# American Institute of Mathematical Sciences

April  2019, 12(2): 129-137. doi: 10.3934/dcdss.2019009

## $G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$

 1 Dipartimento di Matematica e Applicazioni, Università di Napoli "Federico Ⅱ", 80126 Napoli, Italy 2 Università degli Studi di Napoli "Parthenope", Via Parisi 13, 80100 Napoli, Italy

* Corresponding author: Luigi D'Onofrio

Dedicated to Vicentiu Rǎdulescu on the occasion of his 60th birthday

Received  August 2017 Revised  January 2018 Published  August 2018

Fund Project: The third and fourth authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The third author has been also supported by "Sostegno alla Ricerca Locale " Università degli studi di Napoli "Parthenope ". The fourth author has been partially supported by FIRB 2013 project "Geometrical and qualitative aspects of PDE's".

The aim of this paper is to prove a reverse Hölder inequality for nonnegative adjoint solutions for elliptic operator in non divergence form in
 $\mathbb R^3$
. As an application we generalize a Theorem due to Sirazhudinov and Zhikov [24] and, under suitable assumptions, we characterize the
 $G$
-limit of a sequence of elliptic operator.The operator
 $N$
 $N[v] = \sum\limits_{i,j = 1}^3 {\frac{{{\partial ^2}({a_{ij}}v)}}{{\partial {x_i}\partial {x_j}}}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$
arises naturally as the formal adjoint of the operator in "non divergence form"
 $L[u] = \sum\limits_{i,j = 1}^3 {{a_{i,j}}} (x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} = Tr(A{D^2}u).\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 2 \right)$
The reason to study the solutions of the adjoint operator is that they are not only important for the solvability of
 $Lu = f$
but for the properties of the Green's function for
 $L$
. There is a long literature in this context, see for example Sÿogren [22], Bauman [2], Fabes and Stroock [12], Fabes, Garofalo, Marĺn-Malavé, and Salsa [11], Escauriaza and Kenig [10], and Escauriaza [9].
Citation: Teresa Alberico, Costantino Capozzoli, Luigi D'Onofrio, Roberta Schiattarella. $G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 129-137. doi: 10.3934/dcdss.2019009
##### References:
 [1] T. Alberico, C. Capozzoli and L. D'Onofrio, $G$-convergence for non-divergence second order elliptic operators in the plane, Diff. Int. Eq., 26 (2013), 1127-1138. [2] P. Bauman, Positive solutions of elliptic equations in nondivergent form and their adjoints, Ark. Mat., 22 (1984), 153-173.  doi: 10.1007/BF02384378. [3] B. Bojarski, L. D'Onofrio, T. Iwaniec and C. Sbordone, $G$-closed classes of elliptic operators in the complex plane, Ricerche Mat., 54 (2005), 403-432. [4] L. Caso, P. Cavaliere and M. Transirico, On the maximum principle for elliptic operators, Math Inequali. Appl., 7 (2004), 405-418.  doi: 10.7153/mia-07-41. [5] P. Cavaliere and M. Transirico, The Dirichlet problem for elliptic equations in the plane, Comment. Math. Univ. Carolinae, 46 (2005), 751-758. [6] P. Cavaliere and M. Transirico, A strong Maximum Principle for linear elliptic operators, Int. J. Pure Appl. Math, 57 (2009), 299-311. [7] F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with $VMO$ coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853.  doi: 10.2307/2154379. [8] L. D'Onofrio and L. Greco, A counter-example in $G$-convergence of non-divergence elliptic operators, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1299-1310.  doi: 10.1017/S0308210500002948. [9] L. Escauriaza, Weak type-$(1,1)$ inequalities and regularity properties of adjoint and normalized adjoint solutions to linear nondivergence form operators with VMO coefficients, Duke Math. J., 74 (1994), 177-201.  doi: 10.1215/S0012-7094-94-07409-7. [10] L. Escauriaza and C. Kenig, Area integral estimates for solutions and normalized adjoint solutions to nondivergence form elliptic equations, Ark. Mat., 31 (1993), 275-296.  doi: 10.1007/BF02559487. [11] E. Fabes, N. Garofalo, S. Marìn-Malavé and S. Salsa, Fatou theorems for some nonlinear elliptic equations, Rev. Mat. Ib., 4 (1988), 227-251.  doi: 10.4171/RMI/73. [12] E. Fabes and D. W. Stroock, The $L^p$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J., 51 (1984), 997-1016.  doi: 10.1215/S0012-7094-84-05145-7. [13] F. Giannetti, L. Kovalev, T. Iwaniec, G. Moscariello and C. Sbordone, On G-compactness of the Beltrami operators, in Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, NATO Sci. Ser. Ⅱ Math. Phys. Chem. 170, Kluwer Acad. Publ., Dordrecht, (2004), 107-138. doi: 10.1007/1-4020-2623-4_5. [14] P. Jones, Extension theorems for BMO, Indiana Univ, Math. J., 29 (1980), 41-66.  doi: 10.1512/iumj.1980.29.29005. [15] G. Moscariello and C. Sbordone, Optimal $L^p$-properties of Green's functions for non-divergence elliptic equations in two dimensions, (English summary), Studia Math., 169 (2005), 133-141.  doi: 10.4064/sm169-2-3. [16] F. Murat, H-convergence, in "Séminaire d'analyse fonctionnelle et numérique, "Université d'Alger, 1977 (multicopied, 34 pp.), English translation, F. Murat and L. Tartar, H-convergence, in: L. Cherkaev, R. H. Kohn (Eds.), Topics in the Mathematical Modelling of Composite Materials, in: Progress in Nonlinear Differential Equations and their Applications, 31, Birkhäauser, Boston, 1997, 21-43. [17] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa, Ser. IV, 5 (1978), 489-507. [18] T. Radice, Regularity result for nondivergence equations with unbounded coefficients, Differential Integral Equations, 23 (2010), 989-1000. [19] T. Radice, A higher-integrability result for nondivergence elliptic equations, Ann. Mat. Pura Appl.(4), 187 (2008), 93-103.  doi: 10.1007/s10231-006-0035-9. [20] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc., 207 (1975), 391-405.  doi: 10.1090/S0002-9947-1975-0377518-3. [21] C. Sbordone, The precise $L^p$-theory of elliptic equations in the plane, Progr. Nonlinear Differential Equations Appl., 63 (2005), 415-421.  doi: 10.1007/3-7643-7384-9_40. [22] P. Sÿogren, On the adjoint of an elliptic linear differential operator and its potential theory, Ark. Mat., 11 (1973), 153-165.  doi: 10.1007/BF02388513. [23] G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304.  doi: 10.1007/BF02414375. [24] V. V. Zhikov and M. M. Sirazhudinov, On $G$-compactness of nondivergence elliptic operators of second order, Math. USSR Izv., 19 (1982), 27-40.

show all references

Dedicated to Vicentiu Rǎdulescu on the occasion of his 60th birthday

##### References:
 [1] T. Alberico, C. Capozzoli and L. D'Onofrio, $G$-convergence for non-divergence second order elliptic operators in the plane, Diff. Int. Eq., 26 (2013), 1127-1138. [2] P. Bauman, Positive solutions of elliptic equations in nondivergent form and their adjoints, Ark. Mat., 22 (1984), 153-173.  doi: 10.1007/BF02384378. [3] B. Bojarski, L. D'Onofrio, T. Iwaniec and C. Sbordone, $G$-closed classes of elliptic operators in the complex plane, Ricerche Mat., 54 (2005), 403-432. [4] L. Caso, P. Cavaliere and M. Transirico, On the maximum principle for elliptic operators, Math Inequali. Appl., 7 (2004), 405-418.  doi: 10.7153/mia-07-41. [5] P. Cavaliere and M. Transirico, The Dirichlet problem for elliptic equations in the plane, Comment. Math. Univ. Carolinae, 46 (2005), 751-758. [6] P. Cavaliere and M. Transirico, A strong Maximum Principle for linear elliptic operators, Int. J. Pure Appl. Math, 57 (2009), 299-311. [7] F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with $VMO$ coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853.  doi: 10.2307/2154379. [8] L. D'Onofrio and L. Greco, A counter-example in $G$-convergence of non-divergence elliptic operators, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1299-1310.  doi: 10.1017/S0308210500002948. [9] L. Escauriaza, Weak type-$(1,1)$ inequalities and regularity properties of adjoint and normalized adjoint solutions to linear nondivergence form operators with VMO coefficients, Duke Math. J., 74 (1994), 177-201.  doi: 10.1215/S0012-7094-94-07409-7. [10] L. Escauriaza and C. Kenig, Area integral estimates for solutions and normalized adjoint solutions to nondivergence form elliptic equations, Ark. Mat., 31 (1993), 275-296.  doi: 10.1007/BF02559487. [11] E. Fabes, N. Garofalo, S. Marìn-Malavé and S. Salsa, Fatou theorems for some nonlinear elliptic equations, Rev. Mat. Ib., 4 (1988), 227-251.  doi: 10.4171/RMI/73. [12] E. Fabes and D. W. Stroock, The $L^p$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J., 51 (1984), 997-1016.  doi: 10.1215/S0012-7094-84-05145-7. [13] F. Giannetti, L. Kovalev, T. Iwaniec, G. Moscariello and C. Sbordone, On G-compactness of the Beltrami operators, in Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, NATO Sci. Ser. Ⅱ Math. Phys. Chem. 170, Kluwer Acad. Publ., Dordrecht, (2004), 107-138. doi: 10.1007/1-4020-2623-4_5. [14] P. Jones, Extension theorems for BMO, Indiana Univ, Math. J., 29 (1980), 41-66.  doi: 10.1512/iumj.1980.29.29005. [15] G. Moscariello and C. Sbordone, Optimal $L^p$-properties of Green's functions for non-divergence elliptic equations in two dimensions, (English summary), Studia Math., 169 (2005), 133-141.  doi: 10.4064/sm169-2-3. [16] F. Murat, H-convergence, in "Séminaire d'analyse fonctionnelle et numérique, "Université d'Alger, 1977 (multicopied, 34 pp.), English translation, F. Murat and L. Tartar, H-convergence, in: L. Cherkaev, R. H. Kohn (Eds.), Topics in the Mathematical Modelling of Composite Materials, in: Progress in Nonlinear Differential Equations and their Applications, 31, Birkhäauser, Boston, 1997, 21-43. [17] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa, Ser. IV, 5 (1978), 489-507. [18] T. Radice, Regularity result for nondivergence equations with unbounded coefficients, Differential Integral Equations, 23 (2010), 989-1000. [19] T. Radice, A higher-integrability result for nondivergence elliptic equations, Ann. Mat. Pura Appl.(4), 187 (2008), 93-103.  doi: 10.1007/s10231-006-0035-9. [20] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc., 207 (1975), 391-405.  doi: 10.1090/S0002-9947-1975-0377518-3. [21] C. Sbordone, The precise $L^p$-theory of elliptic equations in the plane, Progr. Nonlinear Differential Equations Appl., 63 (2005), 415-421.  doi: 10.1007/3-7643-7384-9_40. [22] P. Sÿogren, On the adjoint of an elliptic linear differential operator and its potential theory, Ark. Mat., 11 (1973), 153-165.  doi: 10.1007/BF02388513. [23] G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304.  doi: 10.1007/BF02414375. [24] V. V. Zhikov and M. M. Sirazhudinov, On $G$-compactness of nondivergence elliptic operators of second order, Math. USSR Izv., 19 (1982), 27-40.
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