publication . Preprint . 2013

Maximum principles for boundary-degenerate linear parabolic differential operators

Feehan, Paul M. N.;
Open Access English
  • Published: 20 Jun 2013
Abstract
We develop weak and strong maximum principles for boundary-degenerate, linear, parabolic, second-order partial differential operators, $Lu := -u_t-\tr(aD^2u)-\langle b, Du\rangle + cu$, with \emph{partial} Dirichlet boundary conditions. The coefficient, $a(t,x)$, is assumed to vanish along a non-empty open subset, $\mydirac_0!\sQ$, called the \emph{degenerate boundary portion}, of the parabolic boundary, $\mydirac!\sQ$, of the domain $\sQ\subset\RR^{d+1}$, while $a(t,x)$ may be non-zero at points in the \emph{non-degenerate boundary portion}, $\mydirac_1!\sQ := \mydirac!\sQ\less\bar{\mydirac_0!\sQ}$. Points in $\mydirac_0!\sQ$ play the same role as those in the ...
Subjects
free text keywords: Mathematics - Analysis of PDEs, Mathematics - Probability, Primary 35B50, 35B51, 35K65, secondary 35D40, 35K85, 60J60
Related Organizations
Funded by
NSF| AMC-SS: Mathematical Finance and Partial Differential Equations Conference - November 2, 2012
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1237722
  • Funding stream: Directorate for Mathematical & Physical Sciences | Division of Mathematical Sciences
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46 references, page 1 of 4

[1] R. A. Adams, Sobolev spaces, Academic Press, Orlando, FL, 1975.

[2] A. Bensoussan and J. L. Lions, Applications of variational inequalities in stochastic control, North-Holland, New York, 1982.

[3] A. Ciomaga, On the strong maximum principle for second-order nonlinear parabolic integro-di erential equations, Adv. Di erential Equations 17 (2012), no. 7-8, 635{671, arXiv:1006.2607. [OpenAIRE]

[4] M. G. Crandall, H. Ishii, and P-L. Lions, User's guide to viscosity solutions of second order partial di erential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 1{67. [OpenAIRE]

[5] P. Daskalopoulos and P. M. N. Feehan, C1;1 regularity for degenerate elliptic obstacle problems in mathematical nance, arXiv:1206.0831.

[6] , Existence, uniqueness, and global regularity for variational inequalities and obstacle problems for degenerate elliptic partial di erential operators in mathematical nance, arXiv:1109.1075.

[7] P. Daskalopoulos and R. Hamilton, C1-regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc. 11 (1998), 899{965.

[8] P. Daskalopoulos and E. Rhee, Free-boundary regularity for generalized porous medium equations, Commun. Pure Appl. Anal. 2 (2003), 481{494.

[9] D. J. Du y, Finite di erence methods in nancial engineering, Wiley, New York, 2006.

[10] E. Ekstrom and J. Tysk, Boundary conditions for the single-factor term structure equation, Ann. Appl. Probab. 21 (2011), 332{350.

[11] C. L. Epstein and R. Mazzeo, Degenerate di usion operators arising in population biology, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2013, arXiv:1110.0032.

[12] L. C. Evans, Partial di erential equations, American Mathematical Society, Providence, RI, 1998.

[13] P. M. N. Feehan, A classical Perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate-parabolic operators via holomorphic maps, in preparation.

[14] , Partial di erential operators with non-negative characteristic form, maximum principles, and uniqueness for boundary value and obstacle problems, Communications in Partial Di erential Equations, to appear, arXiv:1204.6613v1.

[15] , Perturbations of local maxima and comparison principles for boundary-degenerate linear di erential equations, arXiv:1305.5098.

46 references, page 1 of 4
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