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Journal articles on the topic 'Higher differentiability'

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Author: Grafiati
Published: 9 March 2023

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1

Menne, Ulrich. "Pointwise differentiability of higher-order for distributions." Analysis & PDE 14, no. 2 (March 20, 2021): 323–54. http://dx.doi.org/10.2140/apde.2021.14.323.

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2

Esposito, L., F. Leonetti, and G. Mingione. "Higher differentiability for minimizers of irregular integrals." Nonlinear Analysis: Theory, Methods & Applications 47, no. 7 (August 2001): 4355–64. http://dx.doi.org/10.1016/s0362-546x(01)00550-8.

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3

Cellina, Arrigo, and Marco Mazzola. "Higher differentiability of solutions to variational problems." Calculus of Variations and Partial Differential Equations 45, no. 1-2 (August 14, 2011): 11–26. http://dx.doi.org/10.1007/s00526-011-0448-1.

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4

Menne, Ulrich. "Pointwise differentiability of higher order for sets." Annals of Global Analysis and Geometry 55, no. 3 (January 14, 2019): 591–621. http://dx.doi.org/10.1007/s10455-018-9642-0.

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5

Cameron, Peter, and Piotr T. Chruściel. "Asymptotic flatness in higher dimensions." Journal of Mathematical Physics 63, no. 3 (March 1, 2022): 032501. http://dx.doi.org/10.1063/5.0083728.

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We show that ( n + 1)-dimensional Myers–Perry metrics, n ≥ 4, have a conformal completion at space-like infinity of C n−3,1 differentiability class and that the result is optimal in even spacetime dimensions. The associated asymptotic symmetries are presented.
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6

Le Merdy, Christian, and Anna Skripka. "HIGHER ORDER DIFFERENTIABILITY OF OPERATOR FUNCTIONS IN SCHATTEN NORMS." Journal of the Institute of Mathematics of Jussieu 19, no. 6 (February 13, 2019): 1993–2016. http://dx.doi.org/10.1017/s1474748019000033.

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We establish the following results on higher order ${\mathcal{S}}^{p}$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert space:(i)$f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable at every bounded self-adjoint operator if and only if $f\in C^{n}(\mathbb{R})$;(ii)if $f^{\prime },\ldots ,f^{(n-1)}\in C_{b}(\mathbb{R})$ and $f^{(n)}\in C_{0}(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable at every self-adjoint operator;(iii)if $f^{\prime },\ldots ,f^{(n)}\in C_{b}(\mathbb{R})$, then $f$ is $n-1$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable and $n$ times Gâteaux ${\mathcal{S}}^{p}$-differentiable at every self-adjoint operator.We also prove that if $f\in B_{\infty 1}^{n}(\mathbb{R})\cap B_{\infty 1}^{1}(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{q}$-differentiable, $1\leqslant q<\infty$, at every self-adjoint operator. These results generalize and extend analogous results of Kissin et al. (Proc. Lond. Math. Soc. (3)108(3) (2014), 327–349) to arbitrary $n$ and unbounded operators as well as substantially extend the results of Azamov et al. (Canad. J. Math.61(2) (2009), 241–263); Coine et al. (J. Funct. Anal.; doi:10.1016/j.jfa.2018.09.005); Peller (J. Funct. Anal.233(2) (2006), 515–544) on higher order ${\mathcal{S}}^{p}$-differentiability of $f$ in a certain Wiener class, Gâteaux ${\mathcal{S}}^{2}$-differentiability of $f\in C^{n}(\mathbb{R})$ with $f^{\prime },\ldots ,f^{(n)}\in C_{b}(\mathbb{R})$, and Gâteaux ${\mathcal{S}}^{q}$-differentiability of $f$ in the intersection of the Besov classes $B_{\infty 1}^{n}(\mathbb{R})\cap B_{\infty 1}^{1}(\mathbb{R})$. As an application, we extend ${\mathcal{S}}^{p}$-estimates for operator Taylor remainders to a broad set of symbols. Finally, we establish explicit formulas for Fréchet differentials and Gâteaux derivatives.
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7

Carozza, Menita, Jan Kristensen, and Antonia Passarelli di Napoli. "Higher differentiability of minimizers of convex variational integrals." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 28, no. 3 (May 2011): 395–411. http://dx.doi.org/10.1016/j.anihpc.2011.02.005.

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8

Tung, L. T. "On higher-order proto-differentiability of perturbation maps." Positivity 24, no. 2 (June 17, 2019): 441–62. http://dx.doi.org/10.1007/s11117-019-00689-x.

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9

Gavioli, Chiara. "Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions." Forum Mathematicum 31, no. 6 (November 1, 2019): 1501–16. http://dx.doi.org/10.1515/forum-2019-0148.

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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.
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10

La Torre and Rocca. "HIGHER ORDER UNIFORM SMOOTHNESS AND DIFFERENTIABILITY OF REAL FUNCTIONS." Real Analysis Exchange 26, no. 2 (2000): 657. http://dx.doi.org/10.2307/44154068.

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11

Santilli, Mario. "Rectifiability and approximate differentiability of higher order for sets." Indiana University Mathematics Journal 68, no. 3 (2019): 1013–46. http://dx.doi.org/10.1512/iumj.2019.68.7645.

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12

Giova, Raffaella. "Higher differentiability for n-harmonic systems with Sobolev coefficients." Journal of Differential Equations 259, no. 11 (December 2015): 5667–87. http://dx.doi.org/10.1016/j.jde.2015.07.004.

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13

Ma, Lingwei, and Zhenqiu Zhang. "Higher differentiability for solutions of nonhomogeneous elliptic obstacle problems." Journal of Mathematical Analysis and Applications 479, no. 1 (November 2019): 789–816. http://dx.doi.org/10.1016/j.jmaa.2019.06.052.

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14

Giannetti, Flavia, Antonia Passarelli di Napoli, and Christoph Scheven. "Higher differentiability for solutions of stationary p ‐Stokes systems." Mathematische Nachrichten 293, no. 11 (August 31, 2020): 2082–111. http://dx.doi.org/10.1002/mana.201800519.

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15

Convent, Alexandra, and Jean Van Schaftingen. "Higher order intrinsic weak differentiability and Sobolev spaces between manifolds." Advances in Calculus of Variations 12, no. 3 (July 1, 2019): 303–32. http://dx.doi.org/10.1515/acv-2017-0008.

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AbstractWe define the notion of higher-order colocally weakly differentiable maps from a manifold M to a manifold N. When M and N are endowed with Riemannian metrics, {p\geq 1} and {k\geq 2}, this allows us to define the intrinsic higher-order homogeneous Sobolev space {\dot{W}^{k,p}(M,N)}. We show that this new intrinsic definition is not equivalent in general with the definition by an isometric embedding of N in a Euclidean space; if the manifolds M and N are compact, the intrinsic space is a larger space than the one obtained by embedding. We show that a necessary condition for the density of smooth maps in the intrinsic space {\dot{W}^{k,p}(M,N)} is that {\pi_{\lfloor kp\rfloor}(N)\simeq\{0\}}. We investigate the chain rule for higher-order differentiability in this setting.
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16

Giannetti, Flavia, Antonia Passarelli di Napoli, and Christoph Scheven. "Higher differentiability of solutions of parabolic systems with discontinuous coefficients." Journal of the London Mathematical Society 94, no. 1 (May 13, 2016): 1–20. http://dx.doi.org/10.1112/jlms/jdw019.

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17

Izumi, Yuki. "Higher order differentiability of solutions to backward stochastic differential equations." Stochastics 90, no. 1 (April 18, 2017): 102–50. http://dx.doi.org/10.1080/17442508.2017.1315119.

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18

Cianchi, Andrea, and Monia Randolfi. "Differentiability properties of functions from higher-order Orlicz–Sobolev spaces." Nonlinear Analysis: Theory, Methods & Applications 75, no. 7 (May 2012): 3322–38. http://dx.doi.org/10.1016/j.na.2011.12.023.

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19

Coine, Clément, Christian Le Merdy, Anna Skripka, and Fedor Sukochev. "Higher order S2-differentiability and application to Koplienko trace formula." Journal of Functional Analysis 276, no. 10 (May 2019): 3170–204. http://dx.doi.org/10.1016/j.jfa.2018.09.005.

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20

Giannetti, Flavia, and Antonia Passarelli di Napoli. "Higher differentiability of minimizers of variational integrals with variable exponents." Mathematische Zeitschrift 280, no. 3-4 (March 13, 2015): 873–92. http://dx.doi.org/10.1007/s00209-015-1453-4.

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21

Clop, Albert, Raffaella Giova, and Antonia Passarelli di Napoli. "Besov regularity for solutions of p-harmonic equations." Advances in Nonlinear Analysis 8, no. 1 (September 15, 2017): 762–78. http://dx.doi.org/10.1515/anona-2017-0030.

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Abstract We establish the higher fractional differentiability of the solutions to nonlinear elliptic equations in divergence form, i.e., {\operatorname{div}\mathcal{A}(x,Du)=\operatorname{div}F,} when {\mathcal{A}} is a p-harmonic type operator, and under the assumption that {x\mapsto\mathcal{A}(x,\xi\/)} belongs to the critical Besov–Lipschitz space {B^{\alpha}_{{n/\alpha},q}} . We prove that some fractional differentiability assumptions on F transfer to Du with no losses in the natural exponent of integrability. When {\operatorname{div}F=0} , we show that an analogous extra differentiability property for Du holds true under a Triebel–Lizorkin assumption on the partial map {x\mapsto\mathcal{A}(x,\xi\/)} .
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22

Salamat, Nadeem, Muhammad Mustahsan, and Malik Saad Missen. "Switching Point Solution of Second-Order Fuzzy Differential Equations Using Differential Transformation Method." Mathematics 7, no. 3 (March 1, 2019): 231. http://dx.doi.org/10.3390/math7030231.

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The first-order fuzzy differential equation has two possible solutions depending on the definition of differentiability. The definition of differentiability changes as the product of the function and its first derivative changes its sign. This switching of the derivative’s definition is handled with the application of min, max operators. In this paper, a numerical technique for solving fuzzy initial value problems is extended to solving higher-order fuzzy differential equations. Fuzzy Taylor series is used to develop the fuzzy differential transformation method for solving this problem. This leads to a single solution for higher-order differential equations.
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23

Kuleshov, Alexander. "The Various Definitions of Multiple Differentiability of a Function f: ℝn→ ℝ." Mathematics 8, no. 11 (November 4, 2020): 1946. http://dx.doi.org/10.3390/math8111946.

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Since the 17-th century the concepts of differentiability and multiple differentiability have become fundamental to mathematical analysis. By now we have the generally accepted definition of what a multiply differentiable function f:Rn→R is (in this paper we call it standard). This definition is sufficient to prove some of the key properties of a multiply differentiable function: the Generalized Young’s theorem (a theorem on the independence of partial derivatives of higher orders of the order of differentiation) and Taylor’s theorem with Peano remainder. Another definition of multiple differentiability, actually more general in the sense that it is suitable for the infinite-dimensional case, belongs to Fréchet. It turns out, that the standard definition and the Fréchet definition are equivalent for functions f:Rn→R. In this paper we introduce a definition (which we call weak) of multiple differentiability of a function f:Rn→R, which is not equivalent to the above-mentioned definitions and is in fact more general, but at the same time is sufficient enough to prove the Generalized Young’s and Taylor’s theorems.
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24

Marinelli, Carlo, and Luca Scarpa. "Fréchet differentiability of mild solutions to SPDEs with respect to the initial datum." Journal of Evolution Equations 20, no. 3 (October 30, 2019): 1093–130. http://dx.doi.org/10.1007/s00028-019-00546-0.

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Abstract We establish n-th-order Fréchet differentiability with respect to the initial datum of mild solutions to a class of jump diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz-continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order Gâteaux differentiability of their solutions with respect to the initial datum, extending previous results by Marinelli, Prévôt, and Röckner in several ways. The differentiability results obtained here are a fundamental step to construct classical solutions to non-local Kolmogorov equations with sufficiently regular coefficients by probabilistic means.
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25

R. Cirmi, G., and S. Leonardi. "Higher differentiability for solutions of linear elliptic systems with measure data." Discrete & Continuous Dynamical Systems - A 26, no. 1 (2010): 89–104. http://dx.doi.org/10.3934/dcds.2010.26.89.

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26

Cellina, Arrigo. "The higher differentiability of solutions to variational problems of quadratic growth." Journal of Differential Equations 268, no. 2 (January 2020): 813–24. http://dx.doi.org/10.1016/j.jde.2019.08.029.

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27

Lang, Julius. "Differentiability of projective transformations in dimension 2." Advances in Geometry 20, no. 4 (October 27, 2020): 553–57. http://dx.doi.org/10.1515/advgeom-2019-0023.

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AbstractIt is proven by elementary methods that in dimension 2, every locally injective continuous map, sending the curves of a Ck-spray to curves of another Ck-spray as oriented point sets, is a Ck-diffeomorphism. This extends the result [1] for dimension three and higher from 1965.
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28

Ahmadi, A., K. S. Surana, R. K. Maduri, A. Romkes, and J. N. Reddy. "Higher Order Global Differentiability Local Approximations for 2-D Distorted Quadrilateral Elements." International Journal for Computational Methods in Engineering Science and Mechanics 10, no. 1 (February 12, 2009): 1–19. http://dx.doi.org/10.1080/15502280802572262.

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29

Maduri, R. K., K. S. Surana, A. Romkes, and J. N. Reddy. "Higher Order Global Differentiability Local Approximations for 2-D Distorted Triangular Elements." International Journal for Computational Methods in Engineering Science and Mechanics 10, no. 1 (February 12, 2009): 20–26. http://dx.doi.org/10.1080/15502280802572270.

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30

Nguyen, D. T., G. Rauchs, and J. P. Ponthot. "The impact of surface higher order differentiability in two-dimensional contact elements." Journal of Computational and Applied Mathematics 246 (July 2013): 195–205. http://dx.doi.org/10.1016/j.cam.2012.10.024.

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31

Kirsch, Andreas, and Andreas Rieder. "Inverse Problems for Abstract Evolution Equations II: Higher Order Differentiability for Viscoelasticity." SIAM Journal on Applied Mathematics 79, no. 6 (January 2019): 2639–62. http://dx.doi.org/10.1137/19m1269403.

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32

de Cristoforis, Massimo Lanza. "Higher order differentiability properties of the composition and of the inversion operator." Indagationes Mathematicae 5, no. 4 (1994): 457–82. http://dx.doi.org/10.1016/0019-3577(94)90018-3.

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33

Giova, Raffaella, and Antonia Passarelli di Napoli. "Regularity results for a priori bounded minimizers of non-autonomous functionals with discontinuous coefficients." Advances in Calculus of Variations 12, no. 1 (January 1, 2019): 85–110. http://dx.doi.org/10.1515/acv-2016-0059.

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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.
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34

Giannetti, Flavia, Antonia Passarelli di Napoli, and Christoph Scheven. "On higher differentiability of solutions of parabolic systems with discontinuous coefficients and (p, q)-growth." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 1 (January 26, 2019): 419–51. http://dx.doi.org/10.1017/prm.2018.63.

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AbstractWe consider weak solutions $u:\Omega _T\to {\open R}^N$ to parabolic systems of the type $$u_t-{\rm div}\;a(x,t,Du) = 0\quad {\rm in}\;\Omega _T = \Omega \times (0,T),$$where the function a(x, t, ξ) satisfies (p, q)-growth conditions. We give an a priori estimate for weak solutions in the case of possibly discontinuous coefficients. More precisely, the partial maps $x\mapsto a(x,t,\xi )$ under consideration may not be continuous, but may only possess a Sobolev-type regularity. In a certain sense, our assumption means that the weak derivatives $D_xa(\cdot ,\cdot ,\xi )$ are contained in the class $L^\alpha (0,T;L^\beta (\Omega ))$, where the integrability exponents $\alpha ,\beta $ are coupled by $$\displaystyle{{p(n + 2)-2n} \over {2\alpha }} + \displaystyle{n \over \beta } = 1-\kappa $$for some κ ∈ (0,1). For the gap between the two growth exponents we assume $$2 \les p < q \les p + \displaystyle{{2\kappa } \over {n + 2}}.$$Under further assumptions on the integrability of the spatial gradient, we prove a result on higher differentiability in space as well as the existence of a weak time derivative $u_t\in L^{p/(q-1)}_{{\rm loc}}(\Omega _T)$. We use the corresponding a priori estimate to deduce the existence of solutions of Cauchy–Dirichlet problems with the mentioned higher differentiability property.
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35

BROWN, LUKE, GIOVANNI FERRER, GAMAL MOGRABY, LUKE G. ROGERS, and KARUNA SANGAM. "HARMONIC GRADIENTS ON HIGHER-DIMENSIONAL SIERPIŃSKI GASKETS." Fractals 28, no. 06 (September 2020): 2050108. http://dx.doi.org/10.1142/s0218348x2050108x.

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We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpiński Gasket and its higher-dimensional variants [Formula: see text], [Formula: see text], proving results that generalize those of Teplyaev [Gradients on fractals, J. Funct. Anal. 174(1) (2000) 128–154]. When [Formula: see text] is equipped with the standard Dirichlet form and measure [Formula: see text] we show there is a full [Formula: see text]-measure set on which continuity of the Laplacian implies existence of the gradient [Formula: see text], and that this set is not all of [Formula: see text]. We also show there is a class of non-uniform measures on the usual Sierpiński Gasket with the property that continuity of the Laplacian implies the gradient exists and is continuous everywhere in sharp contrast to the case with the standard measure.
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36

Roy, Priyanka, and Geetanjali Panda. "Expansion of Generalized Hukuhara Differentiable Interval Valued Function." New Mathematics and Natural Computation 15, no. 03 (October 7, 2019): 553–70. http://dx.doi.org/10.1142/s1793005719500327.

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In this paper, the concept of [Formula: see text]-monotonic property of interval valued function in higher dimension is introduced. Expansion of interval valued function in higher dimension is developed using this property. Generalized Hukuhara differentiability is used to derive the theoretical results. Several examples are provided to justify the theoretical developments.
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37

Piersanti, Paolo. "On the improved interior regularity of a boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle." Discrete & Continuous Dynamical Systems 42, no. 2 (2022): 1011. http://dx.doi.org/10.3934/dcds.2021145.

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<p style='text-indent:20px;'>In this paper we show that the solution of an obstacle problem for linearly elastic elliptic membrane shells enjoys higher differentiability properties in the interior of the domain where it is defined.</p>
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38

Bulíček, Miroslav, Giovanni Cupini, Bianca Stroffolini, and Anna Verde. "Existence and regularity results for weak solutions to (p,q)-elliptic systems in divergence form." Advances in Calculus of Variations 11, no. 3 (July 1, 2018): 273–88. http://dx.doi.org/10.1515/acv-2016-0054.

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AbstractWe prove existence and regularity results for weak solutions of non-linear elliptic systems with non-variational structure satisfying {(p,q)}-growth conditions. In particular, we are able to prove higher differentiability results under a dimension-free gap between p and q.
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39

BURGUÉS, J. M., and J. CUFÍ. "REGULARITY OF CURVES WITH A CONTINUOUS TANGENT LINE." Journal of the Australian Mathematical Society 86, no. 1 (February 2009): 17–26. http://dx.doi.org/10.1017/s1446788708000402.

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AbstractThis note contains a proof of the fact that a Jordan curve in ℝ2 with a continuous tangent line at each point admits a regular reparameterization. We extend the result both to more general curves in ℝn and to higher orders of differentiability.
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40

Brasco, Lorenzo, and Filippo Santambrogio. "A sharp estimate à la Calderón–Zygmund for the p-Laplacian." Communications in Contemporary Mathematics 20, no. 03 (February 21, 2018): 1750030. http://dx.doi.org/10.1142/s0219199717500304.

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We consider local weak solutions of the Poisson equation for the [Formula: see text]-Laplace operator. We prove a higher differentiability result, under an essentially sharp condition on the right-hand side. The result comes with a local scaling invariant a priori estimate.
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41

Gavioli, Chiara. "Higher differentiability for a class of obstacle problems with nearly linear growth conditions." Rendiconti Lincei - Matematica e Applicazioni 31, no. 4 (February 15, 2021): 767–89. http://dx.doi.org/10.4171/rlm/914.

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42

Kirsch, Andreas, and Andreas Rieder. "Erratum: Inverse Problems for Abstract Evolution Equations II: Higher Order Differentiability for Viscoelasticity." SIAM Journal on Applied Mathematics 81, no. 1 (January 2021): 282–83. http://dx.doi.org/10.1137/20m1372160.

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43

Foralli, Niccolò, and Giovanni Giliberti. "Higher differentiability of solutions for a class of obstacle problems with variable exponents." Journal of Differential Equations 313 (March 2022): 244–68. http://dx.doi.org/10.1016/j.jde.2021.12.028.

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44

Leonardi, S., and J. Stara. "HIGHER DIFFERENTIABILITY FOR SOLUTIONS OF A CLASS OF PARABOLIC SYSTEMS WITH L1, - DATA." Quarterly Journal of Mathematics 66, no. 2 (November 14, 2014): 659–76. http://dx.doi.org/10.1093/qmath/hau031.

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45

Surana, Karan S., Celso H. Carranza, and Sri Sai Charan Mathi. "k-Version of Finite Element Method for BVPs and IVPs." Mathematics 9, no. 12 (June 9, 2021): 1333. http://dx.doi.org/10.3390/math9121333.

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The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) are always p-version hierarchical that permit use of any desired p-level without effecting global differentiability. HGDA/DG are true Ci, Cij, Cijk, hence the dofs at the nonhierarchical nodes of the elements are transformable between natural and physical coordinate spaces using calculus. This is not the case with tensor product higher order continuity elements discussed in this paper, thus confirming that the tensor product approximations are not true Ci, Cijk, Cijk approximations. It is shown that isogeometric analysis for a domain with more than one patch can only yield solutions of class C0. This method has no concept of finite elements and local approximations, just patches. It is shown that compariso of this method with k-version of the finite element method is meaningless. Model problem studies in R2 establish accuracy and superior convergence characteristics of true Cijp-version hierarchical local approximations presented in this paper over tensor product approximations. Convergence characteristics of p-convergence, k-convergence and pk-convergence are illustrated for self adjoint, non-self adjoint and non-linear differential operators in BVPs. It is demonstrated that h, p and k are three independent parameters in all finite element computations. Tensor product local approximations and other published works on k-version and their limitations are discussed in the paper and are compared with present work.
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46

Foss, Mikil, and Joe Geisbauer. "Higher differentiability in the context of Besov spaces for a class of nonlocal functionals." Evolution Equations & Control Theory 2, no. 2 (2013): 301–18. http://dx.doi.org/10.3934/eect.2013.2.301.

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47

Xiao, Ti-Jun, and Jin Liang. "Pazy-type characterization for differentiability of propagators of higher order Cauchy problems in Banach spaces." Discrete & Continuous Dynamical Systems - A 5, no. 3 (1999): 651–62. http://dx.doi.org/10.3934/dcds.1999.5.651.

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48

Passarelli di Napoli, Antonia. "Higher Differentiability of Solutions of Elliptic Systems with Sobolev Coefficients: The Case p = n = 2." Potential Analysis 41, no. 3 (January 29, 2014): 715–35. http://dx.doi.org/10.1007/s11118-014-9390-0.

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49

Surana, K. S., S. Allu, P. W. Tenpas, and J. N. Reddy. "k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions." International Journal for Numerical Methods in Engineering 69, no. 6 (2007): 1109–57. http://dx.doi.org/10.1002/nme.1801.

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50

Fook, Tiaw Kah, and Zarina Bibi Ibrahim. "Block Backward Differentiation Formulas for solving second order Fuzzy Differential Equations." MATEMATIKA 33, no. 2 (December 27, 2017): 215. http://dx.doi.org/10.11113/matematika.v33.n2.868.

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Abstract:
In this paper, we study the numerical method for solving second order Fuzzy Differential Equations (FDEs) using Block Backward Differential Formulas (BBDF) under generalized concept of higher-order fuzzy differentiability. Implementation of the method using Newton iteration is discussed. Numerical results obtained by BBDF are presented and compared with Backward Differential Formulas (BDF) and exact solutions. Several numerical examples are provided to illustrate our methods.
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