Entire solutions of nonlocal elasticity models for composite materials

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Preface: Recent progresses in the theory of nonlinear nonlocal problems

June 2018, 11(3): 357-377. doi: 10.3934/dcdss.2018020

Entire solutions of nonlocal elasticity models for composite materials

Giuseppina Autuori and Patrizia Pucci ^,

Department of Mathematics and Informatics, University of Perugia, Via Vanvitelli, 1, 06123, Perugia, Italy

* Corresponding author: Patrizia Pucci

Received May 2017 Revised August 2017 Published October 2017

Fund Project: This research has been developped within a scientific project Nonlocal Elasticity Models for Composite Materials with Professors F. Cluni and V. Gusella of the Dipartimento di Ingegneria Civile ed Ambientale of the Università degli Studi di Perugia.

Many structural materials, which are preferred for the developing of advanced constructions, are inhomogeneous ones. Composite materials have complex internal structure and properties, which make them to be more effectual in the solution of special problems required for civil and environmental engineering. As a consequence of this internal heterogeneity, they exhibit complex mechanical properties. In this work, the analysis of some features of the behavior of composite materials under different loading conditions is carried out. The dependence of nonlinear elastic response of composite materials on loading conditions is studied. Several approaches to model elastic nonlinearity such as different stiffness for particular type of loadings and nonlinear shear stress–strain relations are considered. Instead of a set of constant anisotropy coefficients, the anisotropy functions are introduced. Eventually, the combined constitutive relations are proposed to describe simultaneously two types of physical nonlinearities. The first characterizes the nonlinearity of shear stress–strain dependency and the latter determines the stress state susceptibility of material properties. Quite satisfactory correlation between the theoretical dependencies and the results of experimental studies is demonstrated, as described in [2,3] as well as in the references therein.

Keywords: Existence theorems, entire solutions, fractional elliptic operators, nonlocal elasticity models, composite materials.

Mathematics Subject Classification: Primary: 35R11, 35A15, 35J60; Secondary: 74B20, 74E30.

Citation: Giuseppina Autuori, Patrizia Pucci. Entire solutions of nonlocal elasticity models for composite materials. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 357-377. doi: 10.3934/dcdss.2018020

References:

[1]	A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.
[2]	G. Autuori, F. Cluni, V. Gusella and P. Pucci, Mathematical models for nonlocal elastic composite materials Adv. Nonlinear Anal. (2017), pages 39. doi: 10.1515/anona-2016-0186.
[3]	G. Autuori, F. Cluni, V. Gusella and P. Pucci, Effects of the fractional Laplacian order on the nonlocal elastic rod response, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering, 3 (2017), 030902, 5pp. doi: 10.1115/1.4036806.
[4]	G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714. doi: 10.1016/j.na.2015.06.014.
[5]	G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, Nonlinear Differ. Equ. Appl., 20 (2013), 977-1009. doi: 10.1007/s00030-012-0193-y.
[6]	L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845. doi: 10.3934/dcds.2016.36.1813.
[7]	H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Universitext Springer, New York, 2011, xiv+599 pp.
[8]	H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.
[9]	L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, 37-52, Abel Symp. 7, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25361-4_3.
[10]	M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129. doi: 10.1007/s10231-016-0555-x.
[11]	X. Chang, Ground states of some fractional Schrödinger equations on $\mathbb R^N$, Proc. Edinb. Math. Soc., 58 (2015), 305-321. doi: 10.1017/S0013091514000200.
[12]	D. C. de Morais Filho, M. A. S. Souto and J. M. do Ó., A compactness embedding lemma, a principle of symmetric criticality and applications to elliptic problems, Proyecciones, 19 (2000), 1-17. doi: 10.4067/S0716-09172000000100001.
[13]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[14]	S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb R^n$ Lecture Notes, Scuola Normale Superiore di Pisa (New Series), 15, Edizioni della Normale, Pisa, 2017, viii+152 pp. doi: 10.1007/978-88-7642-601-8.
[15]	M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961. doi: 10.1088/0951-7715/28/6/1937.
[16]	A. Fiscella and P. Pucci, p-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378. doi: 10.1016/j.nonrwa.2016.11.004.
[17]	A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011.
[18]	Y. Lei, Critical conditions and finite energy solutions of several nonlinear elliptic PDEs in $\mathbb R^n$, J. Differential Equations, 258 (2015), 4033-4061. doi: 10.1016/j.jde.2015.01.043.
[19]	P.-L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., 49 (1982), 315-334. doi: 10.1016/0022-1236(82)90072-6.
[20]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations 2nd edition, Grundlehren der Mathematischen Wissenschaften, 342, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-15564-2.
[21]	R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.
[22]	P. Piersanti and P. Pucci, Entire solutions for critical p-fractional Hardy Schrödinger Kirchhoff equations, Publ. Mat., 62 (2018), pages 34.
[23]	P. Pucci and R. Servadei, Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights, Ann. Inst. H. Poincaré A.N.L., 25 (2008), 505-537. doi: 10.1016/j.anihpc.2007.02.004.
[24]	P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $R^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806. doi: 10.1007/s00526-015-0883-5.
[25]	X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differential Equations, 258 (2015), 1106-1128. doi: 10.1016/j.jde.2014.10.012.
[26]	C. E. Torres Ledesma, Multiplicity result for non-homogeneous fractional Schrödinger-Kirchhoff type equations in $\mathbb R^n$, Adv. Nonlinear Anal., 5 (2016), 133-146. doi: 10.1515/anona-2015-0076.
[27]	Z. Wang and H.-S. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 499-508. doi: 10.3934/dcds.2016.36.499.

show all references

References:

[1]	A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.
[2]	G. Autuori, F. Cluni, V. Gusella and P. Pucci, Mathematical models for nonlocal elastic composite materials Adv. Nonlinear Anal. (2017), pages 39. doi: 10.1515/anona-2016-0186.
[3]	G. Autuori, F. Cluni, V. Gusella and P. Pucci, Effects of the fractional Laplacian order on the nonlocal elastic rod response, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering, 3 (2017), 030902, 5pp. doi: 10.1115/1.4036806.
[4]	G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714. doi: 10.1016/j.na.2015.06.014.
[5]	G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, Nonlinear Differ. Equ. Appl., 20 (2013), 977-1009. doi: 10.1007/s00030-012-0193-y.
[6]	L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845. doi: 10.3934/dcds.2016.36.1813.
[7]	H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Universitext Springer, New York, 2011, xiv+599 pp.
[8]	H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.
[9]	L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, 37-52, Abel Symp. 7, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25361-4_3.
[10]	M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129. doi: 10.1007/s10231-016-0555-x.
[11]	X. Chang, Ground states of some fractional Schrödinger equations on $\mathbb R^N$, Proc. Edinb. Math. Soc., 58 (2015), 305-321. doi: 10.1017/S0013091514000200.
[12]	D. C. de Morais Filho, M. A. S. Souto and J. M. do Ó., A compactness embedding lemma, a principle of symmetric criticality and applications to elliptic problems, Proyecciones, 19 (2000), 1-17. doi: 10.4067/S0716-09172000000100001.
[13]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[14]	S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb R^n$ Lecture Notes, Scuola Normale Superiore di Pisa (New Series), 15, Edizioni della Normale, Pisa, 2017, viii+152 pp. doi: 10.1007/978-88-7642-601-8.
[15]	M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961. doi: 10.1088/0951-7715/28/6/1937.
[16]	A. Fiscella and P. Pucci, p-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378. doi: 10.1016/j.nonrwa.2016.11.004.
[17]	A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011.
[18]	Y. Lei, Critical conditions and finite energy solutions of several nonlinear elliptic PDEs in $\mathbb R^n$, J. Differential Equations, 258 (2015), 4033-4061. doi: 10.1016/j.jde.2015.01.043.
[19]	P.-L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., 49 (1982), 315-334. doi: 10.1016/0022-1236(82)90072-6.
[20]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations 2nd edition, Grundlehren der Mathematischen Wissenschaften, 342, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-15564-2.
[21]	R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.
[22]	P. Piersanti and P. Pucci, Entire solutions for critical p-fractional Hardy Schrödinger Kirchhoff equations, Publ. Mat., 62 (2018), pages 34.
[23]	P. Pucci and R. Servadei, Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights, Ann. Inst. H. Poincaré A.N.L., 25 (2008), 505-537. doi: 10.1016/j.anihpc.2007.02.004.
[24]	P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $R^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806. doi: 10.1007/s00526-015-0883-5.
[25]	X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differential Equations, 258 (2015), 1106-1128. doi: 10.1016/j.jde.2014.10.012.
[26]	C. E. Torres Ledesma, Multiplicity result for non-homogeneous fractional Schrödinger-Kirchhoff type equations in $\mathbb R^n$, Adv. Nonlinear Anal., 5 (2016), 133-146. doi: 10.1515/anona-2015-0076.
[27]	Z. Wang and H.-S. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 499-508. doi: 10.3934/dcds.2016.36.499.

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