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An inverse problem originating from magnetohydrodynamics. (English) Zbl 0732.76096
The authors obtain a rather interesting result for the inverse problem for \(\Delta u=-f(u)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\). In particular, suppose \(\Omega\) is a bounded polygonal domain in \(R^ 2\) and that u and v are classical solutions of the Dirichlet problems for \(\Delta u=-f(u)\) and \(\Delta v=-g(v)\) where f and g are smooth and nonnegative. Under a mild condition on the boundary angles, the authors show that, if \(\partial u/\partial n\) and \(\partial v/\partial n\) agree on a portion of \(\partial \Omega\), then in fact, there must be a level of agreement between f and g in the sense that \(f^{(k)}(0)=g^{(k)}(0)\) for \(k=0,1,..\).

76X05 Ionized gas flow in electromagnetic fields; plasmic flow
35R30 Inverse problems for PDEs
35J60 Nonlinear elliptic equations
Full Text: DOI
[1] S. Agmon, A. Douglis & L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math., 17 (1964), pp. 35-92. · Zbl 0123.28706 · doi:10.1002/cpa.3160170104
[2] L. Bers, F. John & M. Schechter, Partial Differential Equations. Interscience Publishers, New York, 1964.
[3] C. M. Bishop & J. B. Taylor, Degenerate toroidal magnetohydrodynamic equilibria and minimum B. Phys. Fluids 29 (1986), pp. 1144-1148. · Zbl 0593.76058 · doi:10.1063/1.865913
[4] J. Blum, T. Gallouet & J. Simon, Existence and control of plasma equilibrium in a Tokamak. SIAM J. Math. Anal. (1986), pp. 1158-1177. · Zbl 0614.35082
[5] J. Blum, Personal communication.
[6] J. Blum, Simulation Numérique et Contrôle Optimal en Physique des Plasmas, Gauthier-Villars, Paris, 1989.
[7] B. J. Braams, The interpretations of Tokamak magnetic diagnostics: status and prospects. Max-Planck-Institut für Plasmaphysik, preprint IPP 5/2, 1985.
[8] B. Gidas, W.-M. Ni & L. Nirenberg, Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), pp. 209-243. · Zbl 0425.35020 · doi:10.1007/BF01221125
[9] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985. · Zbl 0695.35060
[10] D. G. Figueiredo, P.-L. Lions & R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations. J. Math. Pures et Appl. 61 (1982), pp. 41-63. · Zbl 0452.35030
[11] V. G. Maz’ya & B. A. Plamenevskii, Estimates in L p and in Holder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. A.M.S. Transl. (2), 123 (1984), pp. 1-56.
[12] E. Miersemann, Asymptotic expansions of solutions of the Dirichlet problem for quasilinear elliptic equations of second order near a conical point. Math. Nachr. 135 (1988), pp. 239-274. · Zbl 0667.35026 · doi:10.1002/mana.19881350120
[13] M. Pilant & W. Rundell, An inverse problem for a nonlinear elliptic differential equation. SIAM J. Math. Anal. 18 (1987), pp. 1801-1809. · Zbl 0647.35081 · doi:10.1137/0518127
[14] J. Serrin, A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43 (1971), pp. 304-318. · Zbl 0222.31007 · doi:10.1007/BF00250468
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