Uniqueness of the critical point of the solutions to some semilinear elliptic boundary value problems in 𝑅²

Sakaguchi, Shigeru

doi:10.1090/S0002-9947-1990-1008702-1

Uniqueness of the critical point of the solutions to some semilinear elliptic boundary value problems in $\textbf {R}^ 2$
HTML articles powered by AMS MathViewer

by Shigeru Sakaguchi PDF

Trans. Amer. Math. Soc. 319 (1990), 179-190 Request permission

Abstract:

We consider some two-dimensional semilinear elliptic boundary value problems over a bounded convex domain in ${{\mathbf {R}}^2}$ and show the uniqueness of the critical point of the solutions.

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
N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249. MR 92067
Luis A. Caffarelli and Avner Friedman, Convexity of solutions of semilinear elliptic equations, Duke Math. J. 52 (1985), no. 2, 431–456. MR 792181, DOI 10.1215/S0012-7094-85-05221-4
Luis A. Caffarelli and Joel Spruck, Convexity properties of solutions to some classical variational problems, Comm. Partial Differential Equations 7 (1982), no. 11, 1337–1379. MR 678504, DOI 10.1080/03605308208820254
Jin Tzu Chen and Wu Hsiung Huang, Convexity of capillary surfaces in the outer space, Invent. Math. 67 (1982), no. 2, 253–259. MR 665156, DOI 10.1007/BF01393817
J.-T. Chen, Uniqueness of minimal point and its location of capillary free surfaces over convex domain, Astérisque 118 (1984), 137–143 (English, with French summary). Variational methods for equilibrium problems of fluids (Trento, 1983). MR 761743
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443, DOI 10.1007/978-3-642-96379-7
Philip Hartman and Aurel Wintner, On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math. 75 (1953), 449–476. MR 58082, DOI 10.2307/2372496
Bernhard Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?, Comm. Partial Differential Equations 10 (1985), no. 10, 1213–1225. MR 806439, DOI 10.1080/03605308508820404
Bernhard Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. MR 810619, DOI 10.1007/BFb0075060
Alan U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J. 34 (1985), no. 3, 687–704. MR 794582, DOI 10.1512/iumj.1985.34.34036
Nicholas Korevaar, Capillary surface convexity above convex domains, Indiana Univ. Math. J. 32 (1983), no. 1, 73–81. MR 684757, DOI 10.1512/iumj.1983.32.32007
Nicholas J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 32 (1983), no. 4, 603–614. MR 703287, DOI 10.1512/iumj.1983.32.32042
Nicholas J. Korevaar and John L. Lewis, Convex solutions of certain elliptic equations have constant rank Hessians, Arch. Rational Mech. Anal. 97 (1987), no. 1, 19–32. MR 856307, DOI 10.1007/BF00279844
Gary M. Lieberman, The conormal derivative problem for elliptic equations of variational type, J. Differential Equations 49 (1983), no. 2, 218–257. MR 708644, DOI 10.1016/0022-0396(83)90013-X
Gary M. Lieberman and Neil S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc. 295 (1986), no. 2, 509–546. MR 833695, DOI 10.1090/S0002-9947-1986-0833695-6
Lawrence E. Payne, On two conjectures in the fixed membrane eigenvalue problem, Z. Angew. Math. Phys. 24 (1973), 721–729 (English, with German summary). MR 333487, DOI 10.1007/BF01597076
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
René P. Sperb, Extension of two theorems of Payne to some nonlinear Dirichlet problems, Z. Angew. Math. Phys. 26 (1975), no. 6, 721–726 (English, with German summary). MR 393835, DOI 10.1007/BF01596076