Sharp Conditions for Korn Inequalities in Orlicz Spaces

Dominic Breit1, Lars Diening2
1Mathematisches Institut der Universität München, München, Germany
2LMU Munich, Theresienstr. 39, 80333, Munich, Germany

Tóm tắt

Từ khóa


Tài liệu tham khảo

Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)

Acerbi A., Mingione G.: Regularity results for stationary electro-rheological fluids. Arch. Rat. Mech. Anal. 164, 213–259 (2002)

Breit, D., Diening, L., Fuchs, M.: Existence results for stationary flows of Prandtl-Eyring fluids in 2D by means of solenoidal Lipschitz truncation. OxPDE Technical Report 11/20, University of Oxford

Breit, D., Fuchs, M.: The nonlinear Stokes problem with general potentials having superquadratic growth. J. Math. Fluid Mech. 13, 371–385 (2011)

Conti S., Faraco D., Maggi F.: A new approach to counterexamples to L1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions. Arch. Rat. Mech. Anal. 175, 287–300 (2005)

Courant R., Hilbert D.: Methoden der mathematischen Physik, vol. II. Springer, Berlin (1937)

Cruz-Uribe, D., Martell, J.M., Pérez, C.: Extensions of Rubio de Francia’s extrapolation theorem. Collect. Math. (Vol. Extra), 195–231 (2006)

Duvaut G., Lions J.L.: Inequalities in Mechanics and Physics. Springer Grundlehren, vol. 219. Springer, Berlin (1976)

Diening L., Růžička M.: Calderon–Zygmund operators on generalized Lebesgue spaces Lpx and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197–220 (2003)

Diening, L., Růžička, M., Schumacher, K.: A decomposition technique for John domains. Ann. Acad. Sci. Fenn. Math. 35(1), 87–114 (2010). http://dx.doi.org/10.5186/aasfm.2010.3506

Èidus D.M.: On a mixed problem of the theory of elasticity. Dokl. Akad. Nauk SSSR 76, 181–184 (1951)

Eyring H.J.: Viscosity, plasticity, and diffusion as example of absolute reaction rates. J. Chem. Phys. 4, 283–291 (1936)

Friedrichs K.: On the boundary value problems of the theory of elasticity and Korn’s inequality. Ann. Math. 48(2), 441–471 (1947)

Frehse, J., Seregin, G.: Regularity of solutions to variational problems of the deformation theory of plasticity with logarithmic hardening. In: Proceedings of St. Petersburg Mathematical Soceity, vol. 5, pp. 184–222/ English translation: Am. Math. Soc. Transl. II, vol. 193, pp. 127–152 (1998/1999)

Fuchs M.: On stationary incompressible Norton fluids and some extensions of Korn’s inequality. Zeitschr. Anal. Anwendungen 13(2), 191–197 (1994)

Fuchs M.: Korn inequalities in Orlicz spaces. Irish Math. Soc. Bull. 65, 5–9 (2010)

Fuchs, M., Seregin, G.: Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids. In: Lecture Notes in Mathematics, vol. 1749. Springer, Berlin (2000)

Gobert J.: Une inéquation fondamentale de la théorie de l’élasticité. Bull. Soc. Roy. Sci. Liege 3-4, 182–191 (1962)

Gobert J.: Sur une inégalité de coercivité. J. Math. Anal. Appl. 36, 518–528 (1971)

Jia H., Li D., Whang L.: Regularity theory in Orlicz spaces for elliptic equations in Reifenberg domains. J. Math. Anal. Appl. 334, 804–817 (2007)

Kirchheim B., Müller S., Svérak V.: Studying nonlinear pde by geometry in matrix space. In: Hildebrandt, S., Karcher, H. (eds) Geometriy Analysis and Nonlinear Partial Differential Equations, pp. 347–395. Springer, Berlin (2003)

de Leeuw K., Mirkil H.: A priori estimates for differential operators in L ∞ norm. Ill. J. Math. 8, 112–124 (1964)

Müller, S.: Variational models for microstructure and phase transitions. In: Bethuel, F. (ed.) Calculus of variations and geometric evolution problems. Springer Lecture Notes in Math, vol. 1713. Springer, Berlin, pp. 85–210 (1999)

Mihlin S.G.: The Problem of the Minimum of a Quadratic Functional. GITTL, Moscow (1952)

Mosolov P.P., Mjasnikov V.P.: On the correctness of boundary value problems in the mechanics of continuous media. Math. USSR Sbornik 17(2), 257–267 (1972)

Müller S., Svérak V.: Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math. 157, 715–742 (2003)

Málek J., Necaˇs J., Rokyta M., Růžička M.: Weak and Measure Valued Solutions to Evolutionary PDEs. Chapman & Hall, London (1996)

Nečas, J.: Sur les normes équivalentes dans $${W_k^p(\Omega)}$$ et sur la coecivité des formes formellement positives. In: Séminaire Equations aus Dérivées Partielles, Les Presses de l’Université de Montréal, pp. 102–128 (1966)

Ornstein D.: A non-inequality for differential operators in the L 1 norm. Arch. Rational Mech. Anal. 11, 40–49 (1964)

Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. In: Lecture Notes in Mathematics, vol. 1748. Springer (2000)

Temam R.: Mathematical Problems in Plasticity. Gauthier Villars, Paris (1985)

Zeidler E.: Nonlinear Functional Analysis II/B-Nonlinear Monotone Operators. Springer, Berlin (1987)