Abstract
This work surveys results on the existence and asymptotic behavior of the fundamental solution for an elliptic operator L in nondivergence form, including recent results for operators whose coefficients are continuous with mild conditions on the modulus of continuity: if the square of the modulus of continuity satisfies the Dini condition, then there is an integral invariant which controls the behavior of solutions of L*u=0 and whether there is a fundamental solution for L that is asymptotic to the fundamental solution for the associated constant coefficient operator.
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References
P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints. Ark. Mat. 22 (1984), 153–173.
P. Bauman, Equivalence of the Green’s function for diffusion operators in ℝn: a counterexample. Proc. Amer. Math. Soc. 91 (1984), 64–68.
P. Bauman, A Wiener test for nondivergence structure, second-order elliptic equations. Indiana Univ. Math. J. 34 (1985), 825–844.
M. Cerutti, L. Escauriaza, E. Fabes, Uniqueness in the Dirichlet problem for some elliptic operators with discontinuous coefficients. Annali di Matematica pura ed applicata 163 (1993), 161–180.
L. Escauriaza, Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form. Comm. Part. Diff. Equat. 25 (2000), 821–845.
E.B. Fabes, D. Stroock, The L p-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations. Duke Math. J. 51 (1984), 977–1016.
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. 2nd Edition, Springer-Verlag, 1983.
M. Grüter, K.-O. Widman, The Green function for uniformly elliptic equations. Manuscripta Math. 37 (1982), 303–342.
L. Hörmander, The Analysis of Linear Partial Differential Operators III. Springer-Verlag, Berlin, 1985.
W. Littman, G. Stampacchia, H. Weinberger, Regular points for elliptic equations with discontinuous coefficients. Ann. Sc. Norm. Super. Pisa 17 (1963), 45–79.
R. Lockhart, R. McOwen, Elliptic differential operators on noncompact manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 12 (1985), 409–447.
V. Maz’ya, R. McOwen, Asymptotics for solutions of elliptic equations in double divergent form. Comm. Part. Diff. Equat. 32 (2007), 1–17.
V. Maz’ya, R. McOwen, On the fundamental solution of an elliptic equation in nondivergence form. To appear in: Nonlinear partial differential equations and related topics, Amer. Math. Soc. Transl., (2010).
V. Maz’ya, B. Plamenevski, Estimates in L p and Hölder classes and the Miranda-Agmon maximum principle solutions of elliptic boundary problems in domains with singular points on the boundary (Russian). Math. Nachr. 81 (1978), 25–82.
R. McOwen, The behavior of the Laplacian on weighted Sobolev spaces. Comm. Pure Appl. Math. 32 (1979), 783–795.
C. Miranda, Partial Differential Equations of Elliptic Type, Springer-Verlag, Berlin, 1970.
M. Taylor, Tools for PDE, Mathematical Surveys and Monographs 81, AMS, 2000.
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To Vladimir Maz’ya for his 70th birthday
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© 2009 Birkhäuser Verlag Basel/Switzerland
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McOwen, R. (2009). On Elliptic Operators in Nondivergence and in Double Divergence Form. In: Cialdea, A., Ricci, P.E., Lanzara, F. (eds) Analysis, Partial Differential Equations and Applications. Operator Theory: Advances and Applications, vol 193. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9898-9_13
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DOI: https://doi.org/10.1007/978-3-7643-9898-9_13
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