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

  • * Corresponding author: Luigi D'Onofrio

    * Corresponding author: Luigi D'Onofrio 

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"

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  • 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].

    Mathematics Subject Classification: Primary: 35B40; Secondary: 35R05.


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  •   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. 
      P. Bauman , Positive solutions of elliptic equations in nondivergent form and their adjoints, Ark. Mat., 22 (1984) , 153-173.  doi: 10.1007/BF02384378.
      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. 
      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.
      P. Cavaliere  and  M. Transirico , The Dirichlet problem for elliptic equations in the plane, Comment. Math. Univ. Carolinae, 46 (2005) , 751-758. 
      P. Cavaliere  and  M. Transirico , A strong Maximum Principle for linear elliptic operators, Int. J. Pure Appl. Math, 57 (2009) , 299-311. 
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      P. Jones , Extension theorems for BMO, Indiana Univ, Math. J., 29 (1980) , 41-66.  doi: 10.1512/iumj.1980.29.29005.
      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.
      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.
      F. Murat , Compacité par compensation, Ann. Scuola Norm. Sup. Pisa, Ser. IV, 5 (1978) , 489-507. 
      T. Radice , Regularity result for nondivergence equations with unbounded coefficients, Differential Integral Equations, 23 (2010) , 989-1000. 
      T. Radice , A higher-integrability result for nondivergence elliptic equations, Ann. Mat. Pura Appl., 187 (2008) , 93-103.  doi: 10.1007/s10231-006-0035-9.
      D. Sarason , Functions of vanishing mean oscillation, Trans. Amer. Math. Soc., 207 (1975) , 391-405.  doi: 10.1090/S0002-9947-1975-0377518-3.
      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.
      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.
      G. Talenti , Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965) , 285-304.  doi: 10.1007/BF02414375.
      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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