Maximum principles for boundary-degenerate second-order linear elliptic differential operators

Preprint English OPEN
Feehan, Paul M. N.;
(2012)
  • Related identifiers: doi: 10.1080/03605302.2013.831446
  • Subject: Quantitative Finance - Pricing of Securities | Mathematics - Probability | Mathematics - Analysis of PDEs | Primary 35B50, 35B51, 35J70, 35J86, 35R45, secondary 49J20, 49J40, 60J60
We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity prope... View more
  • References (71)
    71 references, page 1 of 8

    Choose vε = u1 − εwε and observe that vε ∈ H01(O ∪ Σ, w) and vε ≥ u1 − uˆw ≥ u1 − uˆ = u1 ∧ u2 ≥ ψ a.e. on O.

    (5) If u1 and u2 are solutions, respectively, for F1 ≥ F2 and ψ1 ≥ ψ2 a.e. on O, and g1 ≥ g2 on ∂O \ Σ in the sense of H1(O, w), then O Z

    [5] , Diffusions with second order boundary conditions. II, Indiana Univ. Math. J. 25 (1976), 403-441.

    [6] 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), 484-520.

    [7] R. F. Bass and A. Lavrentiev, The submartingale problem for a class of degenerate elliptic operators, Probab. Theory Related Fields 139 (2007), 415-449.

    [8] R. F. Bass and E. A. Perkins, Degenerate stochastic differential equations with Ho¨lder continuous coefficients and super-Markov chains, Trans. Amer. Math. Soc. 355 (2003), 373-405.

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

    [10] S. Bonafede, A weak maximum principle and estimates of ess supΩ u for nonlinear degenerate elliptic equations, Czechoslovak Math. J. 46 (1996), 259-269.

    [11] M. Borsuk, Second-order degenerate elliptic boundary value problems in nonsmooth domains, Sovrem. Mat. Fundam. Napravl. 13 (2005), 3-137.

    [12] M.-F. Bru, Wishart processes, J. Theoret. Probab. 4 (1991), 725-751.

  • Metrics
    No metrics available
Share - Bookmark