Solving semilinear partial differential equations with probabilistic potential theory
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- by Joseph Glover and P. J. McKenna PDF
- Trans. Amer. Math. Soc. 290 (1985), 665-681 Request permission
Abstract:
Techniques of probabilistic potential theory are applied to solve $- Lu + f(u) = \mu$, where $\mu$ is a signed measure, $f$ a (possibly discontinuous) function and $L$ a second order elliptic or parabolic operator on ${R^d}$ or, more generally, the infinitesimal generator of a Markov process. Also formulated are sufficient conditions guaranteeing existence of a solution to a countably infinite system of such equations.References
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
- Walter Allegretto, Nonnegative solutions for a weakly nonlinear elliptic equations, Canad. J. Math. 36 (1984), no. 1, 71–83. MR 733708, DOI 10.4153/CJM-1984-006-4
- Pierre Baras and Michel Pierre, Singularités éliminables d’équations elliptiques semi-linéaires, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 9, 519–522 (French, with English summary). MR 685014
- Philippe Benilan, Haim Brezis, and Michael G. Crandall, A semilinear equation in $L^{1}(R^{N})$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4, 523–555. MR 390473
- R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757
- H. Brezis, Semilinear equations in $\textbf {R}^N$ without condition at infinity, Appl. Math. Optim. 12 (1984), no. 3, 271–282. MR 768633, DOI 10.1007/BF01449045
- Haïm Brézis and Walter A. Strauss, Semi-linear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan 25 (1973), 565–590. MR 336050, DOI 10.2969/jmsj/02540565
- Kai Lai Chung, Lectures from Markov processes to Brownian motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 249, Springer-Verlag, New York-Berlin, 1982. MR 648601
- C. Dellacherie, Potentiels de Green et fonctionnelles additives, Séminaire de Probabilités, IV (Univ. Strasbourg, 1968/69) Lecture Notes in Math., Vol. 124, Springer, Berlin, 1970, pp. 73–75 (French). MR 0293720
- Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel. Chapitres V à VIII, Revised edition, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1385, Hermann, Paris, 1980 (French). Théorie des martingales. [Martingale theory]. MR 566768
- Ronald K. Getoor, Markov processes: Ray processes and right processes, Lecture Notes in Mathematics, Vol. 440, Springer-Verlag, Berlin-New York, 1975. MR 0405598
- Ronald K. Getoor, Multiplicative functionals of dual processes, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 2, 43–83 (English, with French summary). MR 331529
- Kiyoshi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-New York, 1965. MR 0199891
- Wei Ming Ni, On the elliptic equation $\Delta u+K(x)u^{(n+2)/(n-2)}=0$, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493–529. MR 662915, DOI 10.1512/iumj.1982.31.31040
- D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 979–1000. MR 299921, DOI 10.1512/iumj.1972.21.21079
- Barry Simon, Functional integration and quantum physics, Pure and Applied Mathematics, vol. 86, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 544188
- Daniel W. Stroock and S. R. Srinivasa Varadhan, Multidimensional diffusion processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233, Springer-Verlag, Berlin-New York, 1979. MR 532498 W. Walter, Differential inequalities, Springer-Verlag, Berlin, Heidelberg and New York, 1967.
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 290 (1985), 665-681
- MSC: Primary 35J60; Secondary 35K55, 60J45
- DOI: https://doi.org/10.1090/S0002-9947-1985-0792818-7
- MathSciNet review: 792818