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On Neumann’s problem for a quasilinear differential system of finite elastostatics type. Local theorems of existence and uniqueness. (English) Zbl 0532.35014

This paper concerns the Neumann’s boundary value problem for a quasilinear differential system of the type of finite elastostatics. More precisely let \(\Omega\) be a bounded open subset of \({\mathbb{R}}^ n\), let \(\nu\) be the unit outward normal to \(\partial \Omega\) and let a:\({\bar \Omega}\times {\mathbb{R}}^{n^ 2}\to {\mathbb{R}}^{n^ 2}\), \(f:\Omega \to {\mathbb{R}}^ n\) and \(g:\partial \Omega \to {\mathbb{R}}^ n\) be given functions with \(a(x,1)=0\), \(\forall x\in \Omega\). Then one looks for a function u:\({\bar \Omega}\to {\mathbb{R}}^ n\) such that \[ (P)\quad div A(u)+\vartheta f=0\quad in\quad \Omega,\quad -A(u)\nu +\vartheta g=0\quad on\quad \partial \Omega, \] where \(A(u)(x)=a(x,1+Du(x))\quad \forall x\in \Omega \quad bu=(D_ ju_ i) i,j=1,...,n\) and \(\vartheta\) is a real parameter. When \(n=3\) this problem corresponds to the ”dead traction problem” of finite elastostatics. The main results are local theorems of existence and uniqueness in Sobolev spaces and in Schauder spaces. The only hypotheses made on the function a are those suggested by the physical problem and thus any artificial assumption has been avoided. The results obtained are essentially generalizations, in various directions, of a result of F. Stoppelli [Ric. Mat. 6, 241-287 (1957); ibid. 7, 71-101, 138-152 (1958; Zbl 0097.173)]. The choice of spaces for solutions and data which are suitable for a local treatment of problem (P) requires a study of problems of differentiability of operators of various types, and of isomorphism problems for a divergence type linear matrix differential operator, in Sobolev spaces and in Schauder spaces.

MSC:

35G30 Boundary value problems for nonlinear higher-order PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
74B20 Nonlinear elasticity

Citations:

Zbl 0097.173
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References:

[1] A. Adams , Soboles Spaces , Academic Press , 1975 . Zbl 0314.46030 · Zbl 0314.46030
[2] S. Agmon - A. DOUGLIS - L. NIRENBERG, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II , Comm. Pure and Appl. Mathematics , 17 ( 1964 ), pp. 35 - 92 . MR 162050 | Zbl 0123.28706 · Zbl 0123.28706 · doi:10.1002/cpa.3160170104
[3] F. Browder , Estimates and existence theorems for elliptic boundary value problems , Proc. Nat. Acad. Sc. U.S.A. , 45 ( 1959 ), pp. 365 - 372 . MR 132913 | Zbl 0093.29402 · Zbl 0093.29402 · doi:10.1073/pnas.45.3.365
[4] H. Cartan , Cours de calcul differentiel , Hermann , Paris , 1977 . MR 223194 | Zbl 0923.58002 · Zbl 0923.58002
[5] G. Fichera , Existence Theorems in Elasticity, Handbuch der Physik , Bd. VIA / 2 , Springer Verlag ( 1972 ).
[6] J. Gobert , Une inégalité fondamentale de la théorie de l’élasticité, Bull. Soc. Roy . Sci. , Liège , 3 - 4 ( 1962 ), pp. 182 - 191 . MR 133684 | Zbl 0112.38902 · Zbl 0112.38902
[7] M.E. Gurtin - S. J. SPECTOR, On Stability and uniqueness in finite elasticity , Arch. Rational Mech. Anal. , 70 ( 1979 ), pp. 153 - 165 . MR 546633 | Zbl 0426.73038 · Zbl 0426.73038 · doi:10.1007/BF00250352
[8] J.L. Lions - E. MAGENES, Problemi ai limiti non omogenei (III) , Annali della Scuola Norm. Sup. , Pisa , XV ( 1961 ), pp. 39 - 101 . Numdam | MR 146526
[9] F. Stoppelli , Un teorema di esistenza ed unicità relativo alle equazioni dell’elastostatica isoterma per deformazioni finite , Ricerche Matematiche , 3 ( 1954 ), pp. 247 - 267 . MR 74237 | Zbl 0058.39701 · Zbl 0058.39701
[10] F. Stoppelli , Su un sistema di equazioni integro-differenziali interessante l’elastostatica , Ricerche Matematiche , 6 ( 1957 ), pp. 11 - 26 . MR 91003 | Zbl 0117.18403 · Zbl 0117.18403
[11] F. Stoppelli , Sull’esistenza di soluzioni delle equazioni dell’elastostatica isoterma nel caso di sollecitazioni dotate di assi di equilibrio I, II, III , Ricerche Matematiche , 6 ( 1957 ), pp. 241 - 287 ; 7 ( 1958 ), pp. 71 - 101 , 138 - 152 . Zbl 0097.17301 · Zbl 0097.17301
[12] J.L. Thompson , Some existence theorems for the traction boundary value problem of linearized elastostatics , Arch. Rational Mech. Anal. , 45 ( 1968 ), pp. 369 - 399 . MR 237130 | Zbl 0175.22108 · Zbl 0175.22108 · doi:10.1007/BF00275646
[13] T. Valent - G. ZAMPIERI, Sulla differenziabilità di un operatore legato a una classe di sistemi differenziali quasi lineari , Rend. Sem. Mat. Univ. Padova , 57 ( 1977 ), pp. 311 - 322 . Numdam | MR 526198 | Zbl 0402.35027 · Zbl 0402.35027
[14] T. Valent , Teoremi di esistenza e unicità in elastostatica finita , Rend. Sem. Mat. Univ. Padova 60 , ( 1979 ), pp. 165 - 181 . Numdam | Zbl 0425.73011 · Zbl 0425.73011
[15] T. Valent , Local existence and uniqueness theorems in finite elastostatics , Proceeding of the IUTAM symposium on finite elasticity, ed. Carlson and Shield, Martinus Nijhoff Publishers , the Hague - Boston - London , 1982 . MR 676677 · Zbl 0512.73038
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