Abstract
We consider themain boundary value problems of linear elastostatics with nonregular data. We prove existence and uniqueness results for bounded and exterior domains of ℝ3 of class Ck (k ⩾ 2). In the case of isotropic body, we prove the results for domains of class C1,α (α ∈ (0, 1]) and of class C1 in the case of the displacement problem.
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References
J. Nečas, Les Méthodes Directes en Théorie des Équations Élliptiques, Masson/Academia, Paris/Prague, 1967.
A. Cialdea, Elastostatics with non absolutely continuous data, J. Elasticity, 23:13–51, 1990.
G. Duvant and J.L. Lions, Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., Vol. 219, Springer- Verlag, Berlin, Heidelberg, 1976.
E.B. Fabes, M. Jodeit Jr., and N.M. Rivière, Potential techniques for boundary value problems on C1 domains, Acta Math., 141:165–185, 1978.
G. Fichera, Sull’esistenza e sul calcolo delle soluzioni dei problemi al contorno, relativi all’equilibrio di un corpo elastico, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III Ser., 4:35–99, 1950.
G. Fichera, Existence theorems in elasticity, in C. Truesedell (Ed.), Handbuch der Physik, Band VIa/2, Festkörpermechanik II, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
E. Giusti, Metodi Diretti nel Calcolo delle Variazioni, Unione Matematica Italiana, Bologna, 1994. English transl.: Direct Methods in the Calculus of Variations,Word Scientific, Singapore, 2004.
M.E. Gurtin, The linear theory of elasticity, in C. Truesedell (Ed.), Handbuch der Physik, Band VIa/2, Festkörpermechanik II, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
F. John, PlaneWaves and SphericalMeans Applied to Partial Differential Equations, Interscience, New York, 1955.
N. Kalton and M. Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Am. Math. Soc., 350(10):3903–3922, 1998.
V.D. Kupradze (Ed.), Three Dimensional Problems of the Mathematical Theory of Elasticity And Thermoelasticity, North-Holland Ser. Appl. Math. Mech., Vol. 25, North-Holland, Amsterdam, 1979.
J.L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, Vol. 1, Grundlehren Math.Wiss., Vol. 181, Springer-Verlag, Berlin, Heidelberg, 1972.
A. Russo and A. Tartaglione, On the contact problem of classical elasticity, J. Elasticity, 99:19–38, 2010.
A. Russo and A. Tartaglione, Strong uniqueness theorems and the Phragmèn–Lindelöf principle in nonhomogeneous elastostatics, J. Elasticity, 102:133–149, 2011.
A. Russo and A. Tartaglione, On the Stokes problem with data in L1, Z. Angew. Math. Phys., 64:1327–1336, 2013.
M. Schechter, Principles of Functional Analysis, Grad. Stud. Math., Vol. 36, AMS, Providence, RI, 2002.
G. Starita and A. Tartaglione, On the Fredholm property of the trace operators associated with the elastic layer potentials, Mathematics, 7:134, 2019.
A. Tartaglione, On the Stokes and Oseen problems with singular data, J. Math. Fluid Mech., 16:407–417, 2014.
A. Tartaglione, A note on the displacement problem of elastostatics with singular boundary values, Axioms, 8(2):46, 2019.
R. Temam, Navier Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1979.
L. Van Hove, Sur l’extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues, Nederl. Akad. Wet., Proc., Ser. A, 50:18–23, 1947.
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Starita, G., Tartaglione, A. Boundary value problems in elastostatics with singular data. Lith Math J 60, 396–409 (2020). https://doi.org/10.1007/s10986-020-09489-3
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DOI: https://doi.org/10.1007/s10986-020-09489-3