Abstract.
We investigate the ‘‘hot–spots’’ property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the ‘‘hot–spots’’ property for the mixed Dirichlet–Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other.
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Atar, R.: Invariant wedges for a two–point reflecting Brownian motion and the ‘‘hot spots’’ problem. Elect. J. of Probab. 6, 18, 1–19 (2001)
Atar, R., Burdzy, K.: On the Neumann eigenfunctions in lip domains. (preprint)
Atar, R., Burdzy, K.: On nodal lines of Neumann eigenfunction. (preprint)
Bañuelos, R., Burdzy, K.: On the ‘‘hot spots’’ conjecture of J. Rauch. J. Funct. Anal. 164, 1–33 (1999)
Bañuelos, R., Pang, M.: An inequality for potentials and the ‘‘hot–spots’’ conjecture, Indiana Math. J. (to appear)
Bañuelos, R., Pang, M.: Lower bound gradient estimates for solutions of Schrödinger operators and heat kernels. Comm in PDE 24, 499–543 (1999)
Bass, R., Burdzy, K.: Fiber Brownian motion and the ‘‘hot spots’ problem. Duke Math J. 105, 25–58 (2000)
Burdzy, K., Werner, W.: A counterexample to the ‘‘hot spots’’ conjecture. Ann. Math. 149, 309–317 (1999)
Duren, P.: Univalent Functions, Springer Verlag New York, 1983
Freitas, P.: Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces. Indiana Univ. Math. J. 51, 305–316 (2002)
Jerison, D., Nadirashvili, N.: The ‘‘hot spots’’ conjecture for domains with two axes of symmetry. J. Amer. Math. Soc. 13, 741–772 (2000)
Kawohl, B.: Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., 1150, Springer, Berlin, 1985
Pascu, M.: Scaling coupling of reflected Brownian motion and the hot spots problem. Trans. Amer. Math. Soc. 354, 4681–4702 (2001)
Stroock, D.: Probability Theory, An Analytic View, Cambridge University Press, 1993
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Supported in part by NSF Grant # 9700585-DMS
Supported in part by NSF Grant # 0203961-DMS
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Banuelos, R., Pang, M. & Pascu, M. Brownian motion with killing and reflection and the ‘‘hot–spots’’ problem. Probab. Theory Relat. Fields 130, 56–68 (2004). https://doi.org/10.1007/s00440-003-0323-x
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DOI: https://doi.org/10.1007/s00440-003-0323-x