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Critical points of solutions of elliptic equations in two variables. (English) Zbl 0649.35026
We consider solutions u to the uniformly elliptic equation \[ \sum^{2}_{i,j=1}a_{ij}u_{x_ ix_ j}+\sum^{2}_{i=1}b_ iu_{x_ i}=0 \] in a bounded simply connected plane domain \(\Omega\). We analyse the problem of obtaining information on the critical points (zeros of the gradient) of u from information on the Dirichlet data \(u=g\) on \(\partial \Omega.\)
Assuming that the coefficients \(a_{ij},b_ i\) are sufficiently smooth, we prove the following.
1) If the set of points of relative maximum of the Dirichlet data g on \(\partial \Omega\) has N connected components, then the interior critical points of u are finite in number and the sum of their multiplicities is at most N-1.
2) If g satisfies the above hypothesis and g and \(\partial \Omega\) are sufficiently smooth, then, on any subdomain \(G\subset \subset \Omega\) we have \(| \text{grad} u| \geq C\prod^{K}_{i=1}| x-x_ i|^{m_ i}\) where \(x_ 1,...,x_ K\); \(m_ 1,...,m_ K\) are the interior critical points of u and their respective multiplicities.
The problem is also treated when no smoothness assumption is made on the coefficients.
Reviewer: G.Alessandrini

MSC:
35J15 Second-order elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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References:
[1] 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 , 623 - 727 ( 1959 ). MR 125307 | Zbl 0093.10401 · Zbl 0093.10401 · doi:10.1002/cpa.3160120405
[2] G. Alessandrini , An identification problem for an elliptic equation in two variables , Ann. Mat. Pura Appl. ( 4 ) 145 , 265 - 296 ( 1986 ). MR 886713 | Zbl 0662.35118 · Zbl 0662.35118 · doi:10.1007/BF01790543
[3] S. Bernstein , Sur la généralization du problème de Dirichlet (I) , Math. Ann. 62 , 253 - 271 ( 1906 ). MR 1511375 | JFM 37.0383.01 · JFM 37.0383.01
[4] L. Bers and L. Nirenberg , On a representation theorem for linear elliptic systems with discontinuous coefficient.s and its applications , in: Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali , Trieste , 111 - 140 , Cremonese , Roma , 1955 . MR 76981 | Zbl 0067.32503 · Zbl 0067.32503
[5] D. Gilbarg and J. Serrin , On isolated singularities of solutions of second order elliptic differential equations , J. Analyse Math. 4 , 309 - 340 ( 1956 ). MR 81416 | Zbl 0071.09701 · Zbl 0071.09701 · doi:10.1007/BF02787726
[6] D. Gilbarg and N.S. Trudinger , Elliptic Partial Differential Equations of Second Order , Springer-Verlag , Berlin , 1983 . MR 737190 | Zbl 0562.35001 · Zbl 0562.35001
[7] P. Hartman and A. Wintner , On the local behaviour of solutions of non-parabolic partial differential equations (I) , Amer. J. Math. 75 , 449 - 476 ( 1953 ). MR 58082 | Zbl 0052.32201 · Zbl 0052.32201 · doi:10.2307/2372496
[8] P. Hartman and A. Wintner , On the local behaviour of solutions of non-parabolic partial differential equations (II) The uniqueness of the Green singularity , Amer. J. Math. 76 , 351 - 361 ( 1954 ). MR 64271 | Zbl 0055.32403 · Zbl 0055.32403 · doi:10.2307/2372577
[9] P. Hartman and A. Wintner , On the local behaviour of solutions of non-parabolic partial differential equations (III) Approximation by spherical harmonics , Amer. J. Math. 77 , 329 - 354 ( 1955 ). MR 76156 | Zbl 0066.08001 · Zbl 0066.08001 · doi:10.2307/2372634
[10] K. Miller , Barriers on cones for uniformly elliptic operators , Ann. Mat. Pura Appl. ( 4 ) 76 , 93 - 105 ( 1967 ). MR 221087 | Zbl 0149.32101 · Zbl 0149.32101 · doi:10.1007/BF02412230
[11] L.A. Peletier and J. Serrin , Gradient bounds and Liouville theorems for quasilinear elliptic equations , Ann. Scuola Norm. Sup. Pisa Cl. Sci. , ( 4 ) 5 , 65 - 104 ( 1978 ). Numdam | MR 481493 | Zbl 0383.35025 · Zbl 0383.35025 · numdam:ASNSP_1978_4_5_1_65_0 · eudml:83779
[12] T. Radó , The problem of the least area and the problem of Plateau , Math. Z. 32 , 763 - 796 ( 1930 ). MR 1545197 | JFM 56.0436.01 · JFM 56.0436.01
[13] J. Serrin , Removable singularities of solutions of elliptic equations , Arc. Rat. Mech. Anal. 17 , 67 - 78 ( 1964 ). MR 170095 | Zbl 0135.15601 · Zbl 0135.15601 · doi:10.1007/BF00283867
[14] R.P. Sperb , Maximum Principles and their Applications , Academic Press , New York , 1981 . MR 615561 | Zbl 0454.35001 · Zbl 0454.35001
[15] G. Talenti , Equazioni lineari ellittiche in due variabili , Matematiche ( Catania ) 21 , 339 - 376 ( 1966 ). MR 204845 | Zbl 0149.07402 · Zbl 0149.07402
[16] J.L. Walsh , The Location of Critical Points of Analytic and Harmonic Functions , American Mathematical Society , New York , 1950 . MR 37350 | Zbl 0041.04101 · Zbl 0041.04101
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