Abstract
In this report we describe our recent existence results [26], [27] for two not quite standard problems for semilinear elliptic equations in general domains.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D.R. Adams, LP potential theory techniques and nonlinear PDE, in Potential theory (Nagoya 1990) 1–15, de Gruyter, Berlin, 1992.
D.R. Adams, Potential and capacity before and after Wiener. Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, MI, 1994), 63–83. Proc. Sympos. Appl. Math. 52, Amer. Math. Soc., Providence, RI, 1997.
D.R. Adams, A. Heard, The necessity of the Wiener test for some semi-linear elliptic equations. Indiana Univ. Math. J. 41 (1992), 109–124.
D.R. Adams, L.I. Hedberg Function spaces and potential theory Springer-Verlag, Berlin, Heidelberg, 1996.
D.R. Adams, M. Pierre, Capacitary strong type estimates in semilinear problems. Ann. Inst. Fourier (Grenoble) 41 (1991), 117–135.
P. Baras, M. Pierre, Removable singularities for semilinear equations. Ann. Inst. Fourier (Grenoble) 34 (1984), 185–206.
P. Bauman, A Wiener test for nondivergence structure, second-order elliptic equations. Indiana Univ. Math. J. 34 (1985), 825–844.
H. Brezis, L. Veron, Removable singularities for some nonlinear elliptic equations. Arch. Rational Mech. Anal. 75 (1980/81), 1–6.
G. Dal Masco, U. Mosco, Wiener criteria and energy decay for relaxed Dirichlet problems. Arch. Rational Mech. Anal. 95 (1986), 345–387.
J.-S. Dhersin, J.-F. Le Gall, Wiener’s test for super-Brownian motion and the Brownian snake. Probab. Theory Related Fields 108 (1997), 103–129.
E.B. Dynkin Diffusions superdiffusions and partial differential equations American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RIto appear
E.B. Dynkin, An introduction to branching measure-valued processes American Mathematical Society, Providence, RI, 1994.
E.B. Dynkin, S.E. Kuznetsov, Superdiffusions and removable singularities for quasi-linear partial differential equations. Comm. Pure Appl. Math. 49 (1996), 125–176.
A.M. Etheridge, An introduction to superprocesses, American Mathematical Society, Providence, RI, 2000.
L.C. Evans, R.F. Gariepy, Wiener’s criterion for the heat equation. Arch. Rational Mech. Anal. 78 (1982), 293–314.
E.B. Fabes, N. Garofalo, E. Lanconelli, Wiener’s criterion for divergence form parabolic operators with Cl-Dini continuous coefficients. Duke Math. J. 59 (1989), 191–232.
E. Fabes, D. Jerison, C. Kenig, The Wiener test for degenerate elliptic equations. Ann. Inst. Fourier (Grenoble) 32 (1982), 151–182.
D.L. Finn, Behavior of positive solutions toA 9 u= u9+Suwith prescribed singularities.Indiana Univ. Math. J., 49 (2000), 177–219.
D.L. Finn, R.C. McOwen, Singularities and asymptotics for the equationA g u —nq=Su.Indiana Univ. Math. J., 42 (1993), 1487–1523.
R. Gariepy, W.P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations.Arch. Rational Mech. Anal. 67 (1977), 25–39.
J. Heinonen, T. Kilpelainen, 0. Martio, Nonlinear potential theory of degenerate elliptic equations. Oxford University Press, New York, 1993.
N.J. Kalton, I.E. Verbitsky, Nonlinear equations and weighted norm inequalities.Trans. Amer. Math. Soc. 351 (1999), 3441–3497.
J.B. Keller, On solutions of Δu = f (u). Comm. Pure Appl. Math.10 (1957), 503–510.
T. Kilpelainen, J. Malt, The Wiener test and potential estimates for quasilinear elliptic equations.Acta Math. 172 (1994), 137–161.
V.A. Kondratiev, V.A. Nikishkin, On positive solutions of singular boundary value problems for the equation Δu = uk. Russian J. Math. Phys.1 (1993), 131–135.
D.A. Labutin, Wiener regularity for large solution of nonlinear equations. Preprint.
D.A. Labutin, Thinness for scalar-negative singular Yamabe metrics.in preparation.
D.A. Labutin, Potential estimates for a class of fully nonlinear elliptic equations.Duke Math. J. to appear.
J.-F. LeGall, A path-valued Markov process and its connections with partial differential equations. First European Congress of Mathematics (Paris 1992) Vol. II 185–212, Birkhauser, Basel, 1994.
J.-F. LeGall, Branching processes, random trees and superprocesses.Proceedings of the International Congress of Mathematicians (Berlin 1998) Doc. Math. Extra Vol. III (1998), 279–289.
J.-F. LeGall, Spatial branching processes, random snakes and partial differential equations. Lectures in Mathematics ETH Zurich. Birkhauser Verlag, Basel, 1999.
P. Lindqvist, 0. Martio Two theorems of N. Wiener for solutions of quasilinear elliptic equations. Acta Math. 155 (1985), 153–171.
W. Littman, G. Stampacchia, H.F. Weinberger, Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 43–77.
C. Loewner, L. Nirenberg Partial differential equations invariant under conformal or projective transformations, Contributions to analysis (a collection of papers dedicated to Lipman Bers) Academic Press, New York, 1974, 245–272.
J. Malt, W.P. Ziemer Fine regularity of solutions of elliptic partial differential equations. American Mathematical Society, Providence, RI, 1997.
M. Marcus, L. Veron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case. Arch. Rational Mech. Anal. 144 (1998), 201–231.
M. Marcus, L. Veron, The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case. J. Math. Pures Appl. 77 (1998), 481–524.
V.G. Maz’ya, On the continuity at a boundary point of solutions of quasilinear equations. Vestnik Leningrad. Univ. 25 (1970), 42–55.
V.G. Maz’ya Sobolev spaces. Springer-Verlag, Berlin-New York, 1985.
V.G. Maz’ya Unsolved problems connected with the Wiener criterion. The Legacy of Norbert Wiener: A Centennial Symposium (Cambridge, MA, 1994), 199–208. Proc. Sympos. Pure Math. 60, Amer. Math. Soc., Providence, RI, 1997.
V.G. Maz’ya, The Wiener test for higher order elliptic equations. Preprint Institut Mittag-Leffler report No. 38,1999/2000.
R. Mazzeo, D. Pollack, K. Uhlenbeck, Moduli spaces of singular Yamabe metrics. J. Amer. Math. Soc. 9 (1996), 303–344.
R.C. McOwen, Results and open questions on the singular Yamabe problem, Dynamical systems and differential equations, Vol. II (Springfield, MO, 1996).Discrete Contin. Dynam. Systems Added Volume II (1998), 123–132.
R. Osserman, On the inequality Δu≥ f (u).Pacific J. Math. 7 (1957), 1641–1647.
E.A. Perkins, Measure-valued branching diffusions and interactions. Proceedings of the International Congress of Mathematicians (Zurich 1994) Vol. 2 1036–1046, Birkhauser, Basel, 1995.
N. Korevaar, R. Mazzeo, F. Pacard, R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities. Invent. Math. 135 (1999), 233–272.
R. Schoen, S.T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), 47–71.
R. Schoen, S.T. Yau, Lectures on differential geometry. International Press, Cambridge, MA, 1994.
N.S. Trudinger, X.-J. Wang, On the weak continuity of elliptic operators and applications to potential theory. Preprint Australian National University MRR00–018, 2000, Amer.J. Math.to appear.
N.S. Trudinger, X.-J. Wang, Hessian Measures III. Preprint Australian National University MRR00–016, 2000,J. Funct. Anal. to appear.
N.S. Trudinger, X.-J. Wang, Hessian Measures II. Ann. of Math. (2) 150 (1999), 579–604.
N.S. Trudinger, X.-J. Wang, Hessian Measures I. Topol. Methods Nonlinear Anal. 10 (1997), 225–239.
L. Veron, Generalized boundary value problems for nonlinear elliptic equations. Electron. J. Diff. Eqns. 6 (2001), 313–342.
L. Veron Singularities of solutions of second order quasilinear equations. Longman, Harlow, 1996.
N. Wiener, The Dirichlet problem. J. Math. and Phys. 3 (1924), 127–146.
N. Wiener, Certain notions in potential theory. J. Math. and Phys. 3 (1924), 24–51.
Open problems in geometry. Differential geometry: partial differential equations on manifolds (Los Angeles CA 1990) 1–28, Proc. Sympos. Pure Math., 54, Amer. Math. Soc., Providence, RI, 1993.
W.P. Ziemer Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Springer-Verlag, New York, 1989.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Basel AG
About this chapter
Cite this chapter
Labutin, D.A. (2003). Potential Estimates for Large Solutions of Semilinear Elliptic Equations. In: Haroske, D., Runst, T., Schmeisser, HJ. (eds) Function Spaces, Differential Operators and Nonlinear Analysis. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8035-0_25
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8035-0_25
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9414-2
Online ISBN: 978-3-0348-8035-0
eBook Packages: Springer Book Archive