\`x^2+y_1+z_12^34\`
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A note on the unique continuation property for fully nonlinear elliptic equations

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  • We establish the strong unique continuation property for solutions to (1.1) where $F$ satisfies the structural assumptions A)-D). This extends a recent result of Armstrong and Silvestre (see [3]) where $F$ was assumed to be independent of $x$. We also establish an analogous unique continuation result at the boundary along the lines of [1] when the domain is $C^{3, \alpha}$.
    Mathematics Subject Classification: Primary: 35J25; Secondary: 35J70.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    V. Adolfsson and L. Escauriaza, $C^{1, \alpha}$ domains and unique continuation at the boundary, Comm. Pure Appl. Math., 50 (1997), 935-969.doi: 10.1002/(SICI)1097-0312(199710)50:10<935::AID-CPA1>3.0.CO;2-H.

    [2]

    N. Aronszajn, A. Krzywicki and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat., 4 (1962), 417-453.

    [3]

    S. N. Armstrong and L. Silvestre, Unique continuation for fully nonlinear elliptic equations, Math. Res. Lett., 18 (2011), 921-926.doi: 10.4310/MRL.2011.v18.n5.a9.

    [4]

    X. Cabre and L. Caffarelli, Fully Nonlinear Elliptic Equations, Volume 43 of American Mathematical Society Colloquium Publications. 43 American Mathematical Society, Providence, RI, (1995). vi+104 pp. ISBN: 0-8218-0437-5.

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    N. Garofalo and F. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268.doi: 10.1512/iumj.1986.35.35015.

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    N. Garofalo and F. Lin, Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math., 40 (1987), 347-366.doi: 10.1002/cpa.3160400305.

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    I. Kukavica and K. Nystrom, Unique continuation at the boundary for Dini domains, Proc. Amer. Math. Soc., 126 (1998), 441-446.doi: 10.1090/S0002-9939-98-04065-9.

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    G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ,(1996). xii+439 pp. ISBN: 981-02-2883-Xdoi: 10.1142/3302.

    [9]

    N. Nadirashvili and S. Vlădut, Singular solutions of Hessian fully nonlinear elliptic equations, Adv. Math., 228 (2011), 1718-1741.doi: 10.1016/j.aim.2011.06.030.

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    O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations, 32 (2007), 557-578.doi: 10.1080/03605300500394405.

    [11]

    L. Silvestre and B. Sirakov, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations, http://arxiv.org/pdf/1306.6672.pdf. doi: 10.1080/03605302.2013.842249.

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