate. These defects can act as nonradiative recombination centers that can degrade the optical and electrical properties of the epitaxially grown GaAs. For this reason, it is important to optimize epilayer growth conditions in order to minimize resulting dislocation densities. The purpose of this paper is to provide an indication of the quality of the epitaxially grown GaAs layers by using transmission electron microscopy (TEM) to examine dislocation type and density as a function of various growth conditions. In this study an intermediate Ge layer was used to avoid nucleation difficulties observed for GaAs growth directly on Si substrates. GaAs/Ge epilayers were grown by molecular beam epitaxy (MBE) on Si substrates in a manner similar to that described previously.

Diffraction from arrays of antiphase boundaries in ordered III-V alloys

1987 ◽  
Vol 45 ◽  
pp. 312-313
Author(s):  
T. S. Kuan
Keyword(s):  
High Surface ◽  
Growth Conditions ◽  
Ternary Alloys ◽  
Local Growth ◽  
Antiphase Boundaries ◽  
Surface Mobility ◽  
Ordered Phases ◽  
Diffraction Patterns ◽  
Atomically Flat ◽  
Degree Of Order

Recent electron diffraction studies have found ordered phases in AlxGa1-xAs, GaAsxSb1-x, and InxGa1-xAs alloy systems, and these ordered phases are likely to be found in many other III-V ternary alloys as well. The presence of ordered phases in these alloys was detected in the diffraction patterns through the appearance of superstructure reflections between the Bragg peaks (Fig. 1). The ordered phase observed in the AlxGa1-xAs and InxGa1-xAs systems is of the CuAu-I type, whereas in GaAsxSb1-x this phase and a chalcopyrite type ordered phase can be present simultaneously. The degree of order in these alloys is strongly dependent on the growth conditions, and during the growth of these alloys, high surface mobility of the depositing species is essential for the onset of ordering. Thus, the growth on atomically flat (110) surfaces usually produces much stronger ordering than the growth on (100) surfaces. The degree of order is also affected by the presence of antiphase boundaries (APBs) in the ordered phase. As shown in Fig. 2(a), a perfectly ordered In0.5Ga0.5As structure grown along the <110> direction consists of alternating InAs and GaAs monolayers, but due to local growth fluctuations, two types of APBs can occur: one involves two consecutive InAs monolayers and the other involves two consecutive GaAs monolayers.

The value of Electron Microscope studies of imperfect natural diamonds

1990 ◽  
Vol 48 (4) ◽  
pp. 194-195
Author(s):  
J C Walmsley ◽  
A R Lang
Keyword(s):  
Electron Microscope ◽  
Classification Scheme ◽  
Natural Diamond ◽  
Sudden Change ◽  
Growth Conditions ◽  
Diamond Growth ◽  
Major Constituent ◽  
Natural Diamonds ◽  
Type Ia ◽  
Platelet Defects

Interest in the defects and impurities in natural diamond, which are found in even the most perfect stone, is driven by the fact that diamond growth occurs at a depth of over 120Km. They display characteristics associated with their origin and their journey through the mantle to the surface of the Earth. An optical classification scheme for diamond exists based largely on the presence and segregation of nitrogen. For example type Ia, which includes 98% of all natural diamonds, contain nitrogen aggregated into small non-paramagnetic clusters and usually contain sub-micrometre platelet defects on {100} planes. Numerous transmission electron microscope (TEM) studies of these platelets and associated features have been made e.g. . Some diamonds, however, contain imperfections and impurities that place them outside this main classification scheme. Two such types are described.First, coated-diamonds which possess gem quality cores enclosed by a rind that is rich in submicrometre sized mineral inclusions. The transition from core to coat is quite sharp indicating a sudden change in growth conditions, Figure 1. As part of a TEM study of the inclusions apatite has been identified as a major constituent of the impurity present in many inclusion cavities, Figure 2.

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