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Tính Tiếp Diễn Độc Nhất Cho Các Phương Trình Elliptic Bậc Phân Fraction
Tóm tắt
Chúng tôi thiết lập tính tiếp diễn độc nhất mạnh mẽ cho các bậc phân của các phương trình elliptic tuyến tính với các hệ số Lipschitz bằng cách thiết lập tính gần như đồng biến cho một chức năng tần số kiểu Almgren thông qua một quy trình mở rộng.
Từ khóa
#tính tiếp diễn độc nhất #phương trình elliptic #bậc phân #hệ số Lipschitz #chức năng tần số AlmgrenTài liệu tham khảo
Almgren, F.: Q-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to dimension two. Bull. Am. Math. Soc. 8(2), 327–328 (1983)
Aronszajn, N.: A unique continuation theorem for solutions of elliptic differential equations or inequalities of second order. J. Math. Pures et Appl. 36, 235–249 (1957)
Armstrong, S., Silvestre, L.: Unique continuation for fully nonlinear elliptic equations. Math. Res. Lett. 18, 921–926 (2011)
Aronszajn, N., Krzywicki, A., Szarski, J.: A unique continuation theorem for exterior differential forms on Riemannian manifolds. Arkiv För Matematik 4(34), 417–453 (1962)
Carleman, T.: Sur un problème d’unicité pour les systemes d’équations aux derivées partielles à deux variables indépendentes. Ark. Mat. 26B, 1–9 (1939)
Caffarelli, L.A., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Part. Differ. Equ. 32(8), 1245–1260 (2007)
Caffarelli, L.A., Stinga, P.R.: Fractional elliptic equations, Caccioppoli estimates ad regularity, Annales de l’Institut Henri Poincare (C) Nonlinear. Analysis 33(3), 767–807 (2016)
Cohen, P.: The non-uniqueness of the Cauchy problem, Tech Report 93. Stanford University, Applied Math. and Stat. Lab. (1960)
Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Commun. PDEs 7(1), 77–116 (1982)
Fabes, E., Kenig,C., Jerison, D.: Boundary behaviour of solutions to degenerate elliptic equations. In: Proceedings of the Conference in Harmonic Analysis in Honor of Zygmund’s 80th Birthday, Wadsworth International Group, pp. 557–589 (1982)
Fabes, E., Stroock, D.: A new proof of Moser’s parabolic harnack inequality using the old ideas of nash. Arch. Rat. Mech. Anal. 96(4), 327–338 (1986)
Fall, M., Felli, V.: Unique continuation property and local asymptotics of solutions of fractional elliptic equations. Commun. Part. Differ. Equ. 39(2), 354–397 (2014)
Garofalo, N.: Unique continuation for a class of elliptic operators which degenerate on an manifold of arbitrary codimension. J. Differ. Equ. 104, 117–146 (1993)
Garofalo, N., Lin, F.H.: Monotonicity properties of variational integrals, \(A_p\) weights and unique continuation. Indiana Univ. Math. J. 35, 245–268 (1986)
Garofalo, N., Lin, F.H.: Unique continuation for elliptic operators: a geometric-variational approach. Commun. Pure. Appl. Math. 40, 347–366 (1987)
Jin, T., Li, Y.Y., Xiong, J.: On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions. J. Eur. Math. Soc. (JEMS) 16(6), 1111–1117 (2014)
Koch, H., Tataru, D.: Carleman estimates and unique continuation for second order elliptic equations. Commun. Pure Appl. Math. 54, 339–360 (2001)
Han, Q., Lin, F.H.: Nodal sets of solutions of elliptic differential equations, book in preparation. http://www3.nd.edu/~qhan/nodal.pdf
Pliš, A.: On non-uniqueness in the Cauchy problem for an elliptic second-order differential equation. Bull. Acad. Pol. Sci. 11, 95–111 (1963)
Rüland, A.: Unique continuation for fractional Schrödinger equations with rough potentials. Commun. Part. Differ. Equ. 40(1), 77–114 (2015)
Savin, O.: Small perturbation solutions for elliptic equations. Commun. Part. Differ. Equ. 32, 557–578 (2007)
Seo, I.: Carleman inequalities for fractional Laplacians and unique continuation. Taiwan. J. Math 19(5), 1533–1540 (2015)
Stinga, P.R.: Fractional powers of second order partial differential operators: extension problem and regularity theory, Ph.D. thesis, Universidad Autónoma de Madrid, Spain (2010)
Stinga, P.R., Torrea, J.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Part. Differ. Equ. 35(11), 2092–2122 (2010)
Stinga, P.R., Zhang, C.: Harnack’s inequality for fractional nonlocal equations. Discr. Contin. Dyn. Syst. 33(7), 3153–3170 (2012)
Tao, X.: Doubling properties and unique continuation at the boundary for elliptic operators with singular magnetic fields. Studia Mathematica 151(1), 31–48 (2002)
Tao, X., Zhang, S.: Weighted doubling properties and unique continuation theorems for the degenerate Schrödinger equations with singular potentials. J. Math. Anal. Appl. 339, 70–84 (2008)
Tataru, D.: Unique continuation for PDEs. IMA Vol. Math. Appl. 137, 239–255 (2003)