AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
On Space-Time Quasiconcave Solutions of the Heat Equation
About this Title
Chuanqiang Chen, Xinan Ma and Paolo Salani
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 259, Number 1244
ISBNs: 978-1-4704-3524-0 (print); 978-1-4704-5243-8 (online)
DOI: https://doi.org/10.1090/memo/1244
Published electronically: April 12, 2019
Keywords: Heat equation,
quasiconcavity,
space-time level set,
constant rank theorem,
space-time quasiconcave solution
MSC: Primary 35K20; Secondary 35B30
Table of Contents
Chapters
- 1. Introduction
- 2. Basic definitions and the Constant Rank Theorem technique
- 3. A microscopic space-time Convexity Principle for space-time level sets
- 4. The Strict Convexity of Space-time Level Sets
- 5. Appendix: the proof in dimension $n=2$
Abstract
In this paper we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, we can obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain our ideas and for completeness, we also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.- Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 357743
- O. Alvarez, J.-M. Lasry, and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl. (9) 76 (1997), no. 3, 265–288 (English, with English and French summaries). MR 1441987, DOI 10.1016/S0021-7824(97)89952-7
- Baojun Bian and Pengfei Guan, A microscopic convexity principle for nonlinear partial differential equations, Invent. Math. 177 (2009), no. 2, 307–335. MR 2511744, DOI 10.1007/s00222-009-0179-5
- Baojun Bian and Pengfei Guan, A structural condition for microscopic convexity principle, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 789–807. MR 2644765, DOI 10.3934/dcds.2010.28.789
- Baojun Bian, Pengfei Guan, Xi-Nan Ma, and Lu Xu, A constant rank theorem for quasiconcave solutions of fully nonlinear partial differential equations, Indiana Univ. Math. J. 60 (2011), no. 1, 101–119. MR 2952411, DOI 10.1512/iumj.2011.60.4222
- Chiara Bianchini, Marco Longinetti, and Paolo Salani, Quasiconcave solutions to elliptic problems in convex rings, Indiana Univ. Math. J. 58 (2009), no. 4, 1565–1589. MR 2542973, DOI 10.1512/iumj.2009.58.3539
- Christer Borell, Brownian motion in a convex ring and quasiconcavity, Comm. Math. Phys. 86 (1982), no. 1, 143–147. MR 678006
- Christer Borell, A note on parabolic convexity and heat conduction, Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), no. 3, 387–393 (English, with English and French summaries). MR 1387396
- Christer Borell, Diffusion equations and geometric inequalities, Potential Anal. 12 (2000), no. 1, 49–71. MR 1745333, DOI 10.1023/A:1008641618547
- Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366–389. MR 450480, DOI 10.1016/0022-1236(76)90004-5
- Luis A. Caffarelli and Avner Friedman, Convexity of solutions of semilinear elliptic equations, Duke Math. J. 52 (1985), no. 2, 431–456. MR 792181, DOI 10.1215/S0012-7094-85-05221-4
- Luis Caffarelli, Pengfei Guan, and Xi-Nan Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations, Comm. Pure Appl. Math. 60 (2007), no. 12, 1769–1791. MR 2358648, DOI 10.1002/cpa.20197
- L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), no. 3-4, 261–301. MR 806416, DOI 10.1007/BF02392544
- L. Caffarelli, L. Nirenberg, and J. Spruck, Nonlinear second order elliptic equations. IV. Starshaped compact Weingarten hypersurfaces, Current topics in partial differential equations, Kinokuniya, Tokyo, 1986, pp. 1–26. MR 1112140
- Luis A. Caffarelli and Joel Spruck, Convexity properties of solutions to some classical variational problems, Comm. Partial Differential Equations 7 (1982), no. 11, 1337–1379. MR 678504, DOI 10.1080/03605308208820254
- A. Chau, B. Weinkove, Counterexamples to quasiconcavity for the heat equation, Int. Math. Res. Not. in press, 2018, DOI 10.1093/imrn/rny243.
- Chuan Qiang Chen and Bo Wen Hu, A microscopic convexity principle for spacetime convex solutions of fully nonlinear parabolic equations, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 4, 651–674. MR 3029282, DOI 10.1007/s10114-012-1495-z
- ChuanQiang Chen and ShuJun Shi, Curvature estimates for the level sets of spatial quasiconcave solutions to a class of parabolic equations, Sci. China Math. 54 (2011), no. 10, 2063–2080. MR 2838121, DOI 10.1007/s11425-011-4277-7
- Andrea Colesanti and Paolo Salani, Quasi-concave envelope of a function and convexity of level sets of solutions to elliptic equations, Math. Nachr. 258 (2003), 3–15. MR 2000041, DOI 10.1002/mana.200310083
- J. I. Diaz and B. Kawohl, On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, J. Math. Anal. Appl. 177 (1993), no. 1, 263–286. MR 1224819, DOI 10.1006/jmaa.1993.1257
- J. I. Diaz and B. Kawohl, On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, J. Math. Anal. Appl. 177 (1993), no. 1, 263–286. MR 1224819, DOI 10.1006/jmaa.1993.1257
- R. M. Gabriel, A result concerning convex level surfaces of $3$-dimensional harmonic functions, J. London Math. Soc. 32 (1957), 286–294. MR 90662, DOI 10.1112/jlms/s1-32.3.286
- Pengfei Guan, Qun Li, and Xi Zhang, A uniqueness theorem in Kähler geometry, Math. Ann. 345 (2009), no. 2, 377–393. MR 2529480, DOI 10.1007/s00208-009-0358-0
- Pengfei Guan, Changshou Lin, and Xi’nan Ma, The Christoffel-Minkowski problem. II. Weingarten curvature equations, Chinese Ann. Math. Ser. B 27 (2006), no. 6, 595–614. MR 2273800, DOI 10.1007/s11401-005-0575-0
- Pengfei Guan and Xi-Nan Ma, The Christoffel-Minkowski problem. I. Convexity of solutions of a Hessian equation, Invent. Math. 151 (2003), no. 3, 553–577. MR 1961338, DOI 10.1007/s00222-002-0259-2
- Pengfei Guan and Lu Xu, Convexity estimates for level sets of quasiconcave solutions to fully nonlinear elliptic equations, J. Reine Angew. Math. 680 (2013), 41–67. MR 3100952, DOI 10.1515/crelle.2012.038
- Bowen Hu and Xinan Ma, A constant rank theorem for spacetime convex solutions of heat equation, Manuscripta Math. 138 (2012), no. 1-2, 89–118. MR 2898749, DOI 10.1007/s00229-011-0485-2
- Kazuhiro Ishige and Paolo Salani, Is quasi-concavity preserved by heat flow?, Arch. Math. (Basel) 90 (2008), no. 5, 450–460. MR 2414248, DOI 10.1007/s00013-008-2437-y
- Kazuhiro Ishige and Paolo Salani, Parabolic quasi-concavity for solutions to parabolic problems in convex rings, Math. Nachr. 283 (2010), no. 11, 1526–1548. MR 2759792, DOI 10.1002/mana.200910242
- Kazuhiro Ishige and Paolo Salani, On a new kind of convexity for solutions of parabolic problems, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), no. 4, 851–864. MR 2746446, DOI 10.3934/dcdss.2011.4.851
- Bernhard Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. MR 810619, DOI 10.1007/BFb0075060
- Alan U. Kennington, Convexity of level curves for an initial value problem, J. Math. Anal. Appl. 133 (1988), no. 2, 324–330. MR 954709, DOI 10.1016/0022-247X(88)90404-0
- Nicholas J. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Partial Differential Equations 15 (1990), no. 4, 541–556. MR 1046708, DOI 10.1080/03605309908820698
- Nicholas J. Korevaar and John L. Lewis, Convex solutions of certain elliptic equations have constant rank Hessians, Arch. Rational Mech. Anal. 97 (1987), no. 1, 19–32. MR 856307, DOI 10.1007/BF00279844
- John L. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal. 66 (1977), no. 3, 201–224. MR 477094, DOI 10.1007/BF00250671
- Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
- Marco Longinetti and Paolo Salani, On the Hessian matrix and Minkowski addition of quasiconvex functions, J. Math. Pures Appl. (9) 88 (2007), no. 3, 276–292 (English, with English and French summaries). MR 2355460, DOI 10.1016/j.matpur.2007.06.007
- Xi-Nan Ma, Qianzhong Ou, and Wei Zhang, Gaussian curvature estimates for the convex level sets of $p$-harmonic functions, Comm. Pure Appl. Math. 63 (2010), no. 7, 935–971. MR 2662428, DOI 10.1002/cpa.20318
- Robert C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math. J. 20 (1973), 373–383. MR 334045
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
- Max Shiffman, On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes, Ann. of Math. (2) 63 (1956), 77–90. MR 74695, DOI 10.2307/1969991
- I. M. Singer, Bun Wong, Shing-Tung Yau, and Stephen S.-T. Yau, An estimate of the gap of the first two eigenvalues in the Schrödinger operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 2, 319–333. MR 829055
- Gábor Székelyhidi and Ben Weinkove, On a constant rank theorem for nonlinear elliptic PDEs, Discrete Contin. Dyn. Syst. 36 (2016), no. 11, 6523–6532. MR 3543597, DOI 10.3934/dcds.2016081
- François Trèves, A new method of proof of the subelliptic estimates, Comm. Pure Appl. Math. 24 (1971), 71–115. MR 290201, DOI 10.1002/cpa.3160240107