On the weak continuity of elliptic operators and applications to potential theory

In this paper, we establish weak continuity results for quasilinear elliptic and subelliptic operators of divergence form, acting on corresponding classes of subharmonic functions. These results are analogous to our earlier results for fully nonlinear k-Hessian operators. From the weak continuity, we derive various potential theoretic results including capacity estimates, potential estimates and the Wiener criterion for regular boundary points. Our methods make substantial use of Harnack inequalities for solutions.

[1]  N. Trudinger,et al.  Hessian measures I , 1997 .

[2]  C. Carathéodory.  Untersuchungen über die Grundlagen der Thermodynamik , 1909 .

[3]  N. Trudinger,et al.  Harnack inequalities for quasi-minima of variational integrals , 1984 .

[4]  J. Serrin.  Isolated singularities of solutions of quasi-linear equations , 1965 .

[5]  A. Sánchez-Calle.  Fundamental solutions and geometry of the sum of squares of vector fields , 1984 .

[6]  Donatella Danielli,et al.  Regularity at the Boundary for Solutions of Nonlinear Subelliptic Equations , 1995 .

[7]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[8]  L. Carleson.  Selected Problems on Exceptional Sets , 1998 .

[9]  W. Beckner,et al.  Conference on Harmonic Analysis in honor of Antoni Zygmund , 1983 .

[10]  Wei-Liang Chow.  Über Systeme von linearen partiellen Differential-gleichungen erster Ordnung , 1941 .

[11]  J. Lions,et al.  Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder , 1964 .

[12]  Neil S. Trudinger,et al.  On harnack type inequalities and their application to quasilinear elliptic equations , 1967 .

[13]  E. Stein,et al.  Balls and metrics defined by vector fields I: Basic properties , 1985 .

[14]  Luca Capogna,et al.  An embedding theorem and the harnack inequality for nonlinear subelliptic equations , 1993 .

[15]  T. Kilpeläinen,et al.  Degenerate elliptic equations with measure data and nonlinear potentials , 1992 .

[16]  Laurent Veron,et al.  Singularities of solutions of second order quasilinear equations , 1996 .

[17]  O. Martio,et al.  Two theorems of N. Wiener for solutions of quasilinear elliptic equations , 1985 .

[18]  M. Gromov.  Carnot-Carathéodory spaces seen from within , 1996 .

[19]  J. Heinonen,et al.  Nonlinear Potential Theory of Degenerate Elliptic Equations , 1993 .

[20]  D. Labutin.  Potential estimates for a class of fully nonlinear elliptic equations , 2002 .

[21]  Bruno Franchi,et al.  Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations , 1991 .

[22]  T. Kilpeläinen,et al.  The Wiener test and potential estimates for quasilinear elliptic equations , 1994 .

[23]  J. Rodrigues.  Obstacle Problems in Mathematical Physics , 1987 .

[24]  Wei-Liang Chow.  Über Systeme von liearren partiellen Differentialgleichungen erster Ordnung , 1940 .

[25]  L. Hörmander.  Hypoelliptic second order differential equations , 1967 .

[26]  Hessian measures II , 1999, math/9909199.

[27]  E. Heine,et al.  Untersuchungen über die Reihe . , 1847 .

[28]  L. Capogna,et al.  Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations , 1996 .

[29]  Richard L. Wheeden,et al.  Sobolev and isoperimetric inequalities for degenerate metrics , 1994 .

[30]  B. A. Taylor,et al.  A new capacity for plurisubharmonic functions , 1982 .

[31]  D. Danielli,et al.  Trace inequalities for Carnot-Carathéodory spaces and applications , 1998 .

[32]  J. Moser.  On Harnack's theorem for elliptic differential equations† , 1961 .

[33]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[34]  Jan Malý,et al.  Fine Regularity of Solutions of Elliptic Partial Differential Equations , 1997 .

[35]  L. Hedberg,et al.  Function Spaces and Potential Theory , 1995 .

[36]  L. Capogna,et al.  Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type , 2002 .

[37]  Lawrence C. Evans,et al.  Weak convergence methods for nonlinear partial differential equations , 1990 .

[38]  E. Stein,et al.  Hypoelliptic differential operators and nilpotent groups , 1976 .

[39]  Paolo Marcellini.  General Growth Conditions and Regularity , 1996 .

[40]  N. Trudinger,et al.  Hessian Measures III , 2002 .

[41]  Wolfhard Hansen,et al.  The Dirichlet problem for sublaplacians on nilpotent Lie groups —Geometric criteria for regularity , 1987 .

[42]  C. Kenig.  Potential theory of non-divergence form elliptic equations , 1993 .

[43]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[44]  D. Jerison.  The Poincaré inequality for vector fields satisfying Hörmander’s condition , 1986 .

[45]  C. Kenig,et al.  The Wiener test for degenerate elliptic equations , 1982 .

[46]  W. Ziemer,et al.  Local behavior of solutions of quasilinear elliptic equations with general structure , 1990 .

[47]  Nicola Garofalo,et al.  ISOPERIMETRIC AND SOBOLEV INEQUALITIES FOR CARNOT-CARATHEODORY SPACES AND THE EXISTENCE OF MINIMAL SURFACES , 1996 .

[48]  Paolo Negrini,et al.  Wiener criterion for a class of degenerate elliptic operators , 1987 .