Abstract
In this note, we obtain sharp bounds for the Green’s function of the linearized Monge–Ampère operators associated to convex functions with either Hessian determinant bounded away from zero and infinity or Monge–Ampère measure satisfying a doubling condition. Our result is an affine invariant version of the classical result of Littman–Stampacchia–Weinberger for uniformly elliptic operators in divergence form. We also obtain the L p integrability for the gradient of the Green’s function in two dimensions. As an application, we obtain a removable singularity result for the linearized Monge–Ampère equation.
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Caffarelli L.A.: Interior W 2,p estimates for solutions to the Monge–Ampère equation. Ann. Math. 131(1), 135–150 (1990)
Caffarelli L.A., Gutiérrez C.E.: Properties of the solutions of the linearized Monge–Ampère equation. Am. J. Math. 119(2), 423–465 (1997)
De Philippis G., Figalli A., Savin O.: A note on interior \({W^{2, 1+\varepsilon}}\) estimates for the Monge–Ampère equation. Math. Ann. 357, 11–22 (2013)
Grüter M., Widman K.O.: The Green function for uniformly elliptic equations. Manuscr. Math. 37(3), 303–342 (1982)
Gutiérrez C.E.: The Monge–Ampère Equation. Birkhaüser, Boston (2001)
Le N.Q., Savin O.: Some minimization problems in the class of convex functions with prescribed determinant. Anal. PDE. 6(5), 1025–1050 (2013)
Littman W., Stampacchia G., Weinberger H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (3) 17, 43–77 (1963)
Maldonado D.: The Monge–Ampère quasi-metric structure admits a Sobolev inequality. Math. Res. Lett. 20(3), 527–536 (2013)
Maldonado D.: On the \({W^{2, 1+\varepsilon}}\) estimates for the Monge–Ampère equation and related real analysis. Calc. Var. Partial Differ. Equ. 50(1-2), 94–114 (2014)
Schmidt T.: \({W^{2, 1 +\varepsilon}}\) estimates for the Monge–Ampère equation. Adv. Math. 240, 672–689 (2013)
Tian G.J., Wang X.J.: A class of Sobolev type inequalities. Methods Appl. Anal. 15(2), 263–276 (2008)
Trudinger, N.S., Wang, X.-J.: The Monge–Ampère equation and its geometric applications. Handbook of geometric analysis. No. 1, 467–524. Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA (2008)
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Le, N.Q. Remarks on the Green’s function of the linearized Monge–Ampère operator. manuscripta math. 149, 45–62 (2016). https://doi.org/10.1007/s00229-015-0766-2
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DOI: https://doi.org/10.1007/s00229-015-0766-2