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March  2015, 14(2): 623-626. doi: 10.3934/cpaa.2015.14.623

A note on the unique continuation property for fully nonlinear elliptic equations

Agnid Banerjee 1,

1. 

Department of Mathematics, University of California, Irvine, CA 92697, United States

Received  June 2014 Revised  August 2014 Published  December 2014

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 $C^{3, \alpha}$.
Keywords: Fully nonlinear equations, Schauder theory, linearization, unique continuation, regularity.
Mathematics Subject Classification: Primary: 35J25; Secondary: 35J7.
Citation: Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

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