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On the very weak solvability of the beam equation

Díaz Díaz, Jesús Ildefonso (2011) On the very weak solvability of the beam equation. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A: Matemáticas, 105 (1). pp. 167-172. ISSN 1578-7303


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We get some necessary and sufficient conditions for the very weak solvability of the beam equation stated in terms of powers of the distance to the boundary, accordingly to the boundary condition under consideration. We get a L(1)-estimate by using an abstract result due to Crandall and Tartar. Applications to some nonlinear perturbed equations and to the eventual positivity of the solution of the parabolic problems are also given.

Item Type:Article
Uncontrolled Keywords:boundary; order; distance; respect; Beam equation; Very weak solutions; Accretive operators; Maximum principle
Subjects:Sciences > Mathematics > Differential equations
ID Code:15036

Bachar I., Māagli H., Masmoudi S., Zribi M.: Estimates for the Green function and singular solutions for polyharmonic nonlinear equation. Abstr. Appl. Anal. 2003(12), 715–741 (2003)

Benilan, Ph., Crandall, M.G., Pazy, A.: Nonlinear Evolution Governed by Accretive Operators. Book Manuscript, Besançon, France (2001)

Bernis F.: On some nonlinear singular boundary value problems of higher order. Nonlinear Anal. Theory Methods Appl. 26(6), 1061–1078 (1996)

Boggio T.: Sulle funzioni di Green d’ordine m. Rendiconti del Circolo Matematico di Palermo Serie II 20, 97–135 (1905)

Brezis, H.: Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam (1972)

Brezis H., Cazenave T., Martel Y., Ramiandrisoa A.: Blow up for u t −Δu = g(u) revisited. Adv. Differ. Equ. 1, 73–90 (1996)

Brezis H., Cabré X.: Some simple nonlinear PDE’s without solutions. Boll. Unione Mat. Ital. 1-B, 223–262 (1998)

Chow S.N., Dunninger D.R., Lasota A.: A maximum principle for fourth order ordinary differential equations. J. Differ. Equ. 14, 101–105 (1973)

Crandall M.G., Tartar L.: Some relations between nonexpansive and order preserving maps. Proc. AMS 78(3), 385–390 (1980)

Díaz, J.I.: Very weak solutions of the beam and other higher order equations, Rev. Real Academia de Ciencias de Zaragoza (2011, to appear)

Díaz J.I., Rakotoson J.M.: On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary. J. Funct. Anal. 257, 807–831 (2009)

Díaz J.I., Rakotoson J.M.: On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete Continuous Dyn. Syst. 27(3), 1037–1058 (2010)

Díaz, J.I., Hernandez, J., Rakotoson, J.M.: Positive very weak solutions of singular semilinear problems (2011, in preparation)

Duffin R.J.: On a question of Hadamard concerning super-biharmonic functions. J. Math. Phys. 27, 253–258 (1949)

Gazzola F., Grunau H.C.: Some new properties of biharmonic heat kernels. Nonlinear Anal. 70, 2965–2973 (2009)

Gupta C.P.: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal. 26, 289–305 (1988)

Hadamard J.: Memoire sur le probleme d’analyse relatif a l’equilibre des plaques elastiques incastrees. Memoires presentées par divers savants a l’Academie des Sciences 33, 1–128 (1908)

Kawohl B., Sweers G.: On ‘anti’-eigenvalues for elliptic systems and a question of McKenna and Walter. Indiana U. Math. J. 51, 1023–1040 (2002)

McKenna P.J., Walter W.: Nonlinear oscillations in a suspension bridge. Arch. Rational Mech. Anal. 98, 167–177 (1987)

Schröder, J.: Zusammenhängende Mengen inverspositiver Differentialoperatoren vierter Ordnung. Math. Z. 96, 89–110 (1967) (in German)

Stakgold I.: Green’s Functions and Boundary Value Problems. Wiley, New York (1998)

Yao Q., Li Y.: Solution and positive solution to nonlinear cantilever beam equations. J. Southwest Jiaotong Univ. Engl. Ed. 16, 51–54 (2008)

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