Abstract
We prove uniqueness of the viscosity solutions of the Dirichlet problem of the spectral equation \(F(u)=f(\lambda[u])=\psi\) where \(\lambda[u]\) is the vector whose components are eigenvalues of a matrix associated with the unknown function u.
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References
Amster P., Cassinelli M., Mariani M.C., Rial D.F. (1999). Existence and regularity of weak solutions to the prescribed mean curvature equation for a nonparametric surface. Abstr. Appl. Anal. 4(1): 61–69
Caffarelli L., Nirenberg L., Spruck J. (1988). Nonlinear second-order elliptic equations. V. The Dirichlet problem for Weingarten hypersurfaces. Comm. Pure Appl. Math. 41(1): 47–70
Crandall M.G., Lions P.-L. (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277(1): 1–42
Cranny T.R. (1996). On the uniqueness of solutions of the homogeneous curvature equations. Ann. Inst. H. Poincar Anal. Non Linaire 13(5): 619–630
Ivochkina, N.M.: Solution of the Dirichlet problem for an equation of curvature of order m. (Russian) Dokl. Akad. Nauk SSSR 299(1), 35–38 (1988); Translation in Soviet Math. Dokl. 37(2), 322–325 (1988)
Ivochkina, N.M.: Local estimates of normal second derivatives for solutions of the Dirichlet problem for the prescribed quotient curvature equations. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 221 (1995), Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsii. 26, 114–126, 257; Translation in J. Math. Sci. (New York) 87(2),3343–3352 (1997)
Ivochkina N., Trudinger N., Wang X.-J. (2004). The Dirichlet problem for degenerate Hessian equations. Comm. Partial Diff. Eqs. 29(1–2): 219–235
Jensen R. (1988). The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rat. Mech. Anal. 101: 1–27
Lin M., Trudinger N.S. (1994). The Dirichlet problem for the prescribed curvature quotient equations. Topol. Method Nonlinear Anal. 3(2): 307–323
Lewis A., Sendov H. (2001). Twice differentiable spectral functions. SIAM J. Matrix Anal. Appl. 23(2): 368–386
Luo Y., Eberhard A. (2005). Comparison principles for viscosity solutions of elliptic equations via Fuzzy Sum Rule. J. Math. Anal. Appl. 307(2): 736–752
Luo Y., Eberhard A. (2006). An application of C 1,1 approximation to comparison principles for viscosity solutions of curvature equations. Nonlinear Anal. 64(6): 1236–1254
Trudinger N.S. (1990). The Dirichlet problem for the prescribed curvature equations. Arch. Rat. Mech. Anal. 111(1): 153–179
Trudinger N.S. (1995). On the Dirichlet problem for Hessian equations. Acta Math. 175(2): 151–164
Urbas J.I.E. (1990). On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations. Indiana Univ. Math. J. 39(2): 355–382
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Luo, Y. On the uniqueness of solutions of spectral equations. J Glob Optim 40, 155–160 (2008). https://doi.org/10.1007/s10898-007-9216-3
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DOI: https://doi.org/10.1007/s10898-007-9216-3