Remarks on the Harnak Inequality for Local-Minima of Scalar Integral Functionals with General Growth Conditions

Abstract

In this paper we proof a Harnack inequality and a regularity theorem for local-minima of scalar intagral functionals with general growth conditions.

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Granucci, T. (2014) Remarks on the Harnak Inequality for Local-Minima of Scalar Integral Functionals with General Growth Conditions. Journal of Applied Mathematics and Physics, 2, 194-203. doi: 10.4236/jamp.2014.25024.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Adams, R. (1975) Sobolev Spaces. Accademic Press, New York.
[2] Ambrosio, L. Lecture Notes on Partial Differential Equations.
[3] Astarita, G. and Marrucci, G. (1974) Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, London.
[4] Bhattacharaya, T. and Leonetti, F. (1993) W^{2,2} Regularity for Weak Solutions of Elleptic Systems with Nonstandard Growth. J. Math. Anal. Appl., 176, 224-234. http://dx.doi.org/10.1006/jmaa.1993.1210
[5] De Giorgi, E. (1957) Sulla differenziabilità e l'analicità delle estremali degli integrali multipli. Mem.Accad. Sci Torino, cl. Sci. Fis. Mat. Nat., 3, 25-43.
[6] Di Benedetto, E. and Trudinger, N. (1984) Harnack Inequalities for Quasi-Minima of Variational Integrals. Ann. Inst. H. Poincaré (analyse non lineaire), 1, 295-308.
[7] Fuchs, M. and Seregin, G. (1998) A Regularity Theory for Variational Intgrals with LlnL-Growth. Calc. Var., 6, 171-187. http://dx.doi.org/10.1007/s005260050088
[8] Fuchs, M. and Mingione, G. (2000) Full C^{1,α}-Regularity for Free and Constrained Local Minimizers of Elliptic Variational Integrals with Nearly Linear Growth. Manuscripta Mathematica, 102, 227-250. http://dx.doi.org/10.1007/s002291020227
[9] Giaquinta, M. and Giusti, E. (1982) On the Regularity of Minima of Variational Integrals. Acta Mathematica, 148, 31-46. http://dx.doi.org/10.1007/BF02392725
[10] Giaquinta, M. and Giusti, E. (1984) Quasi-Minima. Ann. Inst. H. Poincarè (Analyse non lineaire), 1, 79-107.
[11] Giaquinta, M. and Martinazzi, L. (2005) An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs. S.N.S. press, Pisa.
[12] Giusti, E. (1994) Metodi diretti nel Calcolo delle Variazioni. U. M. I., Bologna.
[13] Gosez, J.P. (1974) Non Linear Elliptic Problems for Equations with Rapidly (or Slowly) Increasing Coefficents. Transactions of the American Mathematical Society, 190, 163-205. http://dx.doi.org/10.1090/S0002-9947-1974-0342854-2
[14] Granucci, T. (2014) An Alternative Proof of the H?lder Continuity of Quasi-Minima of Scalar Integral Functionals with General Growths. Afr. Mat., 25,197-212. http://dx.doi.org/10.1007/s13370-012-0109-3
[15] Granucci, T. An Alternative Proof of the H?lder Continuity of Quasi-Minima of Scalar Integral Functionals with General Growths. Part II. Submit to Afrika Matematika.
[16] Granucci, T. Observations on LΦ-L∞ Estimations and Applications to Regularity. Submit to Indian Journal of Pure and Applied Mathematics.
[17] Lieberman, G.M. (1991) The Natural Generalization of the Natural Conditions of Ladyzhenskaya and Ural'tseva for Elliptic Equations. Communications in Partial Differential Equations, 16, 331-361. http://dx.doi.org/10.1080/03605309108820761
[18] Klimov, V.S. (2000) Embedding Theorems and Continuity of Generalized Solutions of Quasilinear Elliptic Equations. Differential Equation, 36, 870-877. http://dx.doi.org/10.1007/BF02754410
[19] Krasnosel’skij, M. A. and Rutickii, Ya.B. (1961) Convex Function and Orlicz Spaces. Noordhoff, Groningen.
[20] Marcellini, P. (1993) Regularity for Elliptic Equations with General Growth Conditions. Journal of Differential Equations, 105, 296-333. http://dx.doi.org/10.1006/jdeq.1993.1091
[21] Maercellini, P. (1996) Regularity for Some Scalar Variational Problems under General Growth. Journal of Optimization Theory and Applications, 1, 161-181. http://dx.doi.org/10.1007/BF02192251
[22] Marcellini, P. (1996) Everywhere Regularity for a Class of Elliptic Systems without Growth Conditions. Annali della Scuola Normale Superiore di Pisa, 23, 1-25.
[23] Mascolo, E. and Papi, G. (1994) Local Bounddeness of Minimizers of Integrals of the Calculus of Variations. Annali di Matematica Pura ed Applicata, 167, 323-339. http://dx.doi.org/10.1007/BF01760338
[24] Mascolo, E. and Papi, G. (1996) Harnack Inequality for Minimizer of Integral Functionals with General Growth Conditions. Nonlinear Differential Equations and Applications, 3, 231-244. http://dx.doi.org/10.1007/BF01195916
[25] Mingione, G. and Siepe, F. (1999) Full C^{1,α} Regularity for Minimizers of Integral Functionals with L logL Growth. Zeitschrift für Analysis und ihre Anwendungen, 18, 1083-1100. http://dx.doi.org/10.4171/ZAA/929
[26] Moscariello, G. and Nania, L. (1991) H?lder Continuity of Minimizers of Functionals with Nonstandard Growth Conditions. Ricerche di Matematica, 15, 259-273.
[27] Nash, J. (1958) Continuity of Solutions of Parabolic and Elliptic Differential Equations. American Journal of Mathematics, 80, 931-953. http://dx.doi.org/10.2307/2372841
[28] Rao, M.M. and Ren, Z.D. (1991) Theory of Orlicz Spaces. Marcel Dekker, New York.
[29] Talenti, G. (1990) Bounddeness of Minimizers. Hokkaido Mathematical Journal, 19, 259-279. http://dx.doi.org/10.14492/hokmj/1381517360

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