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Global weak solutions to the stochastic Ericksen–Leslie system in dimension two
Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains
Department of Mathematics, Indiana University, Bloomington, 831 E 3rd St, Bloomington, IN 47405, USA |
By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as $ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $ with zero boundary data, have unexpected degenerate nature.
References:
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K. J. Böröczky, E. Lutwak, D. Yang and G. Zhang,
The logarithmic Minkowski problem, J. Amer. Math. Soc., 26 (2013), 831-852.
doi: 10.1090/S0894-0347-2012-00741-3. |
[2] |
L. A. Caffarelli,
A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math., 131 (1990), 129-134.
doi: 10.2307/1971509. |
[3] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.
doi: 10.1002/cpa.3160370306. |
[4] |
E. Calabi, Complete affine hyperspheres. I. Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971), Academic Press, London, 1972.
![]() ![]() |
[5] |
H. Chen and G. Huang,
Existence and regularity of the solutions of some singular Monge-Ampère equations, J. Differential Equations, 267 (2019), 866-878.
doi: 10.1016/j.jde.2019.01.030. |
[6] |
S. Y. Cheng and S. T. Yau,
On the regularity of the Monge-Ampère equation det(∂2u = ∂xi∂xj) = F (x, u), Comm. Pure Appl. Math., 30 (1977), 41-68.
doi: 10.1002/cpa.3160300104. |
[7] |
S. Y. Cheng and S.-T. Yau,
Complete affine hypersurfaces. I. The completeness of affine metrics, Comm. Pure Appl. Math., 39 (1986), 839-866.
doi: 10.1002/cpa.3160390606. |
[8] |
K.-S. Chou and X.-J. Wang,
The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83.
doi: 10.1016/j.aim.2005.07.004. |
[9] |
A. Figalli, The Monge-Ampère Equation and its Applications, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), 2017.
doi: 10.4171/170. |
[10] |
C. E. Gutiérrez, The Monge-Ampère Equation, Second edition. Birkhäuser, Boston, 2001.
doi: 10.1007/978-1-4612-0195-3. |
[11] |
H. Jian and Y. Li,
Optimal boundary regularity for a singular Monge-Ampère equation, J. Differential Equations, 264 (2018), 6873-6890.
doi: 10.1016/j.jde.2018.01.051. |
[12] |
H. Jian and Y. Li,
A singular Monge-Ampère equation on unbounded domains, Sci. China Math., 61 (2018), 1473-1480.
doi: 10.1007/s11425-018-9351-1. |
[13] |
H. Jian and X.-J. Wang,
Bernstein theorem and regularity for a class of Monge-Ampère equations, J. Differential Geom., 93 (2013), 431-469.
doi: 10.4310/jdg/1361844941. |
[14] |
N. Q. Le,
The eigenvalue problem for the Monge-Ampère operator on general bounded convex domains, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1519-1559.
|
[15] |
F. H. Lin and L. Wang,
A class of fully nonlinear elliptic equations with singularity at the boundary, J. Geom. Anal., 8 (1998), 583-598.
doi: 10.1007/BF02921713. |
[16] |
P.-L. Lions,
Two remarks on Monge-Ampère equations, Ann. Mat. Pura Appl., 142 (1985), 263-275.
doi: 10.1007/BF01766596. |
[17] |
E. Lutwak,
The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150.
doi: 10.4310/jdg/1214454097. |
[18] |
K. Tso,
On a real Monge-Ampère functional, Invent. Math., 101 (1990), 425-448.
doi: 10.1007/BF01231510. |
show all references
References:
[1] |
K. J. Böröczky, E. Lutwak, D. Yang and G. Zhang,
The logarithmic Minkowski problem, J. Amer. Math. Soc., 26 (2013), 831-852.
doi: 10.1090/S0894-0347-2012-00741-3. |
[2] |
L. A. Caffarelli,
A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math., 131 (1990), 129-134.
doi: 10.2307/1971509. |
[3] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.
doi: 10.1002/cpa.3160370306. |
[4] |
E. Calabi, Complete affine hyperspheres. I. Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971), Academic Press, London, 1972.
![]() ![]() |
[5] |
H. Chen and G. Huang,
Existence and regularity of the solutions of some singular Monge-Ampère equations, J. Differential Equations, 267 (2019), 866-878.
doi: 10.1016/j.jde.2019.01.030. |
[6] |
S. Y. Cheng and S. T. Yau,
On the regularity of the Monge-Ampère equation det(∂2u = ∂xi∂xj) = F (x, u), Comm. Pure Appl. Math., 30 (1977), 41-68.
doi: 10.1002/cpa.3160300104. |
[7] |
S. Y. Cheng and S.-T. Yau,
Complete affine hypersurfaces. I. The completeness of affine metrics, Comm. Pure Appl. Math., 39 (1986), 839-866.
doi: 10.1002/cpa.3160390606. |
[8] |
K.-S. Chou and X.-J. Wang,
The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83.
doi: 10.1016/j.aim.2005.07.004. |
[9] |
A. Figalli, The Monge-Ampère Equation and its Applications, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), 2017.
doi: 10.4171/170. |
[10] |
C. E. Gutiérrez, The Monge-Ampère Equation, Second edition. Birkhäuser, Boston, 2001.
doi: 10.1007/978-1-4612-0195-3. |
[11] |
H. Jian and Y. Li,
Optimal boundary regularity for a singular Monge-Ampère equation, J. Differential Equations, 264 (2018), 6873-6890.
doi: 10.1016/j.jde.2018.01.051. |
[12] |
H. Jian and Y. Li,
A singular Monge-Ampère equation on unbounded domains, Sci. China Math., 61 (2018), 1473-1480.
doi: 10.1007/s11425-018-9351-1. |
[13] |
H. Jian and X.-J. Wang,
Bernstein theorem and regularity for a class of Monge-Ampère equations, J. Differential Geom., 93 (2013), 431-469.
doi: 10.4310/jdg/1361844941. |
[14] |
N. Q. Le,
The eigenvalue problem for the Monge-Ampère operator on general bounded convex domains, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1519-1559.
|
[15] |
F. H. Lin and L. Wang,
A class of fully nonlinear elliptic equations with singularity at the boundary, J. Geom. Anal., 8 (1998), 583-598.
doi: 10.1007/BF02921713. |
[16] |
P.-L. Lions,
Two remarks on Monge-Ampère equations, Ann. Mat. Pura Appl., 142 (1985), 263-275.
doi: 10.1007/BF01766596. |
[17] |
E. Lutwak,
The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150.
doi: 10.4310/jdg/1214454097. |
[18] |
K. Tso,
On a real Monge-Ampère functional, Invent. Math., 101 (1990), 425-448.
doi: 10.1007/BF01231510. |
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