Parabolic and elliptic equations with singular or degenerate coefficients: The Dirichlet problem

Dong, Hongjie; Phan, Tuoc

doi:10.1090/tran/8397

Parabolic and elliptic equations with singular or degenerate coefficients: The Dirichlet problem
HTML articles powered by AMS MathViewer

by Hongjie Dong and Tuoc Phan PDF

Trans. Amer. Math. Soc. 374 (2021), 6611-6647 Request permission

Abstract:

We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb {R}^d_+$, where the coefficients are the product of $x_d^\alpha , \alpha \in (-\infty , 1),$ and a bounded uniformly elliptic matrix of coefficients. Thus, the coefficients are singular or degenerate near the boundary $\{x_d =0\}$ and they may not be locally integrable. The novelty of the work is that we find proper weights under which the existence, uniqueness, and regularity of solutions in Sobolev spaces are established. These results appear to be the first of their kind and are new even if the coefficients are constant. They are also readily extended to systems of equations.

References

S. R. Athreya, M. T. Barlow, R. F. Bass, and E. A. Perkins, Degenerate stochastic differential equations and super-Markov chains, Probab. Theory Related Fields 123 (2002), no. 4, 484–520. MR 1921011, DOI 10.1007/s004400100191
L. A. Caffarelli and I. Peral, On $W^{1,p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), no. 1, 1–21. MR 1486629, DOI 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N
Luis Caffarelli and Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245–1260. MR 2354493, DOI 10.1080/03605300600987306
Dat Cao, Tadele Mengesha, and Tuoc Phan, Weighted-$W^{1,p}$ estimates for weak solutions of degenerate and singular elliptic equations, Indiana Univ. Math. J. 67 (2018), no. 6, 2225–2277. MR 3900368, DOI 10.1512/iumj.2018.67.7533
P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc. 11 (1998), no. 4, 899–965. MR 1623198, DOI 10.1090/S0894-0347-98-00277-X
P. Daskalopoulos, R. Hamilton, and K. Lee, All time $C^\infty$-regularity of the interface in degenerate diffusion: a geometric approach, Duke Math. J. 108 (2001), no. 2, 295–327. MR 1833393, DOI 10.1215/S0012-7094-01-10824-7
Hongjie Dong and Doyoon Kim, Parabolic and elliptic systems in divergence form with variably partially BMO coefficients, SIAM J. Math. Anal. 43 (2011), no. 3, 1075–1098. MR 2800569, DOI 10.1137/100794614
Hongjie Dong and Doyoon Kim, Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains, J. Funct. Anal. 261 (2011), no. 11, 3279–3327. MR 2835999, DOI 10.1016/j.jfa.2011.08.001
Hongjie Dong and Doyoon Kim, Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces, Adv. Math. 274 (2015), 681–735. MR 3318165, DOI 10.1016/j.aim.2014.12.037
Hongjie Dong and Doyoon Kim, On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights, Trans. Amer. Math. Soc. 370 (2018), no. 7, 5081–5130. MR 3812104, DOI 10.1090/tran/7161
Hongjie Dong and Doyoon Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math. 345 (2019), 289–345. MR 3899965, DOI 10.1016/j.aim.2019.01.016
Hongjie Dong and Tuoc Phan, Weighted mixed-norm $L_p$-estimates for elliptic and parabolic equations in non-divergence form with singular degenerate coefficients, Rev. Mat. Iberoam., accepted, DOI: 10.4171/rmi/1233, arXiv:1811.06393.
Hongjie Dong and Tuoc Phan, Regularity for parabolic equations with singular or degenerate coefficients, Calc. Var. Partial Differential Equations 60 (2021), no. 1, Paper No. 44, 39. MR 4204570, DOI 10.1007/s00526-020-01876-5
Hongjie Dong and Tuoc Phan, On parabolic and elliptic equations with singular or degenerate coefficients, arXiv:2007.04385.
Charles L. Epstein and Camelia A. Pop, Transition probabilities for degenerate diffusions arising in population genetics, Probab. Theory Related Fields 173 (2019), no. 1-2, 537–603. MR 3916114, DOI 10.1007/s00440-018-0840-2
Charles L. Epstein and Rafe Mazzeo, Degenerate diffusion operators arising in population biology, Annals of Mathematics Studies, vol. 185, Princeton University Press, Princeton, NJ, 2013. MR 3202406, DOI 10.1515/9781400846108
Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
Paul M. N. Feehan and Camelia A. Pop, Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations, J. Differential Equations 256 (2014), no. 3, 895–956. MR 3128929, DOI 10.1016/j.jde.2013.08.012
Gaetano Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 5 (1956), 1–30 (Italian). MR 89348
Gaetano Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order, Boundary problems in differential equations, Univ. Wisconsin Press, Madison, Wis., 1960, pp. 97–120. MR 0111931
Mariano Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. MR 1239172
Huaiyu Jian and Xu-Jia Wang, Bernstein theorem and regularity for a class of Monge-Ampère equations, J. Differential Geom. 93 (2013), no. 3, 431–469. MR 3024302
Huaiyu Jian and Xu-jia Wang, Optimal boundary regularity for nonlinear singular elliptic equations, Adv. Math. 251 (2014), 111–126. MR 3130337, DOI 10.1016/j.aim.2013.10.009
Steven L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud. 6 (1993), no. 2, 327–343. MR 3929676, DOI 10.1093/rfs/6.2.327
M. V. Keldyš, On certain cases of degeneration of equations of elliptic type on the boundry of a domain, Doklady Akad. Nauk SSSR (N.S.) 77 (1951), 181–183 (Russian). MR 0042031
Doyoon Kim and N. V. Krylov, Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others, SIAM J. Math. Anal. 39 (2007), no. 2, 489–506. MR 2338417, DOI 10.1137/050646913
Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients, Potential Anal. 26 (2007), no. 4, 345–361. MR 2300337, DOI 10.1007/s11118-007-9042-8
Doyoon Kim, Seungjin Ryu, and Kwan Woo. Parabolic equations with unbounded lower-order coefficients in Sobolev spaces with mixed norms. arXiv:2007.01986.
Kyeong-Hun Kim and N. V. Krylov, On the Sobolev space theory of parabolic and elliptic equations in $C^1$ domains, SIAM J. Math. Anal. 36 (2004), no. 2, 618–642. MR 2111792, DOI 10.1137/S0036141003421145
Ildoo Kim, Kyeong-Hun Kim, and Kijung Lee, A weighted $L_p$-theory for divergence type parabolic PDEs with BMO coefficients on $C^1$-domains, J. Math. Anal. Appl. 412 (2014), no. 2, 589–612. MR 3147235, DOI 10.1016/j.jmaa.2013.10.079
N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations 32 (2007), no. 1-3, 453–475. MR 2304157, DOI 10.1080/03605300600781626
N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Related Fields 98 (1994), no. 3, 389–421. MR 1262972, DOI 10.1007/BF01192260
N. V. Krylov, Weighted Sobolev spaces and Laplace’s equation and the heat equations in a half space, Comm. Partial Differential Equations 24 (1999), no. 9-10, 1611–1653. MR 1708104, DOI 10.1080/03605309908821478
N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. Math. Anal. 31 (1999), no. 1, 19–33. MR 1720129, DOI 10.1137/S0036141098338843
N. V. Krylov, On divergence form SPDEs with VMO coefficients in a half space, Stochastic Process. Appl. 119 (2009), no. 6, 2095–2117. MR 2519358, DOI 10.1016/j.spa.2008.11.003
Fang-Hua Lin, On the Dirichlet problem for minimal graphs in hyperbolic space, Invent. Math. 96 (1989), no. 3, 593–612. MR 996556, DOI 10.1007/BF01393698
Fang Hua Lin and Lihe Wang, A class of fully nonlinear elliptic equations with singularity at the boundary, J. Geom. Anal. 8 (1998), no. 4, 583–598. MR 1724206, DOI 10.1007/BF02921713
O. A. Oleĭnik and E. V. Radkevič, Second order equations with nonnegative characteristic form, Plenum Press, New York-London, 1973. Translated from the Russian by Paul C. Fife. MR 0457908
Yannick Sire, Susanna Terracini, and Stefano Vita, Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions, Comm. Partial Differential Equations 46 (2021), no. 2, 310–361. MR 4207950, DOI 10.1080/03605302.2020.1840586
Yannick Sire, Susanna Terracini, and Stefano Vita, Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions, Math. Eng. 3 (2021), no. 1, Paper No. 5, 50. MR 4144100, DOI 10.3934/mine.2021005
Pablo Raúl Stinga and José L. Torrea, Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation, SIAM J. Math. Anal. 49 (2017), no. 5, 3893–3924. MR 3709888, DOI 10.1137/16M1104317
Chunpeng Wang, Lihe Wang, Jingxue Yin, and Shulin Zhou, Hölder continuity of weak solutions of a class of linear equations with boundary degeneracy, J. Differential Equations 239 (2007), no. 1, 99–131. MR 2341551, DOI 10.1016/j.jde.2007.03.026
Fu Zhang and Kai Du, Krylov-Safonov estimates for a degenerate diffusion process, Stochastic Process. Appl. 130 (2020), no. 8, 5100–5123. MR 4108483, DOI 10.1016/j.spa.2020.02.012